Curvature as a measure of change of direction

The standard unit tangent vector can be thought of a giving the direction of the curve. If we tell how that is changing as the arc length changes we will get a notion of how sharply the curve bends.

Definition 2.1   The curvature of a curve $\v{p}(t)$ is

$$\kappa = \left\vert\frac{d}{ds} \v{\bf T}\right\vert$$

where the derivative is with respect to arc length.

Now because $\v{\bf T}(t)$ is always of length 1 it will always be perpendicular to its derivative. Scaling everything so that the parameter is arc length we get

$$\frac{d\v{\bf T}}{ds}= \kappa \v{\bf N}$$

where $\v{\bf N}$ is a standard normal vector to the curve.

We can relate the curvature to the tangential and normal components of acceleration by observing that

$$\v{p'}(t) = \frac{ds}{dt} \v{\bf T}$$

so that

$$\begin{eqnarray*}\v{p''}(t) &=& \frac{d^2s}{dt^2} \v{\bf T} + \frac{ds}{dt}\frac... ...}{dt^2} \v{\bf T} + \left(\frac{ds}{dt}\right)^2 \kappa \v{\bf N}\end{eqnarray*}$$