# Euler formula

From Encyclopedia of Mathematics

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) |

**How to Cite This Entry:**

Euler formula.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Euler_formula&oldid=32596

This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article