دانلود رایگان مقاله محاسبات بر پایه μ از سطوح قانونمند جبری

Abstract

We find a μ-basis for a rational ruled surface, starting from its implicit representation. A parametrization for this ruled surface is then deduced form this μ-basis. This parametrization does not have any non-generic base points and its directrix has the lowest possible degree. A complete interchange graph is built for the algebraic equation, the μ-basis and the parametric equation of a rational ruled surface.

5. Conclusion and future work

In recent years, μ-bases have been shown to have interesting properties. Once a μ-basis of a rational curve or surface is obtained, we can efficiently compute both the implicit equation and a parametrization. Other researchers achieve some additional important results including theorems, algorithms and applications for μ-bases (Chen, 2003; Chen et al., 2005; Chen and Wang, 2003a, 2003b; Deng et al., 2005; Jia, 2014). Nevertheless, there are still open problems, for instance, computing a μ-basis from an implicit equation. This motivates us to consider the first problem on ruled surfaces which are popular in practical applications. The current results are almost all for symbolic computations. However, in applications, the main computations are numerical computations. There are gaps between symbolic algorithms and numerical implements. In the future we hope to fill in some of these gaps with between symbolic and numeric computations.