| Since the computer was born,it has played an important role in people’s life,learning,industrial production.With the rapid development of modern computer technology,the Computer Aided Geometric Design(CAGD)has been widely used in the research of theory and Application.The surface modeling method is a way of constructing a model surface in CAGD.Among many methods of the surface modeling,the subdivision scheme has an absolute advantage that the traditional surface modeling can not match,and it develops rapidly.The subdivision scheme is a representation of the discrete curve surface based on the mesh refinement,which generates subdivision surface by defining the subdivision rules on the control mesh.This method has become a hot research topic in CG and Artificial Intelligence.Although the subdivision scheme has been widely used in engineering practice,but because of its own theory is not perfect,resulting in its application and development in engineering practice has been constrained.In this thesis,the part about theories of the subdivision scheme has been studied.Subdivision of the surface is the final result of the initial control mesh,the whole surface of the process are simply weighted the original vertex way to generate a new vertex,the process of mathematics in line with the thought of interpolation.The De Boor algorithm is a method of expression based on discretization,therefore,we first generalized the De Boor algorithm to the 3D space,and got the extended De Boor algorithm.Secondly,combination the subdivision process and the interpolation theory,the De Boor interpolation method is used to describe the subdivision scheme,and the thesis presents a method to solve undetermined coefficient curve and the mathematical expression of Catmull-Clark subdivision surface,and established the mathematical models of the two methods.In the process of solving the Catmull-Clark subdivision surface model,the multi-group solution of the surface equation is obtained,and it needs to be analyzed and verified.In this thesis,the solution of the system of equations is verified from the generation process and the geometric mapping of the subdivision curve.In the end,only two sets of solutions meet the requirements of subdivision.Finally,according to the model,we studied the continuity,weight,affine,and boundary properties of subdivision surfaces. |
