Curvature computations for degenerate surface patches

This paper presents a method to compute curvatures of a surface patch S obtained from a degenerate representation defined over a rectangular domain. To compute curvatures at a point 4 in S where the surface representation is degenerate one has to assume that the point sef S has a tangent plane and well defined curvatures at the point 4 where the curvatures are to be computed. Our computation method employs a local height function representation of S in a neighborhood of the point 4 where the height function h is defined over the tangent plane of S at the point 4. In our method the second order partial derivatives of the height function /I at 4 are computed by using second order derivatives of any three surfaces curves on S which end up in (I with pairwise linearly independent tangent directions. The curvature entities of S at 4 are then computed by using the second order partial derivatives of the height function h at 4. We also show how the method is extended to compute partial derivatives of any order of the function II at 4. For this purpose we show (in Theorem I) how the partial derivatives up to nth order of a surface at a point 4 can be computed using derivatives up to order n of n + 1 surface curves emanating from 4. We also present a definition of a concept of generalized surface curvatures under weaker assumptions appropriate for degenerate surfaces. Under those weaker assumptions one does not require a locally C’-smooth height function representation of the surface over the tangent plane at the degenerate point Q, nor a locally well defined (single-valued) height function representation of the surface in a neighborhood of the point 4. One only needs the existence of a unique local second order approximation of the degenerate surface with a quadratic function defined over the tangent plane at 4. The surface curvatures of the corresponding approximating quadratic surface then define the curvatures of the approximated degenerate surface in the contact point Q.