Frénet Formulas

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Frénet Formulas


formulas giving the derivatives, with respect to arc length S, of the unit tangent vector t, the unit normal vector n, and the unit binormal vector b of the arbitrary curve L along the vectors t, n, and b. Ifk and σ are the curvature and torsion of L, then the Frénet formulas have the form

Frénet formulas are used to study the differential-geometric properties of curves and, in kinematics, to investigate the motion of a mass point along a curvilinear trajectory.

The French mathematician F. Frénet used the formulas as early as 1847 but did not publish them until 1852. They were first published by the French mathematician J. Serret in 1851 and consequently are often called the Serret-Frénet, or Frénet-Serret, formulas.

References in periodicals archive ?
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 hold.
For their derivatives the following Frenet formulas hold [9]
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