Frénet Formulas

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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 ?
Now we calculate the partial derivatives of (22) with respect to s and k; using Frenet formulas, we get
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 hold.
For their derivatives the following Frenet formulas hold [9]
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