Meusnier theorem

If $\gamma$ is a curve lying on a surface and $P$ is a point on $\gamma$, then the curvature $k$ of $\gamma$ at $P$, the curvature $k_N$ of the normal section of the surface by the plane passing through both the unit tangent vector to $\gamma$ at $P$ and the unit normal vector to the surface, and the angle $\alpha$ between the referred plane of $\gamma$ at $P$ and the osculating plane, satisfy the relation

$$ k_N = k \cos \alpha . $$

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
[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)