A Surface Interpolation Method Based On Catmull-Clark Subdivision With Shape Control

The subdivision surface method represents surface modeling by defining the initial control network and corresponding subdivision regulations. As a new discrete modeling technology, it combines the polygon mesh representation method and the parametric surface representation method and could process 3D geometric model with complicated topology. It has become one of the hot spots in the researches into computer-aided geometry and computer graphics. The application of subdivision methods usually requires that a subdivision surface intepolates its initial control vertexes and its shape could be controlled. As a generalization of the cubic B-spline surface, Catmull-Clark subdivision surface has been one of the most widely used subdivision method. Therefore, based on Catmull-Clark subdivision, this paper studies the problems about interpolation and shape control for subdivision surfaces, aiming at expanding the modeling capacity of subdivision surface to meet the practical demand.The studies of this paper are mainly to construct interpolation-type Catmull-Clark subdivision surface based on quadrilateral mesh and combining with the progressive-iterative approximation, and to realize the shape control and the local interpolation meanwhile. The purpose is to enrich the modeling methods of Catmull-Clark subdivision surface. The major work of the paper is summarized as follows:1. A progressive interpolation scheme based on Catmull-Clark subdivision is proposed to interpolate the vertexes of the quadrilateral mesh with any topology and the convergence of this method is proven. This subdivision interpolation method changes the vertexes of the initial mesh to generate a new mesh progressively, such that the limit surface of the new mesh obtained through the vertex-based two-step Catmull-Clark subdivision interpolates the vertexes of the initial control mesh finally. Two-step Catmull-Clark subdivision attributes parametric values to each mesh vertex and these parametric values offer the degree of freedoms for adjusting the shape of interpolation surface.2. A local progressive interpolation scheme based on Catmull-Clark subdivision is proposed to realize the local shape control of the interpolation surface and the convergence of this method is proven. Firstly, this method selects several control vertexes of the initial mesh for the iterative adjustment by using the local properties of the progressive iteration approximation method, and maintains the other vertexes unchanged, which results the limit subdivision surface generated finally interpolating the adjusted vertexes of the initial mesh. Secondly, this method is based on the two-step subdivision and the added parametric values of it offer the degree of freedoms for the shape control of the local interpolation surface.3. Under the Visual Studio 2012 compilation environment, a progressive interpolation Catmull-Clark subdivision system with shape control function is developed by using C# and OpenGL. The system realizes the all algorithms given in this paper. The experimental results show that the system could well realize the global or local interpolation functions and control shape for the interpolation-type Catmull-Clark subdivision surfaces with satisfactory effect.