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Meusnier's theorem[mən′yāz ‚thir·əm]
a theorem in differential geometry that establishes the property of curvatures of plane sections of a surface. Let TT be an arbitrary plane through the tangent MT at the point M to the surface S (see Figure 1), θ be the angle that
the plane makes with the normal MN to the surface, and 1/R be the curvature at M of the curve DMC along which the surface S is intersected by the plane cr passing through the normal MN and the line MT (DMC is called the normal section of the surface). Then the curvature 1/ρ at M of the curve AMB along which the surface S is intersected by the plane π is related to the curvature 1/R of the normal section by
This formula expresses Meusnier’s theorem. Meusnier’s theorem was established by J. Meusnier in 1776 but published only in 1785.