Difference between revisions of "Meusnier theorem"
From Encyclopedia of Mathematics
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| − | If | + | 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. | 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, | + | <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><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> | |
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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
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