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
Canonical name	LineOfCurvature
Date of creation	2013-03-22 18:08:44
Last modified on	2013-03-22 18:08:44
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	5
Author	pahio (2872)
Entry type	Definition
Classification	msc 53A05
Classification	msc 26B05
Classification	msc 26A24
Synonym	curvature line
Related topic	TiltCurve