# line of curvature

A line $\gamma$ on a surface $S$ is a line of curvature of $S$, if in every point of $\gamma$ one of the principal sections has common tangent with $\gamma$.

By the parent entry (http://planetmath.org/NormalCurvatures), a surface  $F(x,\,y,\,z)=0$,  where $F$ has continuous first and partial derivatives, has two distinct families of lines of curvature, which families are orthogonal (http://planetmath.org/ConvexAngle) to each other.

For example, the meridian curves and the circles of latitude are the two families of the lines of curvature on a surface of revolution.

On a developable surface, the other family of its curvature lines consists of the generatrices of the surface.

A necessary and sufficient condition for that the surface normals of a surface $S$ set along a curve $c$ on $S$ would form a developable surface, is that $c$ is a line of curvature of $S$.

Title line of curvature LineOfCurvature 2013-03-22 18:08:44 2013-03-22 18:08:44 pahio (2872) pahio (2872) 5 pahio (2872) Definition msc 53A05 msc 26B05 msc 26A24 curvature line TiltCurve