A Refineable Space of Smooth Spline Surfaces of Arbitrary Topological Genus Original Research Article

It is shown that parametrical smoothness conditions are sufficient for modeling smooth spline surfaces of arbitrary topology if degenerate surface segments are accepted. In general, degeneracy, i.e., vanishing partial derivatives at extraordinary points, is leading to surfaces with geometrical singularities. However, if the partial derivatives of higher order satisfy certain conditions, the existence of a regular smooth reparametrization can be guaranteed. So, degeneracy is no fundamental obstacle to generating surfaces which are smooth in the sense of differential geometry. Besides its striking simplicity the approach presented here admits the construction of smooth spline spaces which have a natural refinement property. Thus, various algorithms based on subdivision of tensor product B-spline surfaces become available for surfaces of general type.