# Curvature line

From Encyclopedia of Mathematics

A line on a surface at each point of which the tangent has one of the principal directions. The curvature lines are defined by the equation

$$\begin{vmatrix}dv^2&-dudv&du^2\\E&F&G\\L&M&N\end{vmatrix}=0,$$

where $E,F,G$ are the coefficients of the first fundamental form of the surface, and $L,M,N$ those of the second fundamental form. The normals to the surface along curvature lines form a developable surface. The curvature lines on a surface of revolution are the meridians and the parallels of latitude. The curvature lines on a developable surface are its generators (which are straight lines) and the lines orthogonal to them.

#### Comments

#### References

[a1] | D.J. Struik, "Differential geometry" , Addison-Wesley (1950) |

**How to Cite This Entry:**

Curvature line.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Curvature_line&oldid=32954

This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article