Dissertation
Dissertation > Mathematical sciences and chemical > Mathematics > Geometry, topology > Algebraic Geometry > Algebraic curves,algebraic surfaces

Blending Algebraic Surfaces by the Method of Dividing Boundaries and Filling Holes with S-patches

Author FangMeiE
Tutor WangGuoZhao
School Zhejiang University
Course Applied Mathematics
Keywords UE-Bézier UE-spline algebraic curves sampling blow up peri-metrically approximating algebraic surfaces blending surfaces modelling Spatches
CLC O187.1
Type PhD thesis
Year 2007
Downloads 110
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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 Bézier-like bases and B-spline-like bases. Based on this, UE-Bézier surfaces and UE-spline 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 B-spline surface belong to this type of parametric surfaces. In this paper, rational Bézier surfaces, B-spline surfaces and S-patches are adopted as blending patches. Again, we substitute rational Bézier surfaces and B-spline surfaces with UE-Bézier surfaces and UE-spline surfaces respectively in 2-way blending problem. It can simplify the form of blending surfaces and strengthen their adjustability to do so.Chapter 2 talks about UE-Bézier basis and UE-spline basis in detail. Introducing a kind of frequency parameter (frequency sequence), we define UE-Bézier basis and UE-spline basis of an arbitrary order by integral and recursive method. They unify and extend Bézier-like and B-spline-like bases constructed over polynomial, trigonometric and hyperbolic spaces. They persist good properties of Bernstein basis (B-spline 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 S-patches. Usual blending frames include space-partition, 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 man-made factor and shape adjustable. In this paper, we adopt the frame of division combined with filling holes with S-patches. The division conforms to a certain rule. Each blending boundary is divided into two pieces. A divide-blending patch, blends two basic surfaces along two adjacent boundaries. For 2-way problem, the blending surface consists of two divide-blending patches. For N-way problem, divideblending patches naturally surround two n-sided holes. Each hole is filled by a S-patch. 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 sample-based 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 UE-spline, blowup sampling of algebraic surfaces and UE-spline 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 human-motion models to demonstrate the effectiveness of our blending method.

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