Curvature

At any point along a curve $C$ with

$$C:x=x(s),y=y(s),z=z(s),0\le s\le L$$

we can define a tangent line but also a tangent circle.

Note that the unit tangent vector does not change its length along the curve but can change its direction.

Curvature of Tangent Circle

The magnitude of $\frac{d\hat{T}}{ds}$ is the curvature of the tangent circle.

Proof

Consider the parametric equations of a circle at $(R,0)$ of radius $R$:

$$x=R+R\cos \left(\frac{s}{R}\right),y=R\sin \left(\frac{s}{R}\right),0\le s\le 2\pi R$$

It follows that

$$\hat{T}=-\sin \left(\frac{s}{R}\right)\hat{i}+\cos \left(\frac{s}{R}\right)\hat{j}$$

and

$$\frac{d\hat{T}}{ds}=-\frac{1}{R}\cos \left(\frac{s}{R}\right)\hat{i}-\frac{1}{R}\sin \left(\frac{s}{R}\right)\hat{j}$$

Note that at the origin, $s=\pi R$. At this point, we also have

$$\frac{d\hat{T}}{ds}=\frac{1}{R}\hat{i}+0\hat{j}$$

For any cricle, the vector above has a fixed direction but the magnitude $\mid \frac{d\hat{T}}{ds}\mid =\frac{1}{R}$ is the inverse of the radius of the circle, also called the curvature. The inverse ($R$) is also called the radius of curvature.

Curvature

$$\kappa =\left|\frac{d\hat{T}}{ds}\right|$$

where $\vec{T}$ is the unit tangent and $s$ is the arc length

Radius of curvature

$$\rho (s)=\frac{1}{\kappa (s)}$$

where $\kappa (s)$ is the curvature defined above.

Circle of curvature

The circle of curvature is the circle in the plane defined by $\hat{T}$ and $\hat{N}$ with center at $\vec{r}(s)+\rho (s)\vec{N}$ and radius $\rho (s)$.