دانلود رایگان مقاله پایه μ صریح برای بخش های مخروطی و مسطح منحنی مکعب گویا

1. Introduction

μ-bases for rational curves are important since we can use μ-bases to implicitize rational planar curves (Chen and Wang, 2003). Moreover, μ-bases for rational curves in arbitrary dimensions can be applied to locate and analyze their singularities (Song et al., 2007; Jia and Goldman, 2009, 2012; Shi and Chen, 2010). There are fast algorithms to compute μ-bases for rational curves based mainly on Gaussian Elimination (Chen and Wang, 2003; Song and Goldman, 2009). However, we do not know what the μ-bases look like before we run the algorithm; this drawback prevents us from finding closed formulas for the singularities or the implicit equation for rational curves based on the method of μ-bases. Here we give explicit formulas for the μ-bases of conic sections and planar rational cubic curves. As fundamental objects in Computer Aided Geometric Design and Computer Graphics, conic sections and planar rational cubic curves have a wide range of applications, for example, in animation control and font design. Implicit equations for planar rational cubic curves will also help us when we deal with intersection algorithms (Sederberg and Parry, 1986; Thomassen, 2005). Special attention has been devoted to computing implicit equations for planar rational cubic curves. Sederberg et al. (1985) treat planar rational cubic curves as monoids, that is, degree n curves with a singular point of order n − 1. This approach simplifies the derivation of the implicit equation. Floater (1995) proves that the implicit equation of every planar rational cubic curve can be represented by linear combinations of six basis functions. Using the results of Busé and Jouanolou (2003), one can also derive a closed formula for the implicit equation of a rational cubic curve P(t) = (x(t), y(t), w(t)) as the determinant of 3 × 3 matrix by inspecting the Bezoutian matrix: M = x · Bez(y(t), w(t)) + y · Bez(w(t), x(t)) + w · Bez(x(t), y(t)) (see Section 4 for further details). In contrast, using our explicit formulas for μ-bases of planar rational cubic curves, we derive two closed formulas – one as the determinant of a sparse 3×3 Sylvester matrix and one as the determinant of a 2×2 Bezoutian matrix – for the implicit equations of planar rational cubic curves. Moreover, we also provide closed formulas for the double point of planar rational cubic curves. Explicit formulas for implicit equations of planar rational cubic Bézier curves are also presented recently in Barrowclough (2014). But our explicit formulas for these implicit equations based on μ-bases are more compact. Moreover, using μ-bases for planar rational cubic curves, we get explicit formulas for their double points in a simpler, more straightforward way.

6. Conclusion

We have constructed explicit formulas for μ-bases of planar rational cubic curves and degree n rational curves in n-dimensions (n ≥ 2). But the formulas for the μ-bases of planar rational cubic curves are not so concise as the formulas for the μ-bases of conic sections. We need to treat several different cases of the planar rational cubic curve. So we can imagine that it will be harder and harder to write down explicit formulas for the μ-bases of rational planar curves of arbitrary degree: there are more and more cases and many more conditions to consider. We also generalized our results to these rational curves expressed in terms of the Bernstein basis, so we can also write down their μ-bases directly without messy conversions between the power basis and the Bernstein basis.