# Frenet-Serret formulas

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## Frenet-Serret formulas

[fre′nā sə′rā ‚fōr·myə·ləz]
(mathematics)
Formulas in the theory of space curves, which give the directional derivatives of the unit vectors along the tangent, principal normal and binormal of a space curve in the direction tangent to the curve. Also known as Serret-Frenet formulas.
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Using the Frenet equations, we have dU/ds = 0, that is, U is a constant vector.
By differentiation with respect to s, using the Frenet equations and since [gamma] is a planar curve, we obtain <T(s), U> = 0 for any s.
The Frenet equations were first established for curves in the 3-dimensional Euclidean space [E.
On the other hand, the Frenet equations were also extended for curves in a pseudo-Finsler space (see [1]).
Differentiating both sides of (2) and considering Frenet equations, we have
This boundary are curve base measure with the help of Serret Frenet Equations
Notable recent works include that by Langer and Singer [5], who were able to integrate the Frenet equations for elastica (with fixed arclength) in spaces of constant curvature with the use of Killing fields.
If there exists a family of Frenet frames {T(s),N(s),B(s)} satisfy the Frenet equations (2.
Differentiating (4) and considering Frenet equations, we have a system of differential equation as follow:
In the same space, Frenet equations for some special null; Partially and Pseudo Null curves are given in [4].
Differentitating both sides of (2) and considering Frenet equations, we get

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