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.