## The Geometry of Curves

# Curvature & Torsion

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The fundamental theorem of space curves states that a regular curve $\gamma\colon I\to\mathbb{R}^3$ is uniquely determined up to isometry by its curvature and torsion.

This small graphing calculator lets you explore the existence direction of this theorem: inside the pulldown menu visible in the upper right, you can type in a function for the curvature $\kappa$ and torsion $\tau$, as a function of the arc length $s$. The program will then compute a curve in $\mathbb{R}^3$ with these invariants, and plot it.

In your functions defining curvature and torsion, feel free to use the symbols $a,b,c$ as parameters: this will allow interactivity, as you can manipulate the sliders controlling these three parameters to change the resulting curve in real time.

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