Approximate μ-Bases of Rational Curves and Surfaces
The μ-bases of rational curves and surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of curves and surfaces. However, exact μ-bases may have high degree with complicated rational coefficients and are often hard to compute (especially for surfaces), and sometimes they are not easy to use in geometric modeling and processing applications. In this paper, we introduce approximate μ-bases for rational curves and surfaces, and present an algorithm to compute approximate μ-bases. The algorithm amounts to solving a generalized eigenvalue problem and some quadratic programming problems with linear constraints. As applications, approximate implicitization and degree reduction of rational curves and surfaces with approximate μ-bases are discussed. Both the parametric equations and the implicit equations of the approximate curves/surfaces are easily obtained by using the approximate μ-bases. As indicated by the examples, the proposed algorithm may be a useful alternative to other methods for approximate implicitization.
Keywordsapproximate μ-bases approximate implicitization
Unable to display preview. Download preview PDF.
- 1.Chen, F.: Approximate implicit representation of rational curves. Chinese J. of Computers (in chinese) 21, 855–959 (1998)Google Scholar
- 3.Chen, F., Wang, W.: The μ-basis of a rational curve — properties and computation. Graphical Models 64, 268–381 (2003)Google Scholar
- 8.Chen, F., Wang, W.: Computing the singular points of a planar rational curve using the μ-bases (preprint, 2006)Google Scholar
- 11.Dokken, T., Thomassen, J.B.: Overview of approximate implicitization. In: Topics in Algebraic Geometry and Geometric Modeling, AMS Cont. Math., pp. 169–184 (2003)Google Scholar
- 15.Weinstein, S.E., Xu, Y.: Degree reduction of Bézier curves by approximation and interpolation. In: Anastassiou, G.A. (ed.) Approximation Theory, pp. 503–512. Dekker, New York (1992)Google Scholar
- 16.Sederberg, T.W., Chen, F.: Implicitization using moving curves and surfaces. In: Proceedings of Siggraph, pp. 301–308 (1995)Google Scholar
- 18.Wurm, E., Jüttler, B.: Approximate implicitization via curve fitting. In: Kobbelt, L., Schöder, P., Hoppe, H. (eds.) Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry Processing, pp. 240–247 (2003)Google Scholar