Identity for Torsion and Radius of Curvature

Theorem

For a curvewhereis the distance along the curve, ifis the torsion andis the radius of curvature, then

Proof

(1) and

Hence(2) whereis the curvature. Differentiating this expression gives

One of the Frenet formulae stateswhereis the binormal vector.

Substituting this expression intogives(3)

Substitute (1), (2) and (3) intoobtaining