The Geometry of the Quadratic Formula

The quadratic formula as an isometry between models of the hyperbolic plane.

We study the quadratic formula $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \left(-b\pm\sqrt{b^2-4ac}\right)/2a$ geometrically, as a map from the space of coefficients to the space of roots for polynomials of degree $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \leq 2$. Our main goal is the following theorem: restricted to real polynomials with complex roots, the quadratic formula realizes an isometry from the projective model to the conformal model of the hyperbolic plane.

Geometry of the map Polynomials $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \longrightarrow$ Roots

Our goal is to understand in detail the geometry of the quadratic formula as a map from coefficients to roots, and from this understanding extract a cool fact: namely that restricted to real coefficients, the relevant geometry includes both a projective and a conformal model of the hyperbolic plane. Here we start the road to viewing the relationship between a polynomial and its roots as a geometric problem by first casting it as a topological one. As our interest lies in the quadratics everything is quite explicit: however just here at the beginning I will talk about general $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$, as it’s my hope that some of these ideas will (with much more work) continue to tell a good story in higher degree.

The fundamental theorem of algebra guarantees every complex nonconstant polynomial has at least one root, or equivalently every degree $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$ polynomial has exactly $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$ roots (counting multiplicity). One recasting of this as a topological statement about the map $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots:Polynomials\mapsto Their\;Roots$, is as follows. Let $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Pol_n\cong\C^\times\times\C^n$ denote the space of degree $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$ polynomials over $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C$ identified with their coefficients, and for each $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f\in\Pol_n$ let $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots(f)$ be its multiset of roots. To describe the space of these multi-sets, its useful recall the notion of symmetric power: the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n^{th}$ symmetric power of a space $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} X$ is the collection of all unordered $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$-tuples, $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(X)=X^n/\Sym(n)$. The map $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon\C^\times\C^n\to\SP^n(\C)$ is continuous, and factors through projectivization ($\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f$ and $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} cf$ have the same roots) to a map sending a monic polynomial to its roots. The result is a continuous bijection; and thus homeomorphism, between the space of polynomials viewed-as-their-coefficients, and the space of polynomials-viewed-as-their-roots.

This familiar setting is not quite where our story will take place however. Instead of projectivizing the space $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C^\times\times\C^n$ of degree $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$ polynomials onto their monic representatives, we will instead include back in the missing polynomials of lower degree, and study the space $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Pol_{\leq n}\cong\C^{n+1}$. This at first seems a difficult task, as the roots map no longer lands in a single space but rather in the union of the symmetric powers $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(\C)\cup \SP^{n-1}(\C)\cup\cdots\cup \SP^1(\C)\cup\SP^0(\C)$. However, it is natural to topologize the union of the first $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$ symmetric powers of a space $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} X$ by using the one-point compactification $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} X\cup\{\infty\}$: if $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} k<n$ we represent the unordered $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} k$-tuple $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{x_1,\ldots, x_k\}$ by the unordered $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$-tuple $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{x_1,\ldots, x_k, \infty,\ldots, \infty\}$, essentially filling in the correct number of empty slots with the ‘filler point’ $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \infty$.

For polynomials, this amounts to (for example) saying that, viewed in relation to cubic polynomials, a linear polynomial has two roots at infinity. This is actually quite natural, as when we write down families of cubics which in the limit become linear, two of their roots escape all bounded sets in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C$. This leads to the following observation: when we topologize the union $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(\C)\cup\SP^{n-1}(\C)\cup\cdots$ via the aforementioned identification with $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(\C\cup\{\infty\})$, the natural extension of the roots map $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon \Pol_{\leq n}\to\SP^n(\C\cup\{\infty\})$ remains continuous. This map again factors through projectivization of the domain to a continuous bijection, and thus homeomorphism from $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^{n}$ onto $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(\CP^1)$.

As each space of lower degree polynomials is naturally built into the construction, we may take the direct limit of this over inclusions for increasing $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n$ to give a third and final topological incarnation of the fundamental theorem:

What does this tell us? Firstly, that there is a uniform, natural way to consider polynomials of lower degree as degenerations of higher degree polynomials, allowing us to union to domains, codomains and root maps for each lower degree into a single morphism $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^n\to\SP^n(\CP^1)$. Using this, from the viewpoint of coefficients the geometry of polynomials is naturally contained in the geometry of complex projective space, but from the viewpoint of roots the geometry of polynomials is naturally contained in the geometry of symmetric powers of the sphere. The roots map provides a way of passing between these geometries: if $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} X\subset \Pol_{\leq n}$ is a collection of polynomials, we may consider both a projective model $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{P}(\mathsf{Coefs}(X))$ and a symmetric powers model $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{Roots}(X)$.

Quadratics

Restricting to (at most) quadratic polynomials, gives the map $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon \CP^2\to\SP^2(\CP^1)$.
We will write the projectivized coefficients of a polynomial as $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} [a:b:c]$ and the roots as $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{z,w\}$. The discriminant of the polynomial $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f(z)=az^2+bz+c$ will be denoted $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta(f)=b^2-4ac$.

In this low degree, an explicit description of the second symmetric power of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ is useful: $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)=(\CP^1\times \CP^1)/\Z_2$ where $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Z_2$ acts by switching the first and second coordinates. We may even draw a useful picture of this, after identifying $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ with the unit sphere in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \R^3$: let $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{pr}\colon\S^2\to I$ be the map sending a point on the sphere to its height; its $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z$ coordinate in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} I=[-1,1]$. Then the map $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{pr}\times\mathsf{pr}\colon\S^2\times\S^2\to I\times I$ gives us a kind of degenerate fibration: the preimage of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (x,y)\in I^2$ is generically a torus, consisting of the latitude circle at height $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} x$ times the latitude at height $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} y$.

$\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^2\times\S^2$ as a degenerate fiber bundle over the square. Generic point preimages are flat tori, the preimage of points along the boundary are circles and corners are points. Together the boundary square has preimage a necklace of 4-spheres onto which $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} T^2\times I^2$ is glued.

The diagonal embedding $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta\colon \S^2\hookrightarrow\S^2\times\S^2$ is the preimage of the diagonal in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} I\times I$, and the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Z_2$ action permuting coordinates acts freely off of here. Along this diagonal the action is trivial, and so geometrically $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\S^2)$ is a locus cone singularities in the quotient $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\S^2)\cong\CP^2$. Subsets of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\S^2)$ can be profitably understood by ‘folding’ a corresponding subset of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^2\times\S^2$ along the diagonal $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^2$.

Forming the symmetric product $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{SP}^2(\CP^1)$ as a quotient of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$ by the map exchanging coordinates. (Note that in many of these cartoon drawings it will appear that $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\CP^1)$ is a boundary of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$. This is of course not true; in reality it is a codimension-2 sphere shaped singular locus.)

It’s instructive to view some important subsets of polynomials in both the roots and coefficient spaces. In the coefficients model, the at-most-linear polynomials show up as the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ at infinity with respect to the affine patch centered at $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} [1:0:0]\simeq z^2$. In the roots model, this same set identifies with the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ worth of points $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{\infty, z\}$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$. This $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ intersects another important $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ in the space of quadrics - the set of polynomials with a double root - in a single point (the constant polynomial, which by our convention has a double root at $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \infty$). In the roots model, the quadratics with a double root are easy to describe, they are the image of the diagonal embedding $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\CP^1)$; the singular locus of the orbifold structure on $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$.

The sphere of at most linear polynomials and the sphere of polynomials having a double root meet only in the constant polynomial.

In the coefficient model this set is more difficult to spell out: it is the projective variety cut out by the discriminant $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta)=V(b^2-4ac)$. To see this is a 2-sphere, we may change variables so that $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} b^2-4ac$ is diagonal, $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} x^2+y^2+z^2$ (all non-degenerate quadratic forms are equivalent over $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C$), and note that in the affine patch $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z=-1$ we have $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} x^2+y^2=1$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C^2$, whose solution set is a cylinder; together with two points (the north and south poles) at infinity with respect to this patch.

The projective variety $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(b^2-4ac)$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2$ is homeomorphic to a sphere; as can be seen by changing coordinates to $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(x^2+y^2+z^2)$ and then computing $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(x^2+y^2=1)$ in an affine patch.

Next we aim to impose some notions of geometry on the space of quadratics. The coefficient model, $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2$, has a natural notion of geometry, with automorphism group $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(3;\C)$. Note that this is different than what we would get considering only monic degree 2-polynomials (which would be the affine group for $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C^2$), and it allows the mixing of linear with quadratic polynomials by sending roots to $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \infty$. To understand the geometry of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$, we first look to its orbifold universal cover $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$. This has automorphism group given by $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(\CP^1)$ on each factor extended by swapping the factors, or $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Z_2\ltimes \PSL(2;\C)\times\PSL(2;\C)$. The automorphisms of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ are those elements above which are well defined (and nontrivial) on the quotient: that is, the diagonal elements $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\PSL(2;\C))$. This has a nice geometric interpretation: automorphisms of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ preserve the singular locus which is itself a copy of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$, and any automorphism of this $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ is admissible. Equivalently, all automorphisms of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ are induced from automorphisms of the underlying extended complex plane, and then applied to multi-sets of cardinality 2.

Automorphisms of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{SP}^2(\mathbb{CP}^1)$ must preserve the singular locus, and thus all arise from automorphisms of the underlying space $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$

Finally, it will be useful to return to the coefficients model, and compute the restricted group of symmetries which ‘play nicely’ with the roots map. As symmetries of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ must preserve the singular locus, the diagonal $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ representing double roots, their realization in the coefficients model must preserve the discriminant locus $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta)$. As $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta$ is a quadratic form on $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C^3$, the automorphisms of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2$ preserving $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta)$ form the complex orthogonal group $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PO(\delta,\C)$. This is just an incarnation of the irreducible representation $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SL(2,\C)\to\SO(3;\C)$, taking the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SL(2,\C)$ action on the extended complex plane, applying it to unordered 2-tuples, then interpreting those as roots of a polynomial and viewing the action on coefficients. The fact that all orthogonal groups are isomorphic over $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C$ tells us that $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta,\C)$ contains subgroups isomorphic to $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(3;\R)$ and $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(2,1;\R)$, whose actions we can understand geometrically. With $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta,\C)\cong\SL(2;\C)$ acting as conformal transformations of the sphere $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta)$, as expected an $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(3)$ subgroup acts as rigid rotations of this sphere, and $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(2,1)$ subgroups act as Möbius transformations preserving some circle on $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta)$. A particular one of these, namely the real points $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta;\R)$ will be important below.

In root space, the automorphisms of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ are the diagonal embedding of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\mathbb{C})$ as we saw above.
In coefficient space, this group appears instead as a complex orthogonal group, $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta,\C)\subset\PSL(3;\C)$.

It is interesting to note that from this perspective, it is still natural to allow automorphisms mixing quadratic with linear polynomials, but no longer mixing polynomials with distinct roots with those having a double root.

Real Quadratics

If we restrict to polynomials with real coefficients, $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Pol_{\leq 2}(\C)$ becomes $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Pol_{\leq 2}(\R)$ and correspondingly the space of projectivized coefficients changes from $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2$ to $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$. It is more work to describe the image under the roots map. As $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{Roots}\colon\CP^2\to\SP^2(\CP^1)$ is a homeomorphism, that restriction is an embedding of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ into $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ is clear.
To understand this embedding, we will realize this $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ as a certain 2-complex in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$, ‘folded over’ the diagonal $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\CP^1)$.

Real quadratics come in two flavors: those with real roots, and those with complex conjugate roots.
In the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$ cover, polynomials with complex conjugate roots correspond to the collection $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (z,\overline{z})$ as $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z$ ranges in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C\cup\infty$, forming a sphere. This sphere intersects the sphere $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\CP^1)$ along the great circle $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{(x,x)\mid x\in\R\cup\infty\}$, and so in the quotient gets folded along this great circle into a disk. The polynomials with real roots correspond to the collection $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{(x,y)\mid x,y\in\R\cup\infty\}$, which forms a torus in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$. This torus also contains the real diagonal $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{(x,x)\mid x\in\R\cup \infty\}$ as a $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (1,1)$ curve and is folded along it onto a Möbius band in the quotient. Thus, in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$ the relevant 2-complex is the union of a sphere and a torus, glued along a circle as in the cartoon below.
In the quotient, the sphere becomes a disk and the torus a Möbius band, glued along their common circle of intersection: this is the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ embedded by the roots map.

The preimage of the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ of real polynomials in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$, viewed in the orbifold cover $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$ and its quotient, a creased $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$.

Note that this $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ contains part of the singular locus of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$. It is constructed from $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^2\cup_{\S^1}T^2$ folded in half along their common $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^1$, which we saw above to consist of real double roots. To remember this, I will draw a crease along this curve when a picture denotes the image of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ under the roots map.

The real quadratic polynomials form an $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ inside of both $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2\cong \SP^2(\CP^1)$. In the root-space, this $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ intersects the singular locus in the circle of polynomials with a double real root.

To calculate the automorphisms of this subset in each the coefficient and roots viewpoints, we find the subgroup which fixes the real points setwise. In the coefficient view this is easy: the smoothly embedded $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2\subset\CP^2$ has automorphism group $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(3;\R)<\PSL(3;\C)$. On the roots side, any automorphism of this creased $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ must send the crease to itself (to see this, note that all automorphisms of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ preserve the singular locus, and so the subgroup preserving real polynomials must preserve the intersection of this singular set with the real points, which is precisely the creased circle). This circle represents the real quadratics with a double root $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\R\cup\infty)\subset\Delta(\CP^1)$ and the automorphisms preserving it setwise are none other than the diagonal embedding of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R)$. Viewed back on the other side, the automorphisms of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2\subset\CP^2$ which preserve the division into polynomials with a double root and those with distinct roots are the intersection of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(\RP^2)$ with $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(V(\delta))$; that is $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta;\R)=\PGL(3;\R)\cap\SO(\delta;\C)$. Over $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \R$, the discriminant $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta=b^2-4ac$ has signature $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (2,1)$, so this is a conjugate of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(2,1)$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PGL(3;\R)$ preserving the division of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ into a disk, Möbius band by the discriminant.

The automorphism group of real quadrics in coefficient space is all of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PGL(3;\R)$. In root space, any automorphism must preserve the singular locus, so the symmetries are reduced to a copy of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R)$. Back in coefficient space, this subgroup is realized as $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta;\R)<\PSL(3;\R)$.

This action of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(2,1)$ is well known - it acts as the isometries of both the Klein (projective) model of the hyperbolic plane when restricted to the disk, and the projective model of de Sitter 1+1 space when restricted to the Möbius band exterior (or anti de Sitter 1+1 space, as the two are isomorphic in this dimension). The $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SL(2;\R)$ action on the real polynomials in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ is maybe not as well known, but comes from gluing together two natural actions on the universal cover $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$. Up here, recall that the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ of real polynomials is double covered by a more interesting object; the union of a sphere and torus along a circle. The sphere consists of pairs $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (z,\overline{z})$ for $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z\in\CP^1$ and the torus of points $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (x,y)$ for each coordinate in the extended reals. The subgroup of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(\CP^1\times\CP^1)$ fixing this sphere a copy of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2,\C)$ embedded in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(\CP^1\times\CP^1)$ as $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} A\mapsto (A,\overline{A})$ and the subgroup fixing the torus is the real points, $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R)\times\PSL(2;\R)$.

Automorphisms of the branched cover of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2_\mathsf{Roots}$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$.

Most of the automorphisms of either component individually do not descend to the quotient $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$; to do so an automorphism must preserve the singular locus, which intersects the picture here in the same $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^1$ common to the sphere and torus. On the sphere, this selects out only conformal automorphisms preserving the equator, which divides $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ into two disks, each of which when equipped with this $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R)$ action is a conformal model of the hyperbolic plane (either the Poincaré disk or upper half plane, depending if the stereographic projection to $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C$ has projection point on, or off the equatorial circle). Thinking of the original $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ as the boundary of hyperbolic $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} 3$-space when equipped with the $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2,\C)$ action, we can see fixing a circle in the boundary as restricting the possible isometries to those that preserve that hyperbolic plane (which we can view conformally by then projecting onto the upper, lower hemispheres of the ideal boundary if we would like). On the torus, the corresponding picture is similar, but for anti-de Sitter space. Much like the sphere is the ideal boundary of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathbb{H}^3$, the torus is the ideal boundary of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{AdS}^3$. Fixing certain curves (in our case the (1,1) curve with respect to the decomposition of isometries as $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R)\times\PSL(2;\R)$ above) correspond to preserving a lower-dimensional anti de Sitter space $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{AdS}^2$, which is isomorphic to de Sitter 2-space as a coincidence of low dimensions and has model an open Möbius band. The $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R)$ action on the creased $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ is what you get from gluing these two actions together along the common ideal boundary of their spaces, where they agree.

The $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ of real polynomials is a union of two natural geometries along their ideal boundary.

The Quadratic Formula

We’ve done all the hard work now; having described the domain and codomain of the roots map assigning coefficients in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2$ to their roots in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$ as well as the symmetries relevant to each side.

The complex roots of a real quadratic come in conjugate pairs, and so are represented in root space by the disk $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{z,\overline{z}\}$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)$. In coefficient space, the real polynomials with complex roots are the negative cone of the discriminant, represented by the disk $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \P\{\delta<0\}$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2\subset\CP^2$. The first of these spaces is naturally a conformal model of the hyperbolic plane (looking in the double cover $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1$, it is a hemisphere of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$ together with the Möbius transformations preserving it), and the second is naturally a projective model (the action of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta,\R)\cong\SO(2,1)$ is by projective transformations preserving the metric given by the cross ratio). Restricting the root map to this disk gives the map $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon \D_\mathsf{Coefs}^2\to\D_{\mathsf{Roots}}^2$, which is equivariant with respect to the action of hyperbolic isometries on each side (that is, if $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} g\in\Isom(\mathbb{H}^2)$ then $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots(g.f)=g.\Roots(f)$ where in the first case $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} g$ acts as an element of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta;\R)$ and in the second as an element of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\PSL(2;\R))$.)

The roots map is an intertwiner for the action of isometries on both sides, and is itself an isometry from the coefficients model to the roots model.

It only remains to show that $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots$ is an isometry. But this is no work at all, thanks to the symmetry we have established along the way. Since $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots$ is $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Isom(\Hyp^2)$ equivariant and this action is transitive, $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots$ is completely determined by its value at any point. Consider $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f=z^2+1$ with $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots(f)=\{i,-i\}$.
Now let $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Phi$ be any isometry between these two models of hyperbolic space equivariant with respect to the given group actions. Without loss of generality we may assume that $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Phi(z^2+1)=\{i,-i\}$ (if not, pre- and post-compose by isometries of the domain / codomain to make it so). But now $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots, \Phi$ are $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Isom(\Hyp^2)$-equivariant maps from $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \D^2_{\mathsf{Coefs}}$ to $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \D^2_{\mathsf{Roots}}$ agreeing on the point $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z^2+1$, and so they are equal.

To finish it off, we will write this map down in coordinates, as an explicit map from a disk in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2$ to a disk in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1$. This also constitutes a second proof of the statement above, in which you are free to ignore everything up to here in this writeup and just directly compare the formula to the standard conversion from the Klein disk to the Upper Half Plane, for instance as found on Wikipedia. Starting with a quadratic $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f=az^2+bz+c$, we represent it in the space of projectivized coefficients as $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} [a:b:c]$, and get at its roots via the familiar quadratic formula. $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots:[a:b:c]\mapsto \left\{ \frac{-b\pm\sqrt{b^2-4ac}}{2a}\right\}$

To get useful coordinates on the codomain, note that as the roots are always a complex conjugate pair, one must be in the upper half plane and the other in the lower. Thus we may unambiguously select the root in the upper half plane and represent our pair of roots by a single number $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} x+iy$ for $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} y>0$. Noting that $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta<0$, we may write this as follows, using the convention that $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \sqrt{-}$ means ‘the unique positive square root of’ when applied to real numbers. $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon[a:b:c]\mapsto \frac{-b}{2a}+i\frac{\sqrt{4ac-b^2}}{2a}$ This provides us with useful coordinates on the codomain, and so our next move is to do something similar for the domain. The first obvious choice is the reduction to monic quadratics using the affine patch $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} a=1$, which lets us think of the domain as the set of points in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \R^2=\{(b,c)\}$ with $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} b^2<4c$:

$\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon (b,c)\mapsto \frac{-b}{2}+i\frac{\sqrt{4c-b^2}}{2}$

The quadratic formula as usually written is a map from a projective model of $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathbb{H}^2$ as a paraboloid onto the upper half plane.

This is nothing more than the quadratic formula as learned in grade school, but we know from the above that we may interpret this as an isometry between two copies of the hyperbolic plane! The copy in the codomain is familiar; by selecting coordinates given by the root with positive imaginary part we have naturally landed in the upper half plane. But what is the model in the domain? This ‘paraboloid’ model is in fact the Klein model in disguise - we have just chosen the wrong affine patch; one that runs parallel to the lightcone instead of transverse to it.

projective model of the hyperbolic plane is a paraboloid, hyperboloid disk depending on choice of affine patch.

To reconstruct the more familiar picture, we need to change the affine patch. The following transformation does the job:

$$\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \pmat{ a\\b\\c }= \pmat{ \tfrac{w+u}{2}\\v\\\frac{w-u}{2} }$$

In these coordinates, the discriminant is $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta=u^2+v^2-w^2$ and the quadratic formula is the map taking the polynomial $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (w+u)z^2+2vz+(w-u)=0$ to its roots:

$\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon [u:v:w]\mapsto\frac{-v}{u+w}+i\frac{\sqrt{w^2-u^2-v^2}}{u+w}$

The disk $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \D^2_\mathsf{Coefs}$ here is fully contained in the affine patch $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} w=1$ and so we may take the domain to be the unit disk centered at $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \vec{0}$ in $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \R^2=\{(u,v)\}$ giving the expression below, which is the usual transformation from the Klein disk $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{(u,v)\mid u^2+v^2<1\}$ to the upper half plane $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{z\mid \mathsf{Im}(z)>0\}$ up to possibly a reflection of the domain/codomain, depending on your source.

$\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon (u,v)\mapsto \frac{-v}{1+u}+i\frac{\sqrt{1-u^2-v^2}}{1+u}$

After a linear change of coordinates, the quadratic formula provides the usual identification of the Klein model with the upper half plane model of hyperbolic space.

Appendix

Here’s the easy lemma that was used in the proof: an equivariant map between two spaces each equipped with a transitive $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} G$ action is determined by its value at a point.

Denote $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f(x)=\phi(x)=y$. Let $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z\in X$, and choose $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} g\in G$ such that $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z=g.x$.
Then evaluating $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f, \phi$ on $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z$ gives $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f(z)=f(g.x)=g.f(x)=g.y$ and similarly $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \phi(z)=\phi(g.x)=g.\phi(x)=g.y$. Thus $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f(z)=\phi(z)$ so $\providecommand{\Roots}{\mathsf{Roots}} \providecommand{\Pol}{\mathsf{Pol}} \providecommand{\SP}{\mathsf{SP}} \providecommand{\Coefs}{\mathsf{Coefs}} \providecommand{\C}{\mathbb{C}} \providecommand{\R}{\mathbb{R}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\SO}{\operatorname{SO}} \providecommand{\CP}{\mathbb{C}\mathsf{P}} \providecommand{\Z}{\mathbb{Z}} \providecommand{\SL}{\operatorname{SL}} \providecommand{\PSL}{\operatorname{PSL}} \providecommand{\RP}{\mathbb{R}\mathsf{P}} \providecommand{\PGL}{\operatorname{PGL}} \providecommand{\PO}{\operatorname{PO}} \providecommand{\D}{\mathbb{D}} \providecommand{\Hyp}{\mathbb{H}} \providecommand{\Isom}{\operatorname{Isom}} \providecommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f=\phi$.