Meusnier’s theorem

Let $P$ be a point of a surface  $F(x,y,z)=0$  where $F$ is twice continuously differentiable in a neighbourhood of $P$.  Set at $P$ a tangent of the surface. At the point $P$, set through this tangent both the normal plane and a skew plane forming the angle (http://planetmath.org/AngleBetweenTwoPlanes) $\omega$ with the normal plane. Let $\varrho$ be the radius of curvature of the normal section and ${\varrho }_{\omega }$ the radius of curvature of the inclined section.

Meusnier proved in 1779 that the equation

$${\varrho }_{\omega }=\varrho \cos \omega$$

between these radii of curvature is valid.

One can obtain an illustrative interpretation for the Meusnier’s theorem, if one thinks the sphere with radius the radius $\varrho$ of curvature of the normal section and with centre the corresponding centre of curvature. Then the equation utters that the circle, which is intersected from the sphere by the inclined plane, is the circle of curvature of the intersection curve of this plane and the surface  $F(x,y,z)=0.$

Title	Meusnier’s theorem
Canonical name	MeusniersTheorem
Date of creation	2013-03-22 17:28:39
Last modified on	2013-03-22 17:28:39
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	9
Author	pahio (2872)
Entry type	Theorem
Classification	msc 53A05
Classification	msc 26B05
Classification	msc 26A24
Synonym	theorem of Meusnier
Related topic	EulersTheorem2
Related topic	ProjectionOfPoint
Related topic	NormalCurvatures
Defines	inclined section