Implicitizing rational surfaces of revolution using μ-bases☆
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 . ► We provide compact expressions for the implicit equation of a surface of revolution.
References (11)
- et al.
The μ-basis and implicitization of a rational parametric surface
Journal of Symbolic Computation
(2005) Shifting planes always implicitize a surface of revolution
Computer Aided Geometric Design
(2009)- et al.
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
(1998) - et al.
The μ-basis of a planar rational curve-properties and computation
Graphical Models
(2003) - et al.
Using Algebraic Geometry
(1998)
Cited by (22)
Using μ-bases to reduce the degree in the computation of projective equivalences between rational curves in n-space
2022, Journal of Computational and Applied MathematicsImplicitizing rational surfaces without base points by moving planes and moving quadrics
2019, Computer Aided Geometric DesignCitation Excerpt :Unfortunately, while extensive experiments show that the method of moving surfaces never fails, there lack explicit constructions of moving surfaces and rigorous proofs of the corresponding determinant being always non-vanishing. Thereafter a lot of work has emerged on the validity, compactness and efficiency of implicitization method by moving surfaces (Adkins et al., 2005; Botbol and Dickenstein, 2016; Busé et al., 2003; Busé and Dohm, 2007; Chen et al., 2001, 2003, 2005, 2007 and 2008; Cox et al., 2000, and 2003; Hong et al., 2017; D'Andrea, 2001; Deng et al., 2005; Dohm, 2009; Jia, 2014; Khetan and D'Andrea, 2006; Lai and Chen, 2016, 2017; Shen and Goldman, 2017a, 2017b; Shi and Goldman, 2012; Wang and Chen, 2012; Zheng et al., 2009; Zhang, 2000). For a more complete review of the most recent developments on surface implicitization, the reader is referred to Chen (2014) and Jia et al. (2018).
μ-Bases for rational canal surfaces
2019, Computer Aided Geometric DesignCombining complementary methods for implicitizing rational tensor product surfaces
2018, CAD Computer Aided DesignCitation Excerpt :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.
Survey on the theory and applications of μ-bases for rational curves and surfaces
2018, Journal of Computational and Applied MathematicsAlgorithms for computing strong μ-bases for rational tensor product surfaces
2017, Computer Aided Geometric DesignCitation Excerpt :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.