Meusnier’s theorem

Let P be a point of a surfaceMathworldPlanetmathF(x,y,z)=0  where F is twice continuously differentiable in a neighbourhood of P.  Set at P a tangentPlanetmathPlanetmathPlanetmath of the surface. At the point P, set through this tangent both the normal planeMathworldPlanetmathPlanetmath and a skew plane forming the angle ( ω with the normal plane. Let ϱ be the radius of curvatureMathworldPlanetmath of the normal sectionMathworldPlanetmathPlanetmath and ϱω the radius of curvature of the inclined section.

Meusnier proved in 1779 that the equation


between these radii of curvatureMathworldPlanetmathPlanetmath is valid.

One can obtain an illustrative interpretation for the Meusnier’s theorem, if one thinks the sphere with radius the radius ϱ 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