Now we calculate the partial derivatives of (22) with respect to s and k; using Frenet formulas
, we get
Then, we have the following Frenet formulas
, (TURHAN; KORPINAR, 2011b):
If [alpha] is a pseudo null curve, the Frenet formulas
have the form [2, 3]
If [k.sub.1] and [k*.sub.1] are the natural curvatures of base curves of M and M* and [k.sub.2] and [k*.sub.2] are the torsions of base curves of M and M*, then from the Frenet formulas
we have [k*.sub.i] = [k.sub.i] [dt/dt*], 1 [less than or equal to] i [less than or equal to] 2.
If [alpha] is a null space curve with a spacelike principal normal [??], then the following Frenet formulas hold
If [alpha] is a pseudonull curve, that is, [alpha] is a spacelike curve with a null principal normal N, then the following Frenet formulas hold:
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.sup.4.sub.1], that is [R.sup.4] with the metric g = -d[x.sup.2.sub.1] + d[x.sup.2.sub.2] + d[x.sup.2.sub.3] + d[x.sup.2.sub.4].
(The Frenet formulas
for non-unit speed curves in [E.sup.3.sub.1]) For a regular curve [alpha] with speed
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