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What is curvature

Curvature describes how small movements along a surface result in changes to the surface normal, roughly, the shape of the surface.

This two-dimensional space (on the right) captures all the possibilities: planar, and cap, ridge, saddle, valley, and cup. This is parameterized by kappa1 and kappa2, the principal curvatures magnitudes, which are two of the degrees of freedom in curvature.

The other degrees of freedom in curvature are the orientation of these lines, the principal curvature directions, which are the directions along which the cross-sections have maximal and minimal curvature.

Curvature is both these magnitudes and these directions. We feel it is simplest to have a general framework for measuring all curvature information, from which we can then extract those components needed for a given problem, rather than rely on the various special-case formula which exist for particular scalar curvature metrics, as is often done with level sets, for example.