Implicitizing rational surfaces of revolution using μ-bases

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Abstract

We provide a new technique for implicitizing rational surfaces of revolution using μ-bases. A degree n rational plane curve rotating around an axis generates a degree 2n rational surface. From a μ-basis p,q of this directrix curve, where μ=deg(p)deg(q)=nμ, and a rational parametrization of the circle r(s)=(2s,1s2,1+s2), we can easily generate three moving planes p,q,r with generic bidegrees (1,μ),(1,nμ),(2,0) that form a μ-basis for the corresponding surface of revolution. We show that this μ-basis is a powerful bridge connecting the parametric representation and the implicit representation of the surface of revolution. To implicitize the surface, we construct a 3n×3n Sylvester style sparse resultant matrix Rs,t for the three bidegree polynomials p,q,r. Applying Gaussian elimination, we derive a 2n×2n sparse matrix Ss,t, and we prove that det(Ss,t)=0 is the implicit equation of the surface of revolution. Using Bezoutians, we also construct a 2(nμ)×2(nμ) matrix Bs,t, and we show that det(Bs,t)=0 is also the implicit equation of the surface of revolution. Examples are presented to illustrate our methods.

Highlights

► We compute a μ-basis for a surface of revolution from a μ-basis for its directrix. ► We construct sparse resultant matrices for three bivariate polynomials of bidegrees (1,μ),(1,nμ),(2,0). ► We provide compact expressions for the implicit equation of a surface of revolution.

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    Here we shall present a simple, global view of the best of these methods, both classical and modern, and we show how to combine and in some cases improve these techniques to develop a general implicitization algorithm that works even in the presence of base points. It is known that for certain simple rational surfaces such as ruled surfaces [7–9], quadric surfaces [10,11], Steiner surfaces [12], surfaces of revolution [13], cyclides [14] and surfaces where one of the surface coordinates depends only on one of the surface parameters [20,21], there are special fast implicitization techniques. For these simple surfaces, we recommend using these special implicitization methods.

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    μ-Bases for rational surfaces are bases for the syzygy module only with respect to affine parameterizations rather than homogeneous parameterizations, the degrees of their elements are not unique and do not necessarily sum to the degree of the parametrization, and although their outer product retrieves the affine parametrization, the resultant of the μ-basis that generates the implicit equation of the surface may contain extraneous factors. Moreover, while there are simple, fast algorithms for computing μ-bases for rational planar curves (Chen and Wang, 2002), fast efficient algorithms for computing μ-bases for rational surfaces are known only for rational ruled surfaces (Chen and Wang, 2003; Shen, 2016), quadric surfaces (Chen et al., 2007; Wang et al., 2008), Steiner surfaces (Wang and Chen, 2012), surfaces of revolution (Shi and Goldman, 2012) and cyclides (Jia, 2014). Algorithms for computing μ-bases for general rational surfaces are neither simple nor fast (Deng et al., 2005).

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This paper has been recommended for acceptance by Ralph Martin.

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