Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 26I / a

A smooth curve $\alpha :I\to {\mathbb{R}}^3$ is regular when ${\alpha }^{\prime }(t)\ne 0$ for every $t\in I$. Its arc-length between parameters $t_0$ and $t_1$ is

Fix $t_0$ and defineRegularity gives $s^{\prime }(t)=|{\alpha }^{\prime }(t)|>0$, so $s$ is strictly increasing and has a smooth inverse $t=t(s)$ on its image. The reparametrized curve $\beta (s)=\alpha (t(s))$ satisfiesThus every regular curve has an arc-length parametrization.

For a unit-speed curve with nonzero curvature, letIts torsion isequivalentlyIf the curve lies in an affine plane, all three derivative vectors lie in the parallel two-dimensional vector plane, so their determinant is zero. Hence $\tau =0$ wherever torsion is defined.