Theorem
If
is a curve with tangent
unit normal
and 'binormal'
then
1.
where
is a scalar (called the radius of curvature)
2.
where
is a scalar called the torsion.
3.
Proof
The tangent vector to
isthe derivative of
and the rate of change of
is
1. If
is the length along the curve, then
and
Hence
and
is 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 give
so
is perpendicular to
(sinceandare perpendicular ubit vectors) so
and
andis then perpendicular to both
andso 
3. Vectors vec N , vec B , vec T form a right handed coordinate system and
Differentiating
gives