Principle Curvatures

Consider the plane curve y = kx²/2. At <0, 0>, d²y/dx² and the curvature are both k. The center of curvature is at <0, 1/k>.

Next we consider a 2D surface embedded in a flat 3D space. Consider the curvatures at <0, 0, 0> of the surface z = x² + 2y². In the x-z plane the curvature is 2 while in the y-z plane it is 4. The principle curvatures are 2 and 4.

In general in n-dimensions at a point on a “smooth” n−1-surface you can find a coordinate system in which to 2nd order x₀ = ∑k_ix_i²/2 where the sum is over 1 to n−1. The k_i are the principle curvatures. They may be of different signs. Some may be zero.

If the surface is the surface of a smooth body then we take the x₀ coördinate to be pointing inwards. If the body is smooth and convex then all of the k’s will be positive.