# Curvature

Curvature measures how fast a curve is changing direction at a given point. So, local bending of the surface is measured by curvature. There are a few types of curvatures useful for you:

Normal curvature is the curvature of the curve projected onto the plane containing the curve’s tangent (T) and surface unit-normal (u) vectors. In other words, it is the curvature of the plane containing both and T. All curves with the same tangent vector will have the same normal curvature.

Taking all possible tangent vectors then the max (k1) and min (k2) value (extremum) of the normal curvature at a point are called principal curvatures.

Mean curvature is equal to average of principal curvatures (kH = (k+ k2) / 2).

Gaussian curvature is product of principal curvatures (kGk1 * k2). Sign of the Gaussian curvature can be used to characterize the surface.

k1 * k2 > 0 tells how locally spherical or elliptic the surface is.

k1 * k2 = 0 tells how locally cylindrical or parabolic the surface is.

k1 * k2 < 0 tells how locally saddle-shaped or hyperbolic the surface is.

For more detail and discretization you can check Discrete Differential Geometry Operators for Triangulated 2-Manifolds by Meyer et al.