line of curvature

A line γ on a surface S is a line of curvature of S, if in every point of γ one of the principal sections has common tangentPlanetmathPlanetmathPlanetmath with γ.

By the parent entry (, a surface  F(x,y,z)=0,  where F has continuousMathworldPlanetmathPlanetmath first and partial derivativesMathworldPlanetmath, has two distinct families of lines of curvature, which families are orthogonalPlanetmathPlanetmathPlanetmath ( 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 surfaceMathworldPlanetmath, 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