Recursive Gk transformations between adjacent Bézier surfaces

Abstract

In this paper, explicit conditions of G k continuity between Bézier surfaces are given. We concentrate on the structures of G k transformations between adjacent Bézier surfaces and show that a general G k transformation can be represented recursively with the composition of k cardinal G k transformations. We can thus construct a new Bézier surface Q from a given Bézier surface R such that Q and R meet with G k continuity by recursively applying simple geometric transformations which have intuitive geometric meaning for k times. When these simple G k transformations are also polynomial preserving, each of them is actually determined by three real constants which are called shape parameters. The structures of G k transformations are explored and described. Since the G k conditions between two Bézier surfaces are finally expressed with the explicit relationship of the related control points, these results can be used directly in closed surface modeling, surface blending and surface connecting.