Algorithms for computing strong μ-bases for rational tensor product surfaces

doi:10.1016/j.cagd.2017.03.001

Computer Aided Geometric Design

Volumes 52–53, March–April 2017, Pages 48-62

https://doi.org/10.1016/j.cagd.2017.03.001 Get rights and content

Highlights

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Present fast algorithms for finding strong μ-bases for rational tensor product surfaces.
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Show the relationship between the strong μ-bases and the number of certain base points counting multiplicity.
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Provide some tables of rational tensor product surfaces with strong μ-bases and the number of simple base point.

Abstract

Implicitizing rational surfaces is a fundamental computational task in Algorithmic Algebraic Geometry. Although the resultant of a μ-basis for a rational surface is guaranteed to contain the implicit equation of the surface as a factor, this resultant may also contain extraneous factors. Moreover, μ-bases for rational surfaces are, in general, notoriously difficult to compute. Here we develop fast algorithms to find μ-bases for rational tensor product surfaces whose resultants are guaranteed to be the implicit equation of the corresponding rational surface with no extraneous factors. We call these μ-bases strong μ-bases. Surfaces with strong μ-bases are relatively rare. We show how these strong μ-bases are related to the number of base points counting multiplicity of the corresponding surface parametrization. In addition, when the base points are simple, we provide tables of rational tensor product surfaces with strong μ-bases based on the bidegree of the rational surface and the number of base points of the parametrization. The bidegrees of the corresponding strong μ-bases are also listed in these tables.

Introduction

μ-Bases are a powerful tool for representing and analyzing rational planar curves. μ-Bases for rational planar curves are bases for the syzygy module with respect to homogeneous parameterizations, the degrees of their elements are unique and sum to the degree of the parametrization, their cross product retrieves the homogeneous parametrization, and their resultant generates the implicit equation of the curve with no extraneous factors (Chen and Wang, 2002). Thus we can easily recover both the parametric equations and implicit equation of a rational planar curve from a μ-basis for the curve. We can also use μ-bases to locate and analyze the singularities of rational planar curves (Jia and Goldman, 2009).

The notion of a μ-basis readily extends from rational curves to rational surfaces (Chen et al., 2005, Chen et al., 2001; Chen and Wang, 2003, Song and Goldman, 2009). Nevertheless, μ-bases for rational surfaces have many properties that are qualitatively different from the characteristic properties of μ-bases for rational curves. 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).

For rational surfaces a class of μ-bases more analogous to the notion of μ-bases for rational curves are the strong μ-bases (Shen and Goldman, 2017). Like μ-bases for rational curves, strong μ-bases for rational surfaces are bases for the syzygy module, the degrees of their elements sum to the degree of the parametrization, and their resultant generates the implicit equation of the surface with no extraneous factors. Based on numerical experiments, we conjecture as well that the degrees of the elements of a strong μ-basis are unique (see Section 5).

The primary goal of this paper is to present fast algorithms for finding strong μ-bases for rational tensor product surfaces by solving a simple system of linear equations. We will also show how these exceptional μ-bases are related to the number of base points counting multiplicity of the corresponding surface parametrization. In addition, when the base points are simple, we provide some tables of rational tensor product surfaces with strong μ-bases based on the bidegree of the rational surface and the number of base points of the parametrization; the bidegrees of the corresponding strong μ-bases are also listed in these tables.

Surfaces with strong μ-bases are relatively rare. We can use these tables to find strong μ-bases in the following manner: If we know the number of simple base points of a rational tensor product surface or the implicit degree of a rational tensor product surface (which is easy to compute – see Section 4.2), these tables predict whether or not the surface has a strong μ-basis. Moreover, if the surface does have a strong μ-basis, then these tables tell us the bidegrees of the elements of this μ-basis, so we can solve a simple system of linear equations to compute the elements of this μ-basis. The resultant of this strong μ-basis will give the implicit equation of the surface without any extraneous factors.

We proceed in the following fashion. In Section 2, we provide the basic background, definitions, and notation for base points, moving planes (i.e. syzygies), strong μ-bases, and resultants that we shall use throughout the remainder of this paper. In Section 3 we investigate the existence of syzygies of different bidegrees for rational tensor product surfaces of fixed bidegrees. Section 4 contains our main results: fast algorithms for finding strong μ-bases for rational tensor product surfaces. Here we also provide tables of rational tensor product surfaces with strong μ-bases. Section 5 states a conjecture about the uniqueness of the bidegrees of the elements of a strong μ-basis for a rational tensor product surface. We conclude in Section 6 with a brief summary of our results along with a few details about our implementations.

Section snippets

Preliminaries: base points, moving planes, strong μ-bases and resultants

A rational tensor product surface $P$ of bidegree $(m, n)$ can be represented by homogeneous parametric equations with bihomogeneous parameters $P ((s, u), (t, v)) = (a ((s, u), (t, v)), b ((s, u), (t, v)), c ((s, u), (t, v)), d ((s, u), (t, v)))$ where $\begin{matrix} a ((s, u), (t, v)) = \sum_{i = 0}^{m} \sum_{j = 0}^{n} a_{i, j} (s^{i} u^{m - i}) (t^{j} v^{n - j}), b ((s, u), (t, v)) = \sum_{i = 0}^{m} \sum_{j = 0}^{n} b_{i, j} (s^{i} u^{m - i}) (t^{j} v^{n - j}), \\ c ((s, u), (t, v)) = \sum_{i = 0}^{m} \sum_{j = 0}^{n} c_{i, j} (s^{i} u^{m - i}) (t^{j} v^{n - j}), d ((s, u), (t, v)) = \sum_{i = 0}^{m} \sum_{j = 0}^{n} d_{i, j} (s^{i} u^{m - i}) (t^{j} v^{n - j}) \end{matrix}$ are bihomogeneous polynomials in $R [s, u; t, v]$ and $\gcd (a, b, c, d) = 1$ . The parametrization can also

Existence of moving planes

Consider a rational parametrization $P (s, t) = (a (s, t), b (s, t), c (s, t), d (s, t))$ with bidegree $(m, n)$ . A moving plane of bidegree $(σ_{1}, σ_{2})$ (7) with blending functions $s^{i} t^{j}, i = 0, \dots, σ_{1}, j = 0, \dots, σ_{2}$ can be written in the form $\begin{matrix} (x y z w \dots s^{σ_{1}} t^{σ_{2}} x s^{σ_{1}} t^{σ_{2}} y s^{σ_{1}} t^{σ_{2}} z s^{σ_{1}} t^{σ_{2}} w) \cdot \\ (A_{0, 0} B_{0, 0} C_{0, 0} D_{0, 0} \dots A_{σ_{1}, σ_{2}} B_{σ_{1}, σ_{2}} C_{σ_{1}, σ_{2}} D_{σ_{1}, σ_{2}}) = 0 . \end{matrix}$ Since $(x, y, z, w) = (a, b, c, d)$ is bidgree $(m, n)$ , there is a $(m + σ_{1} + 1) (n + σ_{2} + 1) \times 4 (σ_{1} + 1) (σ_{2} + 1)$ matrix MP such that $(x y z w \dots s^{σ_{1}} t^{σ_{2}} x s^{σ_{1}} t^{σ_{2}} y s^{σ_{1}} t^{σ_{2}} z s^{σ_{1}} t^{σ_{2}} w) = (1 \dots s^{m + σ_{1}} \dots t^{n + σ_{2}} \dots s^{m + σ_{1}} t^{n + σ_{2}}) \cdot M P .$ The columns of the matrix MP are

Computing strong μ-bases

We can easily implicitize a rational parametric surface if we can find a strong μ-basis for this surface. We are now ready to find a strong μ-basis if one exists for a given rational tensor product surface $P (s, t)$ with bidegree $(m, n)$ that has base points in general position.

Conjecture

The degrees of the elements of a μ-basis for a fixed rational curve are unique, but the degrees of the elements of a μ-basis for a fixed rational surface need not be unique. Indeed there are examples of rational surfaces that have two very different μ-bases: one parametrically and algebraically strong and another neither parametrically nor algebraically strong (Shen and Goldman, 2017). As we observed in the Introduction, arbitrary μ-bases for rational surfaces fail to share many of the

Conclusion

We presented fast algorithms for finding strong μ-bases when they exist for rational tensor product surfaces. Using these algorithms, we generated tables of rational tensor product surfaces with strong μ-bases based on the bidegree of the rational surface and the number of simple base points or the degree of the implicit equation of the parametrization.

We have implemented these algorithms using Maplesoft 2015 on an Acer Aspire S7 Ultrabook with RAM 4G and Intel(R) Core(TM)@1.8GHz. We can

Acknowledgements

The first author would thank Dr. Chun-ming Yuan for helpful discussions concerning the proof of Theorem 3. The authors wish to thank the anonymous reviewers for their comments and suggestions.

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Algorithms for computing strong μ-bases for rational tensor product surfaces

Highlights

Abstract

Introduction

Section snippets

Preliminaries: base points, moving planes, strong μ-bases and resultants

Existence of moving planes

Computing strong μ-bases

Conjecture

Conclusion

Acknowledgements

J. Symb. Comput.

J. Symb. Comput.

Graph. Models

J. Symb. Comput.

Comput. Aided Geom. Des.

Comput. Aided Geom. Des.

Comput. Aided Geom. Des.

Comput. Aided Geom. Des.

Comput. Aided Geom. Des.

Comput. Aided Geom. Des.