Shape Representations with Blossoms and Buds
Polynomials, either on their own or as components of splines, play a fundamental role for shape representations in computer-aided geometric design (CAGD) and computer graphics. This paper shows that any polynomial p(t) of degree d ≤n can be represented in the form of a blossom of another polynomial b(t) of degree d evaluated off the diagonal at the linear functions X j (t), j=1, ..., n, chosen under some conditions expressed in terms of the elementary symmetric functions. The polynomial b(t) is called a bud of the polynomial p(t). An algorithm for finding a bud b(t) of a given polynomial p(t) is presented. Successively, a bud of b(t) can be computed and so on, to form a sequence of representations. The information represented by the original polynomial is preserved in its buds. This scheme can be used for encoding/decoding geometric design information.
KeywordsComputer Graphic Geometric Design Decode Process Shape Representation Control Polygon
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- 3.de Casteljau, P.: Shape Mathematics and CAD. Kogan Page, London (1986)Google Scholar
- 4.Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Academic Press, London (2002)Google Scholar
- 6.Ramshaw, L.: Blossoming: A Connect-the-Dots Approach to Splines. Digital Equipment Corporation, SRC Report, June 21 (1987)Google Scholar
- 7.Stefanus, L.Y.: Blossoming off the Diagonal. Ph.D. Thesis, Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada (1991)Google Scholar
- 8.Stefanus, L.Y., Goldman, R.N.: Discrete Convolution Schemes. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods in Computer Aided Geometric Design II. Academic Press, London (1992)Google Scholar
- 10.Stefanus, L.Y.: Surface Representations using Blossoms and Buds (in preparation)Google Scholar