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

$$\kappa_g=\sqrt{EG-F^2}[-\Gamma_{11}^2u'^3+\Gamma_{22}^1v'^3-... ...}^1)u'^2v' +(2\Gamma_{12}^1-\Gamma_{22}^2)u'v'^2+u''v'-v''u'],$$

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

References

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