Difference between revisions of "Curvature line"
From Encyclopedia of Mathematics
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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 | 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 | + | 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. |
Revision as of 10:40, 15 August 2014
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
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