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Some Researches on Minimal Surface Modelling

Author: HaoYongXia
Tutor: WangRenHong; LiChongJun
School: Dalian University of Technology
Course: Computational Mathematics
Keywords: Curve and Surface Geometric Modeling Subdivision scheme Plateau prob-lem minimal surface Mean curvature Multiresolution B-spline Minimal subdivision sur-face
CLC: TP391.7
Type: PhD thesis
Year: 2013
Downloads: 24
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Abstract


Many problems in the modern product design and manufacturing process can be translated into the problems of curve and surface modeling, such as auto appearance de-sign、design of airplane body、design and process of cloth、and the appearance design of buildings, and so on. The design of curve and surface is also the core content of Computer Aided Geometric Design. Especially in the real applications, the design of curve and surface meeting certain properties and functions is of particular importance. In this paper, our re-search focuses on the minimal surface, a kind of surface important in differential geometry and computational geometry, we also present a simple curve subdivision scheme, and con-sider to combine the research on minimal surface and subdivision scheme. The main work includes the following several aspects:1. Polynomial is an important representation for curve and surface in computational ge-ometry design. However, we can only find very few parametric polynomial minimal surface in CAGD, therefore searching for the parametric polynomial minimal surface and introducing them into CAGD are of great significance. Using a classic result in differential geometry, we present the general formula of parametric polynomial minimal surface, obtain the sufficient and necessary conditions for the corresponding coefficients, analyze their symmetric prop-erties, give the conjugate minimal surfaces, and illustrate concrete examples for polynomial minimal surface of degree6、7、8、9.2. Bezier curve and Bezier surface are basic representation tools in Computer Graph-ics and Computer Aided Geometric Design, while quasi-Bezier is a generalization of Bezier and the space it spans not only includes polynomial space also trigonometric space and hyperbolic space. Using Dirichlet energy, we consider the Plateau problem under the rep-resentation of quasi-Bezier, i.e., Plateau-quasi-Bezier problem. In this case, the boundary curve can not only be polynomial curve also circular curve and catenary. Moreover, we also discuss the harmonic and biharmonic quasi-Bezier surface.3. Plateau problem is always an important problem for minimal surface. In this problem, the given curve is required to be the whole bound of the solution surface, however in applications the boundary information we get maybe only part boundary curve. Based on this, we consider the quasi-Plateau problem:find the surface of minimal area among all the surfaces bounded by the given curve and defined on the rectangular domain, the ’given curve’here can be the whole boundary or part boundary. Replacing the area functional with Dirichlet energy, and using the Ritz-Galerkin method, the quasi-Plateau problem changes into solving a simple sparse linear equations, and finally is summarized as four algorithms according to different boundary conditions. Examples demonstrate that our algorithms are simple and effective.4. Many surface design problems, such as the design of car body、airplane hull and the machine parts, require the surface to satisfy certain continuity conditions at the given bound. Therefore, we consider the C1and C2quasi-Plateau problem, i.e., the surface of minimal area not only is bounded by the given curve, but satisfies the C1or C2continuity. Still using the Dirichlet energy and Ritz-Galerkin method, we obtain six algorithms corresponding to different boundary conditions to get the approximation surface in different levels. Examples demonstrate that these algorithms are simple and effective.5. Subdivision scheme is also an important design method for curve and surface in CAGD. We present a simple binary six point curve subdivision method, this scheme not only has simple form, but also has many good geometric properties at the same time, such as polynomial reproduction%convexity preserving and high continuity.6. Based on the importance of subdivision scheme and minimal surface, we consider to combine the both aspects to construct the quasi-minimal subdivision surface satisfying the given initial boundary control points. Using Dirichlet energy, we obtain the results on inner control points for general subdivision scheme, and illustrate our results by Loop subdivision scheme with several examples.

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