## Alternative Form for the Frenet Equations

Theorem

The Frenet equations can be written in the form

Proof

For a curvethe Frenet equations are

1.whereis a scalar (called the radius of curvature) andis the unit tangent to the curve.

2whereis a scalar called the torsion and

3. whereis a unit normal to the curve.

Equating the expressions forgives

Substituteto give

The vectorsform a right handed coordinate system so and we can writeashence

Hence(Eq 1. above) and

Subsituteintoto obtainwhich is equation 3.

Subsituteintoto obtainwhich is Eq 2. above.