In the paper An Introduction to Surface Curvature Analysis the analysis of the previous blog entry is generalized to methods that apply to any surface. The paper recounts the fundamental results regarding the computation of the principal curvatures and directions at a point on a surface. It includes several different approaches to these results including derivations using basic definitions, a directional derivative approach, a linear algebra approach, and an eigenvalue/eigenvector approach. The latter is seen to be quite efficient and easy to remember. In bringing these various approaches together in one paper, I hope to save the reader the work involved in sorting through each one independently. The references include the sources I used to gather these approaches.
In the end, just as in the example of the previous blog entry, we set up a coordinate sytem with origin at the point of interest, this time with the principal directions as two axes and the third axis as the normal direction making the trio a right hand system. The key result that the theory establishes is that relative to this coordinate system, the surface can be approximated in a neighborhood of the (new) origin by a mapping z=(x^2 k_1 + y^2 k_2)/2 where k_1 and k_2 are the maximum and minimum curvatures. Thus the surface can be characterized by the signs of the principal curvatures at the point of interest. The theory establishes methods for computing these principal curvatures and their associated principal directions.
The paper includes detailed examples employing the various approaches recounted in the paper. These examples constitute nearly half the paper.
As always, any shortcomings in notation, logic, or lack of clarity are welcomed feedback.
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