Singularity

The Mathematics

To be singular simply means to fail to be smooth — and there are many ways a variety can do so. It might have a cusp, a node, or a cross-cap; it might be the union of pieces of differing dimension. This piece is a subset of a larger project — hanging in my office at USF — to display this whole variety of singular behavior in dimension three.

Technique

The first step is searching the literature — and querying algebraic geometers — for explicit polynomials that exemplify each singularity type. With an implicit equation in hand, I wrote a custom path tracer that estimates the distance to the variety from a linearization of its gradient, using an adaptive step size to ray-march even singular surfaces at high resolution.