Euler's theorem (differential geometry)
# Euler's theorem (differential geometry)

In the mathematical field of differential geometry, Euler's theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated principal directions which give the directions in which the surface curves the most and the least. The theorem is named for Leonhard Euler who proved the theorem in (Euler 1760).

More precisely, let M be a surface in three-dimensional Euclidean space, and p a point on M. A normal plane through p is a plane passing through the point p containing the normal vector to M. Through each (unit) tangent vector to M at p, there passes a normal plane $P_X$ which cuts out a curve in M. That curve has a certain curvature $κ_X$ when regarded as a curve inside $P_X.$ Provided not all $κ_X$ are equal, there is some unit vector $X_1$ for which $k_1$ = $κ_{X1}$ is as large as possible, and another unit vector $X_2$ for which $k_2$ = $κ_{X2}$ is as small as possible. Euler's theorem asserts that $X_1$ and $X_2$ are perpendicular and that, moreover, if X is any vector making an angle θ with $X_1,$ then

    $\kappa _{X}=k_{1}\cos ^{2}\theta +k_{2}\sin ^{2}\theta .\,$         (1)

The quantities $k_1$ and $k_2$ are called the principal curvatures, and $X_1$ and $X_2$ are the corresponding principal directions. Equation (1) is sometimes called Euler's equation (Eisenhart 2004, p. 124).

