§ Shape operator [TODO ]
§ Principal curvature
- take a point . Consider the normal to the surface at the point, .
- Take any normal plane: a plane which contains . This plane (which is normal to the surface, since it contains the normal) intersecs the surface at a curve (intuitively, since a plane in 3D is defined by 1 eqn, intersection with plane imposes 1 equation on the surface, cutting it down to 1D).
- The curvature of this curve (normal plane intersection surface ) at point is the normal curvature of the normal plane .
- The maximum and minimum such normal curvatures at a point (max, min taken across all possible normal planes ) are the principal curvatures.
§ Shape operator has principal curvatures as eigenvalues
§ Shape operator in index notation
- Let be tangent vectors at point , be normal to surface at point . The shape operator is determined by the equation: