Centre of curvature

Question

How do we define a centre of curvature for a space (3D) curve? Is it defined from some evolute? Which one? My question is provoked by the statement of the theorem of Meusnier.

Answer 1

The evolute of a plane curve is defined as the locus of the centers of the osculating circles to the curve. The space curve analog of an osculating circle is an osculating sphere.

Let $\pmb\alpha(t):(a,b)\to \Bbb R^3$ be a regular space curve, and let $a < t_0 < b$ be such that $\kappa[\alpha](t_0)\neq 0$ and $\tau[\alpha](t_0)\neq 0$, where $\kappa\neq 0$ is the curvature of the curve $\pmb \alpha(t)$ and $\tau$ is the torsion. Then the osculating sphere of $\pmb \alpha(t)$ at $\pmb \alpha(t_0)$ is the unique sphere which has at least order 3 contact with $\pmb \alpha$ at $\pmb \alpha(t_0)$.

Then the osculating sphere at $\pmb \alpha(t_0)$ is a sphere of radius $$ R= \sqrt{\left(\frac{1}{\kappa[\alpha](t_0)}\right)^2+\left(\frac{\kappa'[\alpha](t_0)}{||\alpha'(t_0)|| \kappa^2[\alpha](t_0)\tau[\alpha](t_0)}\right)^2} $$ and center $$ \pmb C=\pmb\alpha(t_0)+\frac{1}{\kappa[\alpha](t_0)}\pmb N(t_0)-\frac{\kappa'[\alpha](t_0)}{||\alpha'(t_0)|| \kappa^2[\alpha](t_0)\tau[\alpha](t_0)}\pmb B(t_0) $$ where $\pmb N(t)$ and $\pmb B(t)$ are the normal and binormal vectors.

We call $R$ radius of curvature an $\pmb C$ centre of curvature of the curve $\alpha$ at $\alpha](t_0)$.

It is natural to define the evolute of a space curve to be the locus of the centers of the osculating spheres.

The evolute of a regular space curve $\pmb\alpha(t):(a,b)\to \Bbb R^3$ is the curve given by $$ \pmb\beta(t)=\pmb\alpha(t)+\frac{1}{\kappa[\alpha](t)}\pmb N(t)-\frac{\kappa'[\alpha](t)}{||\alpha'(t)|| \kappa^2[\alpha](t)\tau[\alpha](t)}\pmb B(t) $$ that is, $\pmb\beta(t)$ is the locus of the centers of the osculating spheres.