Subdivision Of NURBS Curves And Surfaces Based On Matrix Representation

The technology of free-form curves and surface is the core of the CAGD,the NURBS method is gaining popularity and has become a standard tool in industry of geometric modeling because it provides a common mathematical form for analytical geometry and free-form curves and surfaces.Though the NURBS method is practical and widely used,we will still meet a series of new problems,which haven't solved in Non-Rational method.The calculation would be more complicated,especially for the problem of weights and parameters, which even now has not been completely solved.Thus,scholars have been studying new representation methods of curves and surfaces.Subdivision is another way of geometric modeling,which generating smooth curves and surfaces recursively.Subdivision is an efficient tool to design,represent and approach arbitrarily topological surfaces.By means of simple rules,efficiency and convenient modification,subdivision has already been widely used in 3D surface modeling,multiresolution analysis and computer graphics,etc.But current schemes of subdivision can only construct limited curves and surfaces and they usually are lack of analytic expressions.In practical application,we usually need to know the type of the surface and design a suitable scheme of subdivision to make it.We hope our research on the NURBS subdivision can promote the combination of NURBS and subdivision,then it can meet the practical application.This thesis discusses the method of subdivision for NURBS curves,quadratic surfaces and NURBS surfaces based on matrix representation.Here are some works in this thesis:(1) Based on NURBS matrix representation,we gain the subdivision expression of NURBS curves.Using the subdivision method,four results of quadratic NURBS curves in different circumstances are given,then we get the subdivision expression of normal quadratic NURBS curves.(2) Based on the subdivision of NURBS curves,we obtain the subdivision of quadratic surfaces and non-rotary surface by the NURBS representation of circle and ellipse.We develop software of quadratic NURBS subdivision surfaces by MATLAB platform,the illustrations in chapter 3 obtains from this software.(3) Based on NURBS surfaces matrix representation,we present a method for NURBS subdivision surfaces.Through solving nonlinear equations,we gain the subdivision explicit expression of quadratic rational B(?)zier surfaces by MATLAB software.Then,it's to illustrate the impact of the subdivision surface when the weights are modified through specific examples.