## Proof of the Frenet Formulae

Theorem

Ifis a curve with tangentunit normaland 'binormal'then

1.whereis a scalar (called the radius of curvature)

2.whereis a scalar called the torsion.

3.

Proof

The tangent vector toisthe derivative ofand the rate of change ofis

1. Ifis the length along the curve, thenand

Henceandis perpendicular toso must be parallel to vec N , sowhereis a scalar.

2.so

Now take the dot product of (1) on with vec T to giveso is perpendicular to

(sinceandare perpendicular ubit vectors) soand andis then perpendicular to bothandso

3. Vectors vec N , vec B , vec T form a right handed coordinate system and

Differentiatinggives