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Frenet Formulas


Also known as the Serret-Frenet formulas, these vector differential equations relate inherent properties of a parametrized curve. In matrix form, they can be written

 [T^.; N^.; B^.]=[0 kappa 0; -kappa 0 tau; 0 -tau 0][T; N; B],

where T is the unit tangent vector, N is the unit normal vector, B is the unit binormal vector, tau is the torsion, kappa is the curvature, and x^. denotes dx/ds.


See also

Centrode, Fundamental Theorem of Space Curves, Natural Equation

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References

Frenet, F. "Sur les courbes à double courbure." Thèse. Toulouse, 1847. Abstract in J. de Math. 17, 1852.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 186, 1997.Kreyszig, E. "Formulae of Frenet." §15 in Differential Geometry. New York: Dover, pp. 40-43, 1991.Serret, J. A. "Sur quelques formules relatives à la théorie des courbes à double courbure." J. de Math. 16, 1851.

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Frenet Formulas

Cite this as:

Weisstein, Eric W. "Frenet Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FrenetFormulas.html

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