PlanetMath: Meusnier's theorem

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Meusnier's theorem

(Theorem)

Let be a point of a surface $F(x,\,y,\,z) = 0$ where is twice continuously differentiable in a neighbourhood of . Set at a tangent of the surface. At the point , set through this tangent both the normal plane and a skew plane forming the angle $\omega$ 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

$\displaystyle \varrho_\omega = \varrho\cos\omega$

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 $F(x,\,y,\,z) = 0.$

"Meusnier's theorem" is owned by pahio.

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Other names:

theorem of Meusnier

Also defines:

inclined section

This object's parent.

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Cross-references: curve, intersection, circle of curvature, circle, centre of curvature, centre, radius, sphere, interpretation, curvature, radii, equation, normal section, radius of curvature, plane, normal plane, tangent, neighbourhood, continuously differentiable, surface, point
There is 1 reference to this entry.

This is version 6 of Meusnier's theorem, born on 2007-08-14, modified 2007-09-24.
Object id is 9863, canonical name is MeusniersTheorem.
Accessed 927 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
	53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)

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