Then, we have the following Frenet formulas
, (TURHAN; KORPINAR, 2011b):
2] are the natural curvature and torsion of [alpha] (t), respectively, then for [alpha] the Frenet formulas are given by
2] are the torsions of base curves of M and M*, then from the Frenet formulas we have [k*.
Finally, Chapter 7 is dedicated to various aspects of extrinsic geometry, such as Frenet formulas
for curves, ruled and developable surfaces, the shape operator, principal curvatures and curvature lines, etc.
We confine ourselves to the case n = 4, but it is obvious that similar Frenet formulas hold for every dimension n.
We study the Frenet frames and Frenet formulas in the Minkovski spacetime [E.
It represents the Frenet formulas for a versor field if the functions [[xi].
The Frenet formulas
for non-unit speed curves in [E.
It is clear to show that there exists a function [tau](s)on D such that the Frenet formulas
For their derivatives the following Frenet formulas hold 
Differentiating both sides of the equation (9) with respect to s and use the Frenet formulas, we obtain
Differentiating both sides of equation (12) with respect to s and using Frenet formulas, we have following equation