Blending Algebraic Surfaces by the Method of Dividing Boundaries and Filling Holes with Spatches 

Author  FangMeiE 
Tutor  WangGuoZhao 
School  Zhejiang University 
Course  Applied Mathematics 
Keywords  UEBézier UEspline algebraic curves sampling blow up perimetrically approximating algebraic surfaces blending surfaces modelling Spatches 
CLC  O187.1 
Type  PhD thesis 
Year  2007 
Downloads  110 
Quotes  0 
Algebraic surfaces, including quadrics, are often used in geometric modelling. Surfaces’ blending is an important topic in CAD/CAM. This paper aims to seek a blending method which is simple, stable and suitable for arbitrary algebraic surfaces. Furthermore, the blending surface should be easily controlled. For these reasons, we research the topics on blending surfaces’ types, blending frames and the parametric approximation of blending boundaries.1. Unify and extend Bézierlike bases and Bsplinelike bases. Based on this, UEBézier surfaces and UEspline surfaces with frequency parameters are produced. Blending surfaces traditionally include parametric surfaces, implicit surfaces, subdivision surfaces and meshes. Those parametric surfaces, whose shapes can be easy to be controlled and adjusted, are advantageous for modelling. For example, classical Bézier surfaces and Bspline surface belong to this type of parametric surfaces. In this paper, rational Bézier surfaces, Bspline surfaces and Spatches are adopted as blending patches. Again, we substitute rational Bézier surfaces and Bspline surfaces with UEBézier surfaces and UEspline surfaces respectively in 2way blending problem. It can simplify the form of blending surfaces and strengthen their adjustability to do so.Chapter 2 talks about UEBézier basis and UEspline basis in detail. Introducing a kind of frequency parameter (frequency sequence), we define UEBézier basis and UEspline basis of an arbitrary order by integral and recursive method. They unify and extend Bézierlike and Bsplinelike bases constructed over polynomial, trigonometric and hyperbolic spaces. They persist good properties of Bernstein basis (Bspline basis) and also have some new properties which are advantageous for modelling.2. Propose the blending frame of dividing boundaries combined with filling holes with Spatches. Usual blending frames include spacepartition, con structing initial meshes, base line combined with a moving circle and sweeping along blending boundaries etc.. A good frame should be easily constructed, insensitive to manmade factor and shape adjustable. In this paper, we adopt the frame of division combined with filling holes with Spatches. The division conforms to a certain rule. Each blending boundary is divided into two pieces. A divideblending patch, blends two basic surfaces along two adjacent boundaries. For 2way problem, the blending surface consists of two divideblending patches. For Nway problem, divideblending patches naturally surround two nsided holes. Each hole is filled by a Spatch. The whole process of constructing the frame is simple, stable and sole. The number and degree of blending patches and continuous order with base surfaces can be determined in advance. At the same time, we set free parameters for each blending patch to adjust its shape intuitively.3. Propose the method of approximating planar algebraic curves with splines. Existing parametric blending methods all assume that blending boundaries have parametric representations. However, most of algebraic curves are difficult or impossible to be parameterized or even approximated. Up to now, there isn’t parametric blending method suitable for arbitrary algebraic surfaces. For this reason, we propose a samplebased method of approximating planar algebraic curves. Its accuracy is higher than that of existing methods. We sample nonsingular curves by improved stochastic sampling method (SSM). High accurate sample points can be obtained rapidly. But of singular curves, SSM has a serious weakness, sample results are often bad around singular points. So we blow up singular curves to get several nonsingular curve, which are birationally equivalent to the original curve. Then we sample these nonsingular curves by improved SSM. The difficulty of sampling singular curves is solved in essence.4. Many methods in this paper are extended from different aspects, including 2nd form UEspline, blowup sampling of algebraic surfaces and UEspline approximation of spacial algebraic curves. Though we emphasize the case of planar blending boundaries, corresponding methods can be used in the case of spacial ones. Furthermore, they can also be used to blend parametric surfaces or meshes after some modifications. In Chapter 6, we make some lamps and humanmotion models to demonstrate the effectiveness of our blending method.