Line of Curvature

A curve on a surface whose tangents are always in the direction of Principal Curvature. The equation of the lines of curvature can be written

$$\left\vert\matrix{g_{11} & g_{12} & g_{22}\cr b_{11} & b_{12} & b_{22}\cr du^2 & -du\,dv & dv^2\cr}\right\vert=0,$$

where $g$ and $b$ are the Coefficients of the first and second Fundamental Forms.

See also Dupin's Theorem, Fundamental Forms, Principal Curvatures