Radius of Curvature in Whewell Form

Theorem
Let $C$ be a curve defined by a real function which is twice differentiable. The radius of curvature $\kappa$ of $C$ at a point $P$ can be expressed in the form of a Whewell equation as:


 * $\rho = \size {\dfrac {\d s} {\d \psi} }$

where:
 * $s$ is the arc length of $C$
 * $\psi$ is the turning angle of $C$
 * $\size {\, \cdot \,}$ denotes the absolute value function.

Proof
By definition, the radius of curvature $\rho$ is given by:
 * $\rho = \dfrac 1 {\size \kappa}$

where $\kappa$ is the curvature, given in Whewell form as:
 * $\kappa = \dfrac {\d \psi} {\d s}$

Hence the result.