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Difference between revisions of "Curvature line"

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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027340/c0273401.png" /></td> </tr></table>
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$$\begin{vmatrix}dv^2&-dudv&du^2\\E&F&G\\L&M&N\end{vmatrix}=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027340/c0273402.png" /> are the coefficients of the first fundamental form of the surface, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027340/c0273403.png" /> 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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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=19095
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article