The Application Of Polar Form In Computer Aided Geometric Design

This paper mainly discuss some application of polar form in Computer Aided Geometry Design. Specifically, we apply the polar form in the conversion between triangular Bezier surfaces and rectangular Bezier surfaces. This problem helps us to understand the theory of polar form in a unified way. At the last half of this paper, we consider the problem of geometric continuity of parametric curves and parametric surfaces.In the first chapter, we introduce the progress of surface modeling, its current status and its future development, which are the most important reasons to choose the topic of this passage. In chapter 1.3, we present some basis knowledge which will be used in this paper.In the second chapter, we introduce the definition of polar form, the three representations of it and how to represent Bezier curves and surfaces by polar form. In chapter 2.5, we brief introduce how to represent Bezier curves and surfaces by shifting operator. In the third chapter, we use functional composition to deal with the conversion between triangular Bezier surfaces and rectangular Bezier surfaces. The polar form is used as the representation for one of the composite functions, whereas the Bernstein representation is used for another function. At the last part of this chapter, we consider the problem about generalized Bezier surface, which is defined in arbitrary convex polygon.In the chapter four, we consider the problem of geometric continuity. The idea of piecewise is often used in complex surface modeling, so the smoothness conditions become an important question for discussion. Geometric continuity is a kind of smooth measurement which is independence of parametric. We should think about geometric continuity at nodes, besides think about geometric continuity between two patches.