Alternative Form for the Frenet Equations
Theorem
The Frenet equations can be written in the form
Proof
For a curve
the Frenet equations are
1.
where
is a scalar (called the radius of curvature) and
is the unit tangent to the curve.
2
where
is a scalar called the torsion and
3.
where
is a unit normal to the curve.
Equating the expressions for
gives
Substituteto give
The vectors
form a right handed coordinate system so
and we can write
as
hence
Hence
(Eq 1. above) and
Subsituteinto
to obtain
which is equation 3.
Subsituteinto
to obtain
which is Eq 2. above.
