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Contour Tracing in Subdivision Schemes

Duifhuis, H.C. (2000) Contour Tracing in Subdivision Schemes. Master's Thesis / Essay, Computing Science.

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Abstract

Surfaces play an important role in three-dimensional computer aided geometric design (CAGD). Flat faces are always represented by polyhedral meshes. A cube, for example, can be represented by 6 quadrilateral polygons, squares. But it is impossible to represent a smooth sphere using these polygons. There are several different ways to represent smoothly curved surfaces. The classical way to do this is to use piecewise polynomial surfaces, like tensor product Bézier surfaces and tensor product B-sphne surfaces. Tensor product non-uniform rational B-splines (NURBS) have become the most standard representation. A major disadvantage of these tensor product surfaces is the requirement that control nets, defining the surface, must consist of a regular rectangular grid of control points. Therefore these surfaces can only represent limited surface topologies (planes, cylinders and tori). Chaikin's use of subdivision to create curves inspired Catmull and Clark and simultaneously Doo and Sabin to use subdivision to create surfaces, called subdivision suilace,. This work provided the first method of constructing smooth surfaces of arbitrary topological type. Subdivision surfaces received little attention from the computer graphics community for many years, but regained interest because of the intimate connection between subdivision and multiresolution analysis. George M. Chaildn [2] was the first to use recursive subdivision in 1974. Chaikin's algorithm can be thought of as a 'corner cutting' procedure to successively smooth down an initial piecewise-linear function to represent a smooth curve, as shown in Figure 1.1. At each step new vertices are placed and of the way between the old vertices. The new curve has twice as many segments, and repeatedly applied this process yields a piecewise-linear curve with a great number of segments that closely approximates a smooth curve. Similarly the subdivision surfaces introduced by Catmull and Clark and also by Doo and Sabin are defined as the limit of an infinite refinement process of a 3D control mesh or control polyhedron, using a specific subdivision algorithm. These algorithms are based on the binary subdivision of uniform B-spline surfaces. The main advantage of using subdivision surfaces is that surfaces of arbitrary topologies can be represented. Because of its recursive structure arbitrary detail can still be reached. Subdivision surfaces are a good compromise between polygonal meshes and spline patches. They can be treated as large collections of small polygons, and behave like composite patches. The simplicity behind the idea of subdivision makes it easy to understand and implement. In Chapter 8 some examples are illustrated. In this work, three different subdivision surfaces will be discussed: 1. Loop surfaces, which are based on three-directional quartic box spline surfaces (see Figure 1.2). 2. Doo-Sabin surfaces, which are generalizations of biquadratic uniform Bspline surfaces. 3. Catmull-Clark surfaces, which are generalizations of bicubic uniform Bspline surfaces.

Item Type: Thesis (Master's Thesis / Essay)
Degree programme: Computing Science
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Feb 2018 07:29
Last Modified: 15 Feb 2018 07:29
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/8829

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