Alternative Form for the Frenet Equations
The Frenet equations can be written in the form
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
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.