Euler formula

A formula expressing the normal curvature of a surface in a given direction $l$ in terms of the principal curvatures $k_1$ and $k_2$:

$$k_l=k_1\cos^2\phi+k_2\sin^2\phi,$$

where $\phi$ is the angle between the direction $l$ and the principal direction corresponding to the principal curvature $k_1$.

This formula was established by L. Euler (1760).

Comments

See also Normal curvature; Principal curvature.

References

[a1]	M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a2]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)