Meusnier theorem

If is a curve lying on a surface and is a point on , then the curvature of at , the curvature of the normal section of the surface by the osculating plane passing through the unit tangent to at and the unit normal vector, and the angle between the referred osculating plane of at and any other osculating plane that does not go through the unit normal vector, satisfy the relation

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

References

[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)