Difference between revisions of "Meusnier theorem"
From Encyclopedia of Mathematics
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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 plane | + | 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 |
<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> | <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> | ||
Revision as of 16:25, 26 July 2014
If 
is a curve lying on a surface and 
is a point on 
, then the curvature 
of 
at 
, the curvature 
of the normal section of the surface by the osculating plane passing through the unit tangent to 
at 
and the unit normal vector, and the angle 
between the referred osculating plane of 
at 
and any other osculating plane that does not go through the unit normal vector, satisfy the relation
 |
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 |
Comments
References
| [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=15630
Meusnier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meusnier_theorem&oldid=15630
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article