Curvature

Topics: Curve - Vector-Valued Function

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(definition)

Let $r:\mathcal{D}\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be a vector-valued function.

Given an arc length parametrisation, we define the curvature of $r$ as:

$$\kappa (s):=\Vert \frac{dT}{ds}\Vert$$

…where $T$ is the tangent unit vector in terms of the arc length parameter (that is, $T(s)=\frac{dr}{ds}$).

(theorem)

Given a non-arc length parametrisation $r(t)$ for a curve $c$, then the curvature of $c$ is:

$$\kappa (t)=\frac{\Vert T^{\prime }(t)\Vert }{\Vert r^{\prime }(t)\Vert }$$