On degenerate surface patches
The authors outline an algorithm to construct a geometrically \(C^ 1\)- surface through a set of given points with prescribed normals at that point. The problem naturally is that one has to start with a triangulation defined by the given points and the normal directions do not relate to any of the patches concurrent at the vertex. They solve the problem by using degenerate polynomial patches, i.e., patches constructed for coalescing base points. They note that the boundary curves are singular in that \(ds \to 0\) at the vertices and at the same time the curvature \(\to \infty\). Since the geometric theory of curvature presupposes nondegenerate \(ds\), one still does obtain a geometrically smooth curve on a geometrically smooth surface. (In fact, these curves have simple cusps at these points, as defined in algebraic geometry. In their example of the plane curve \(y = | x|^{2-\alpha}\), the curve is composed of two arcs fitting together smoothly at the origin; for \(x > 0\) the curve really represents a branch of \(y^ 2 = x^{4-2\alpha}\), with two branches forming a cusp at 0. Since the patches stop at the vertices, the other branch then is discarded).
scientific article; zbMATH DE number 510229