Blending Pipe Surfaces With Toric Patches

Construction of transition surface to blend pipe surfaces is a hot topic in Computer Aided Geometric Design(CAGD). One method is to construct a single multisided surface patch to blend the pipes. The disadvantages of this method are that the degree of surface is high and the shape of surface is not easy to control. Another method is to construct piecewise patches of lower degrees with smoothness. Compared with the single patch method, the conditions for continuity between these patches must be considered, which are difficult to solve generally. Thus, a very interesting problem is how to construct the blending surface by fewer patches with lower degrees.Toric patch is multisided surface defined by Krasauskas in 2002. The classical tensor product and triangular rational Bezier surfaces are special forms of toric patches. In this paper, we present a method for constructing the surface consists of two pieces of toric patches to blend pipe surfaces. The main idea of the method is:Firstly, the parametric domains of toric patches are subdivided into regular decompositions after they determined by the geometric property of pipes. Then, based on the degeneration of toric patches and the geometric continuity conditions between rational Bezier patches, the G1 continuity conditions for control points of the blending surface are derived. The advantages of our method are simple, effective, flexible, and without solving any system of equations.The paper is organized as follows:In chapter 1, we summarize the background and the main research developments of the blending surface.In chapter 2, the basics of Bezier surfaces and the condition for G1 continuity are introduced.In chapter 3, we introduce the basics of toric patches and the degeneration theory.In chapter 4, we present the method and examples of blending pipe surfaces with toric patches.