Frenet Formulas

Also known as the Serret-Frenet Formulas

$$\left[{\matrix{\dot{\bf T}\cr \dot{\bf N}\cr \dot{\bf B}\cr}... ...ht] \left[{\matrix{{\bf T}\cr {\bf N}\cr {\bf B}\cr}}\right],$$

where ${\bf T}$ is the unit Tangent Vector, ${\bf N}$ is the unit Normal Vector, ${\bf B}$ is the unit Binormal Vector, $\tau$ is the Torsion, $\kappa$ is the Curvature, and $\dot{\bf x}$ denotes $d{\bf x}/ds$.

See also Centrode, Fundamental Theorem of Space Curves, Natural Equation

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. Boca Raton, FL: CRC Press, p. 126, 1993.

Kreyszig, E. ``Formulae of Frenet.'' §15 in Differential Geometry. New York: Dover, p. 40-43, 1991.

Serret, J. A. ``Sur quelques formules relatives à la théorie des courbes à double courbure.'' J. de Math. 16, 1851.