D.1. Frenet Serret Formula for a Space Curve

Let a space curve C be defined by x( s) in with s as the arc-length parameter. Then, the unit tangent vector is given by

Differentiating the equation ? t , t ? = 1 with respect to s, we have ? t , (d t /d s) ? = 0. Hence, d t /d s is orthogonal to t and also to curve C, and the vector d t/d s defines a unique direction (if d t/d s ? 0) in a plane normal to C at x called the direction of principal normal, represented by

where n is the vector of unit principal normal and k( s) is the curvature. Then, we can define the unit binormal vector b by the equation, b = t n, which is normal to the osculating plane spanned by t and n. Thus, we have a local right-handed orthonormal frame ( t , n , b) at each point x.

Analogously to d t /d s ? t, the vector d n /d s is orthogonal to n. Hence, it may be written as n ? = ? t ? ? b, where the prime denotes d /d s, and ?, ? ? .