Bézier Surface Design Based On PDE Method

Bezier surface method is one of the basic surface modeling tools in CAD/CAM system. It uses the Bernstein basis functions, control points and weights to represent the surface. Bezier surface has some good mathematical properties, and can meet certain requirements of the smoothness. But for complex surface design, Bezier surface will produce a large number of control points which are often unevenly distributed across the surface. When adjusting the control points, the relationship between the changes in geometry upon the manipulation of the control points is not intuitive. The process to modify surface shape is tedious, making it very difficult for surface design, thus the surface designers are often inefficient. What's more, it is necessary for Bezier surface designers to master the mathematical principles about Bezier surface technique.In recent years, partical differential equation(PDE) method of surface modeling is introduced into computer-aided design area as an efficient design tool. The main idea of PDE is regarding surface modeling as boundary problem of partical differential equation and the solution of equation is what we need. Surface adjustment is performed by modifying the boundary conditions. This method can be used to construct blending surface, to solve interactive surface design problem and other issues. PDE method generates a smooth surface. The methods for solving partial differential equations are relatively mature, so it is easy to implement PDE method.There are many people research on Bezier surface method and PDE method at home and abroad, but few people study on Bezier surfaces design based on PDE method. In this paper, the theory and algorithms of Bezier surfaces design based on PDE method are researched. If we only give some Bezier control points as the surface boundary conditions, a Bezier surface can be constructed to satisfy the given boundary conditions by finding a polynomial solution of the selected partial differential equation.The main structure and layout are as fallows:The first chapter reviews the development course of surface modeling, and describes the research status of PDE method. The second chapter firstly gives a brief introduction to partial differential equations. It also describes the principles of surface modeling method based on partial differential equations, and then focuses on describing analytical solution and numerical solution method for solving partial differential equations to construct PDE surface. Analytical solution method is faster and can be used to analyse surface properties, but this method needs strict boundary condition. Numerical solution method is slower but has a wide range of usage and is easy to implement. The third chapter introduces the basic principles of Bezier curves and surfaces, then researches the principle and algorithm based on harmonic partial differential equation to generate C0 continuous Bezier surface that satisfies two boundary Bezier curves, or based on biharmonic partial differential equation to generate C0 continuous Bezier surface that satisfy four boundary Bezier curves. C'continuous Bezier surface that satisfy two boundary Bezier curves and their first derivation Bezier curves is researched based on biharmonic-like partial differential equation. C2 continuous Bezier surface is researched based on triharmonic partial differential equation. It can satisfy two boundary Bezier curves, and also satisfy their first derivation Bezier curves and second derivation Bezier curves. Chapter four applys Bezier surfaces design based on PDE method to other boundary conditions, and extends PDE method to a general vector form of 4-order partial differential equation.