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Difference between revisions of "Meusnier theorem"

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m0637401.png" /> is a curve lying on a surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m0637402.png" /> is a point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m0637403.png" />, then the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m0637404.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m0637405.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m0637406.png" />, the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m0637407.png" /> of the normal section of the surface by the osculating plane passing through the unit tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m0637409.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m06374010.png" /> and the unit normal vector, and the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m06374011.png" /> between the referred osculating plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m06374012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063740/m06374013.png" /> and any other osculating plane that does not go through the unit normal vector, satisfy the relation
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If $\gamma$ is a curve lying on a surface and $P$ is a point on $\gamma$, then the curvature $k$ of $\gamma$ at $P$, the curvature $k_N$ of the normal section of the surface by the plane passing through both the unit tangent vector to $\gamma$ at $P$ and the unit normal vector to the surface, and the angle $\alpha$ between the referred plane of $\gamma$ at $P$ and the osculating plane, satisfy the relation
  
<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/m/m063/m063740/m06374015.png" /></td> </tr></table>
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$$ k_N = k \cos \alpha . $$
  
 
In particular, the curvature of every inclined section of the surface can be expressed in terms of the curvature of the normal section with the same tangent.
 
In particular, the curvature of every inclined section of the surface can be expressed in terms of the curvature of the normal section with the same tangent.
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Meusnier,   ''Mém. prés. par div. Etrangers. Acad. Sci. Paris'' , '''10'''  (1785)  pp. 477–510</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J. Meusnier, ''Mém. prés. par div. Etrangers. Acad. Sci. Paris'' , '''10'''  (1785)  pp. 477–510</TD></TR><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 142</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Blaschke, K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR>
 
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</table>
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 142</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR></table>
 

Latest revision as of 18:43, 4 May 2023

If $\gamma$ is a curve lying on a surface and $P$ is a point on $\gamma$, then the curvature $k$ of $\gamma$ at $P$, the curvature $k_N$ of the normal section of the surface by the plane passing through both the unit tangent vector to $\gamma$ at $P$ and the unit normal vector to the surface, and the angle $\alpha$ between the referred plane of $\gamma$ at $P$ and the osculating plane, satisfy the relation

$$ k_N = k \cos \alpha . $$

In particular, the curvature of every inclined section of the surface can be expressed in terms of the curvature of the normal section with the same tangent.

This theorem was proved by J. Meusnier in 1779 (and was published in [1]).

References

[1] J. Meusnier, Mém. prés. par div. Etrangers. Acad. Sci. Paris , 10 (1785) pp. 477–510
[a1] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 142
[a2] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
How to Cite This Entry:
Meusnier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meusnier_theorem&oldid=32553
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article