Survey on the theory and applications of \(\mu\)-bases for rational curves and surfaces. (English) Zbl 1388.13035
\(\mu\)-bases are special bases for the syzygy modules for the parametrizations of rational curves and surfaces. Geometrically they represent a set of moving lines or moving planes with special properties.
As the authors describe in the abstract, the technique of \(\mu\)-bases has proved to be a useful tool in solving problems in geometric modeling, such as fast implicitization, singularity computation, reparametrization and point inversion.
This survey paper offers an overview of the development of \(\mu\)-bases theory of rational curves and surfaces which includes definitions, existence, properties, algorithms and applications. The authors review the more recent results of \(\mu\)-bases theory and propose unsolved problems for future research.
As the authors describe in the abstract, the technique of \(\mu\)-bases has proved to be a useful tool in solving problems in geometric modeling, such as fast implicitization, singularity computation, reparametrization and point inversion.
This survey paper offers an overview of the development of \(\mu\)-bases theory of rational curves and surfaces which includes definitions, existence, properties, algorithms and applications. The authors review the more recent results of \(\mu\)-bases theory and propose unsolved problems for future research.
Reviewer: Carlos Hermoso Ortíz (Madrid)
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 14Q05 | Computational aspects of algebraic curves |
| 14Q10 | Computational aspects of algebraic surfaces |
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 68U07 | Computer science aspects of computer-aided design |
Keywords:
rational curve/surface; \(\mu\)-basis; Syzygy; parametrization; implicitization; singularity computation; point inversionReferences:
| [1] | Sederberg, T. W.; Anderson, D. C.; Goldman, R. N., Implicit representation of parametric curves and surfaces, Comput. Vis. Graph. Image Process., 28, 72-84 (1984) · Zbl 0601.65008 |
| [3] | Manocha, D.; Canny, J. F., Algorithm for implicitizing rational parametric surfaces, Comput. Aided Geom. Design, 9, 25-50 (1992) · Zbl 0817.65012 |
| [4] | Chionh, E. W.; Goldman, R. N., Implicitizing rational surfaces with base points by applying perturbations and factors of zero theorem, (Mathematical Methods in Computer Aided Geometric Design II (1992), Academic Press: Academic Press London, Boston), 101-110 |
| [5] | Chionh, E. W.; Goldman, R. N., Elimination and resultants, part I: Elimination and bivariate resultants, IEEE Comput. Graph. Appl., 15, 1, 69-77 (1995) |
| [6] | Chionh, E. W.; Goldman, R. N., Elimination and resultants, part II: Multivariate resultants, IEEE Comput. Graph. Appl., 15, 2, 60-69 (1995) |
| [9] | Chionh, E. W.; Zhang, M.; Goldman, R. N., Fast computation of the Bezout and Dixon resultant matrices, J. Symbolic Comput., 33, 1, 13-29 (2002) · Zbl 0996.65046 |
| [10] | Buchberger, B., Gröbner basis: an algorithmic method in polynomial ideal theory, (Bose, N. K., Recent Trends in Multidimensional Systems Theory (1985), D. Reidel Publishing Company), 431-436 · Zbl 0587.13009 |
| [11] | Buchberger, B., Applications of Gröbner basis in Non-linear computational geometry, Trends Comput. Algebra, 296, 52-80 (1988) · Zbl 0653.13012 |
| [12] | Cox, D.; Little, J.; O’Shea, D., Using Algebraic Geometry (1998), Springer · Zbl 0920.13026 |
| [13] | Cox, D.; Little, J.; O’Shea, D., Ideal, Varieties and Algorithms (2000), Springer |
| [14] | Becker, T.; Weispfenning, V.; Kredel, H., Gröbner Bases: A Computational Approach to Commutative Algebra (1993), Springer-Verlag: Springer-Verlag London, UK · Zbl 0772.13010 |
| [15] | Sederberg, T. W.; Chen, F., Implicitization using moving curves and surfaces, (Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques. Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH’95 (1995), ACM), 301-308 |
| [16] | Sederberg, T. W.; Saito, T.; Qi, D.; Klimaszewski, K. S., Curve implicitization using moving lines, Comput. Aided Geom. Design, 11, 687-706 (1994) · Zbl 0875.68827 |
| [17] | Sederberg, T. W.; Goldman, R.; Du, H., Implicitizing rational curves by the method of moving algebraic curves, J. Symbolic Comput., 23, 153-175 (1997) · Zbl 0872.68193 |
| [18] | Zhang, M.; Chionh, E. W.; Goldman, R. N., On a relationship between the moving line and moving conic coefficient matrices, Comput. Aided Geom. Design, 16, 517-527 (1999) · Zbl 0997.65019 |
| [19] | Cox, D.; Goldman, R.; Zhang, M., On the validity of implicitization by moving quadrics for rational surfaces with no base points, J. Symbolic Comput., 29, 419-440 (2000) · Zbl 0959.68124 |
| [20] | D’Andrea, C., Resultants and moving surfaces, J. Symbolic Comput., 31, 585-602 (2001) · Zbl 1008.65010 |
| [21] | Cox, D., Equations of parametric curves and surfaces via syzygies, (Symbolic Computation Solving Equations in Algebra Geometry and Engineering (2001)), 1-20 · Zbl 1009.68174 |
| [22] | Chen, F.; Sederberg, T., A new implicit representation of a planar rational curve with high order singularity, Comput. Aided Geom. Design, 19, 151-167 (2002) · Zbl 0995.68134 |
| [23] | Busé, L.; Cox, D.; D’Andrea, C., Implicitization of surfaces in \(P^3\) in the presence of base points, J. Algebra Appl., 189-214 (2003) · Zbl 1068.14066 |
| [24] | Cox, D., Curves, surfaces and syzygies, (Topics. in Algebraic Geometry and Geometric Modeling: Contemporary Mathematics, Vol. 334 (2003)), 131-149 · Zbl 1058.14072 |
| [25] | Adkins, W. A.; Hoffmann, J. W.; Wang, H., Equations of parametric surfaces with base points via Syzygies, J. Symbolic Comput., 39, 73-101 (2005) · Zbl 1120.14051 |
| [26] | Busé, L.; Dohm, M., Implicitization of bihomegenous parametrizations of algebraic surfaces via linear syzygies, (Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation. Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, ISSAC’07 (2007), ACM), 69-76, ISSAC · Zbl 1190.14063 |
| [27] | Zheng, J.; Sederberg, T. W.; Chionh, E.-W.; Cox, D. A., Implicitizing rational surfaces with base points using the method of moving surfaces, Contemp. Math., 151-168 (2003) · Zbl 1058.14074 |
| [28] | Lai, Y.; Chen, F., Implicitizing rational surfaces using moving quadrics constructed from moving planes, J. Symbolic Comput., 77, 127-161 (2016) · Zbl 1338.65050 |
| [29] | Botbol, N.; Dickenstein, A., Implicitization of rational hypersurfaces via linear syzygies: a practical overview, J. Symbolic Comput., 74, 493-512 (2016) · Zbl 1375.14199 |
| [30] | Shen, L.; Goldman, R., Implicitizing rational tensor product surfaces using the resultant of three moving planes, ACM Trans. Graph., 36, 5 (2017), Article 167 |
| [31] | Chen, F., Recent advances on surface implicitization, J. Univ. Sci. Technol. China, 44, 5, 345-361 (2014) |
| [32] | Cox, D. A.; Sederberg, T. W.; Chen, F., The moving line ideal basis of planar rational curves, Comput. Aided Geom. Design, 15, 803-827 (1998) · Zbl 0908.68174 |
| [33] | Chen, F.; Zheng, J.; Sederberg, T. W., The \(\mu \)-basis of a rational ruled surface, Comput. Aided Geom. Design, 18, 61-72 (2001) · Zbl 0972.68158 |
| [34] | Chen, F., Reparametrization of a rational ruled surface using the \(\mu \)-basis, Comput. Aided Geom. Design, 20, 11-17 (2003) · Zbl 1069.65510 |
| [35] | Chen, F.; Wang, W., Revisiting the \(\mu \)-basis of a rational ruled surface, J. Symbolic Comput., 36, 699-716 (2003) · Zbl 1037.14014 |
| [36] | Chen, F.; Cox, D.; Liu, Y., The \(\mu \)-basis and implicitization of a rational parametric surface, J. Symbolic Comput., 39, 689-706 (2005) · Zbl 1120.14054 |
| [37] | Chen, F.; Shen, L.; Deng, J., Implicitization and parametrization of quadratic and cubic surfaces by \(\mu \)-bases, Computing, 79, 131-142 (2007) · Zbl 1116.65023 |
| [38] | Wang, X.; Chen, F.; Deng, J., Implicitization and parameterization of quadratic surfaces with one simple base point, (Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation. Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation, ISSAC’08 (2008), ACM), 31-38 · Zbl 1297.68274 |
| [39] | Wang, X.; Chen, F., Implicitization, parameterization and singularity computation of Steiner surface using moving surfaces, J. Symbolic Comput., 47, 733-750 (2012) · Zbl 1245.65021 |
| [40] | Shi, X.; Goldman, R., Implicitizing rational surfaces of revolution using \(\mu \)-bases, Comput. Aided Geom. Design, 29, 348-362 (2012) · Zbl 1256.65016 |
| [41] | Shi, X.; Wang, X.; Goldman, R., Using \(\mu \)-bases to implicitize rational surfaces with a pair of orthogonal directories, Comput. Aided Geom. Design, 29, 541-554 (2012) · Zbl 1256.65017 |
| [42] | Jia, X., Role of moving planes and moving spheres following Dupin cyclides, Comput. Aided Geom. Design, 31, 168-181 (2014) · Zbl 1295.65014 |
| [43] | Dohm, M., The implicit equation of a canal surface, J. Symbolic Comput., 44, 111-130 (2009) · Zbl 1152.14311 |
| [44] | Wang, X.; Goldman, R., Quaternion rational surfaces: rational surfaces generated from the quaternion product of two rational space curves, Graph. Models, 81, 18-32 (2015) |
| [45] | Chen, F., Approximate implicitization of rational curves, Chinese J. Comput., 21, 9, 855-859 (1998) |
| [46] | Song, N.; Chen, F.; Goldman, R., Axial moving lines and singularities of rational planar curves, Comput. Aided Geom. Design, 24, 200-209 (2007) · Zbl 1171.65352 |
| [47] | Chen, F.; Wang, W.; Liu, Y., Computing singular points of plane rational curves, J. Symbolic Comput., 43, 2, 92-117 (2008) · Zbl 1130.14039 |
| [48] | Jia, X.; Goldman, R., \( \mu \)-bases and singularities of rational planar curves, Comput. Aided Geom. Design, 26, 970-988 (2009) · Zbl 1205.14033 |
| [49] | Jia, X.; Goldman, R., Using Smith normal forms and \(\mu \)-bases to compute all the singularities of rational planar curves, Comput. Aided Geom. Design, 29, 296-314 (2012) · Zbl 1298.65028 |
| [50] | Busé, L.; D’Andrea, C., Singular factors of rational plane curves, J. Algebra, 357, 322-346 (2012) · Zbl 1261.14032 |
| [51] | Shi, X.; Chen, F., Computing the singularities of rational space curves, (Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation. Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC’10 (2010), ACM), 171-178 · Zbl 1321.14043 |
| [52] | Shi, X.; Jia, X.; Goldman, R., Using a bihomogeneous resultant to find the singularities of rational space curves, J. Symbolic Comput., 53, 1-25 (2013) · Zbl 1268.14053 |
| [53] | Chen, F.; Wang, W., The \(\mu \)-basis of a planar rational curve — properties and computation, Graph. Models, 64, 14, 368-381 (2003) · Zbl 1055.68117 |
| [54] | Zheng, J.; Sederberg, T., A direct approach to computing the \(\mu \)-basis of planar rational curves, J. Symbolic Comput., 31, 619-629 (2001) · Zbl 0978.65008 |
| [55] | Schonhage, A., Quasi-GCD computations, J. Complexity, 1, 1, 118-137 (1985) · Zbl 0586.68031 |
| [56] | Emiris, I.; Galligo, A.; Lombardi, H., Certified approximate univariate GCDs, J. Pure Appl. Algebra, 117, 229-251 (1997) · Zbl 0891.65015 |
| [57] | Corless, R.; Watt, S.; Zhi, L., QR factoring to computing the GCD of univariate approximate polynomials, IEEE Trans. Signal Process., 52, 3394-3402 (2004) · Zbl 1372.65120 |
| [58] | Chionh, E. W.; Sederberg, T. W., On the minors of the implicitization Bézout matrix for a rational plane curve, Comput. Aided Geom. Design, 18, 1, 21-36 (2001) · Zbl 0972.68156 |
| [59] | Cox, D.; Kustin, A.; Polini, C.; Ulrich, B., A study of singularities on rational curves via syzygies, Mem. Amer. Math. Soc., 222, 1045, 116 (2013) · Zbl 1305.14014 |
| [60] | Song, N.; Goldman, R., \( \mu \)-bases for polynomial systems in one variable, Comput. Aided Geom. Design, 26, 217-230 (2009) · Zbl 1205.65043 |
| [61] | Hong, H.; Hough, Z.; Kogan, I. A., Algorithm for computing \(\mu \)-bases of univariate polynomials, J. Symbolic Comput., 80, 844-874 (2017) · Zbl 1429.13013 |
| [62] | Jia, X.; Wang, H.; Goldman, R., Set-theoretic generators of rational space curves, J. Symbolic Comput., 45, 414-433 (2010) · Zbl 1184.14052 |
| [63] | Wang, H.; Jia, X.; Goldman, R., Axial moving planes and singularities of rational space curves, Comput. Aided Geom. Design, 26, 300-316 (2009) · Zbl 1205.14077 |
| [64] | Shen, L., Computing \(\mu \)-bases from algebraic ruled surfaces, Comput. Aided Geom. Design, 46, 125-130 (2016) · Zbl 1418.65027 |
| [65] | Jia, X.; Chen, F.; Deng, J., Computing self-intersection curves of rational ruled surfaces, Comput. Aided Geom. Design, 26, 287-299 (2009) · Zbl 1205.65119 |
| [66] | Schenck, H.; Seceleanu, A.; Validashti, J., Syzygies and singularities of tensor product surfaces of bidegree (2,1), Math. Comp., 83, 287, 1337-1372 (2013) · Zbl 1286.13013 |
| [67] | Duarte, E.; Schenck, H., Tensor product surfaces and linear syzygies, Proc. Amer. Math. Soc., 144, 65-72 (2015) · Zbl 1327.14225 |
| [68] | Deng, J.; Chen, F.; Shen, L., Computing \(\mu \)-bases of rational curves and surfaces using polynomial matrix factorization, (Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation. Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation, ISSAC’05 (2005), ACM), 132-139 · Zbl 1360.14141 |
| [69] | Shen, L.; Goldman, R., Strong \(\mu \)-bases for rational tensor product surfaces and extraneous factors associated to bas base points and anomalies at infinity, SIAM J. Appl. Algebra Geometry, 1, 1, 328-351 (2017) · Zbl 1371.68300 |
| [70] | Shen, L.; Goldman, R., Algorithms for computing strong \(\mu \)-bases for rational tensor product surfaces, Comput. Aided Geom. Design, 52-53, 48-62 (2017) · Zbl 1366.65033 |
| [71] | Busé, L.; Jouanolou, J. P., On the closed image of a rational map and the implicitization problem, J. Algebra, 265, 312-357 (2003) · Zbl 1050.13010 |
| [72] | Busé, L.; Chardin, M., Implicitizing rational hypersurfaces using approximation complexes, J. Symbolic Comput., 40, 1150-1168 (2005) · Zbl 1120.14052 |
| [73] | Chardin, M., Implicitization using approximation complexes, (Elkadi, M.; Mourrain, B.; Piene, R., Algebraic Geometry and Geometric Modeling (2006), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 23-35 · Zbl 1116.14049 |
| [74] | Dohm, M., Implicitization of rational algebraic surfaces with syzygy-based methods, J. Signal Process. Syst., 244, 1-12 (2008) |
| [75] | Busé, L., Implicit matrix representation of rational Bézier curves and surfaces, Comput. Aided Des., 46, 14-24 (2014) |
| [76] | Busé, L.; Ba, T. L., Matrix-based implicit representations of rational algebraic curves and applications, Comput. Aided Geom. Design, 27, 681-699 (2010) · Zbl 1211.65020 |
| [77] | Wang, X.; Goldman, R., \( \mu \)-bases for complex rational curves, Comput. Aided Geom. Design, 30, 623-635 (2013) · Zbl 1286.65027 |
| [78] | Cox, D. A., The moving curve ideal and the Rees algebra, Theoret. Comput. Sci., 392, 23-36 (2008) · Zbl 1170.13004 |
| [79] | Cox, D.; Hoffman, J. W.; Wang, H., Syzygies and the Rees algebra, J. Pure Appl. Math., 212, 1787-1796 (2008) · Zbl 1151.13012 |
| [80] | Busé, L., On the equations of the moving curve ideal of a rational algebraic plane curve, J. Algebra, 321, 6, 2317-2344 (2009) · Zbl 1168.14027 |
| [81] | Hoffman, J. W.; Wang, H., Defining equations of the Rees algebra of certain parametric surfaces, J. Algebra Appl., 9, 6, 1033-1049 (2010) · Zbl 1211.14063 |
| [82] | Benítez, T. C.; D’Andrea, C., The Rees algebra of a monomial plane parametrization, J. Symbolic Comput., 70, C, 71-105 (2015) · Zbl 1327.13018 |
| [83] | Kustin, A. R.; Polini, C.; Ulrich, B., Rational normal scrollsand the defining equations of Rees algebras, J. Reine Angew. Math., 650, 4, 23-65 (2011) · Zbl 1211.13005 |
| [84] | Hoffman, J. W.; Wang, H.; Jia, X.; Goldman, R., Minimal genrators for the Rees algebra of rational space curves of type (1,1,d-2), Eur. J. Pure Appl. Math., 3, 4, 602-632 (2010) · Zbl 1209.14049 |
| [85] | Morey, S.; Ulrich, B., Rees algebras of ideals with low codimension, Proc. Amer. Math. Soc., 124, 3653-3661 (1996) · Zbl 0882.13003 |
| [86] | Benítez, T. C.; D’Andrea, C., Minimal generators of the defining ideal of the Rees algebra associated to monoid parameterizations, Comput. Aided Geom. Design, 27, 6, 461-473 (2010) · Zbl 1210.65036 |
| [87] | Busé, L.; Chardin, M.; Simis, A., Elimination and nonlinear equations of Rees algebras, J. Algebra, 324, 6, 1314-1333 (2010) · Zbl 1210.13008 |
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