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Geodesic Curvature

For a unit speed curve on a surface, the length of the surface-tangential component of acceleration is the geodesic curvature $\kappa_g$. Curves with $\kappa_g=0$ are called Geodesics. For a curve parameterized as $\boldsymbol{\alpha}(t)={\bf x}(u(t),v(t))$, the geodesic curvature is given by

...}^1)u'^2v' +(2\Gamma_{12}^1-\Gamma_{22}^2)u'v'^2+u''v'-v''u'],

where $E$, $F$, and $G$ are coefficients of the first Fundamental Form and $\Gamma_{ij}^k$ are Christoffel Symbols of the Second Kind.

See also Geodesic


Gray, A. ``Geodesic Curvature.'' §20.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 402-407, 1993.

© 1996-9 Eric W. Weisstein