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 | ||
| − | $$\begin{vmatrix}dv^2&-dudv&du^2\\E&F&G\\L&M&N\end{vmatrix}=0,$$ | + | $$ |
| + | \begin{vmatrix} | ||
| + | dv^2&-dudv&du^2\\ | ||
| + | E&F&G\\ | ||
| + | L&M&N\end{vmatrix}=0, | ||
| + | $$ | ||
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. | 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. | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J. Struik, "Differential geometry" , Addison-Wesley (1950)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J. Struik, "Differential geometry" , Addison-Wesley (1950)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 05:53, 8 May 2024
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.
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