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If we have a parametric surface , we can take its taylor expansion

We can now reparametrize such that lies in the origin and the plane is parallel to



Since this matrix is symmetric and real-valued, we can do an eigendecomposition:

From the eigenvalues and , we can derive two characteristics for local curvature:
Mean Curvature:
Gaussian Curvature:
Which has the following interpretations:
: Cap / Peak / Valley (elliptic)
: Cylinder (parabolic)
: Saddle (hyperbolic)

The eigenvectors corresponding to and represent the respective curvature directions and are orthogonal.

Shape Operator
Approximates curvature for triangle meshes.

For each edge : atomic shape operator

where is the angle between the triangle normals .

Around a vertex, we compute the atomic shape operators of all edges in the vicinity

where measures the length of the portion of the edge inside .

From this, we can compute:
The eigenvector to the largest eigenvalue of points into the direction of minimum curvature.
The direction of maximum curvature is orthogonal to that and can be found by taking the cross product with the normal vector.
We cannot derive the exact curvature values and , but it is