Construction And Application Of Polyhedral Splines

The progress and development of Computer Science has brought great revolution into the design and manufacture of aerospace industry, automobile industry and other modern industry.Consequently,representation,analysis and processing of industrial product shape become a hot research field. Computational geometry develops rapidly in this environment and turns into a new discipline.Computational geometry mainly focuses on the geometry representation of curves,surfaces and solids which provides a reliable mathematical basis for storage, display and processing of geometry objects.In the past forty years, after the appearances of Bezier method, B-spline method and the unified NURBS method, the different kinds of curves and surfaces can be expressed as parametric spline curves and surfaces, which can be used conveniently in the actual design and production.Curve representation theory has been mature, while tensor product spline achieves some good results in the expression of special two-dimensional surfaces, for the case of arbitrary segmentation in planar area bivariate spline theory is still preliminary.Polyhedral spline is a generalization of B-spline with very excellent properties such as smoothness,local support property, flexibility, etc.It can be fully applied in the regular or arbitrary grid.Under the different grids, this paper tries to construct the polyhedral splines and applies them in the generation of surfaces.The main content of this paper is as follows:1. Introduces the fundamental concepts, theories and methods of computational ge-ometry;2. Polyhedral spline theory is summarized in this section, including the definition and properties of simplex spline, simplex spline recursive evaluation and analysis of polyhedral spline;3. In this section we will interpret how to construct polyhedral spline by simplex s-pline in the case of regular uniform type-2 triangular mesh,and analyze the proper-ty of these splines.In the end of this section,an example will be given to show the appearance of polyhedral spline surface;4. In the case of arbitrary parametric grids, polyhedral splines will be designed ac-cordingly.We will combine these splines and the open source library CGAL to im-plement a routine which will generate the candide 3 surface.