Euler's theorem (differential geometry)

{{short description|Orthogonality of the directions of the principal curvatures of a surface}}

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 {{harv|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 X₁ for which k₁ = κ_X1 is as large as possible, and another unit vector X₂ for which k₂ = κ_X2 is as small as possible. Euler's theorem asserts that X₁ and X₂ are perpendicular and that, moreover, if X is any vector making an angle θ with X₁, then

{{NumBlk|:|$\backslash kappa\_X\; =\; k\_1\backslash cos^2\backslash theta\; +\; k\_2\backslash sin^2\backslash theta.\backslash ,$|{{EquationRef|1}}}}

The quantities k₁ and k₂ are called the principal curvatures, and X₁ and X₂ are the corresponding principal directions. Equation ({{EquationNote|1}}) is sometimes called Euler's equation {{harv|Eisenhart|2004|p=124}}.