Efficient computation of moving planes for rational parametric surfaces with base points using Dixon resultants (2024)

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1. Buchberger, 1988 Buchberger B., Applications of Gröbner bases in non-linear computational geometry, in: Trends in Computer Algebra, Springer Berlin Heidelberg, Berlin, Heidelberg, 1988, pp. 52–80.Google Scholar
2. Busé, 2014 Busé L., Implicit matrix representations of rational Bézier curves and surfaces, Comput. Aided Des. 46 (2014) 14–24.Google ScholarDigital Library
3. Busé and Chen, 2021 Busé L., Chen F., Determinantal tensor product surfaces and the method of moving quadrics, Trans. Am. Math. Soc. 374 (7) (2021) 4931–4952.Google Scholar
4. Chen et al., 2005 Chen F., Cox D., Liu Y., The μ-basis and implicitization of a rational parametric surface, J. Symb. Comput. 39 (6) (2005) 689–706.Google Scholar
5. Chen et al., 2008 Chen F., Wang W., Liu Y., Computing singular points of plane rational curves, J. Symb. Comput. 43 (2) (2008) 92–117.Google Scholar
6. Chionh and Goldman, 1992 Chionh E.-W., Goldman R.N., Implicitizing rational surfaces with base points by applying perturbations and the factors of zero theorem, in: Mathematical Methods in Computer Aided Geometric Design II, Academic Press, 1992, pp. 101–110.Google Scholar
7. Chionh et al., 2000 Chionh E.-W., Zhang M., Goldman R.N., Implicitization by Dixon a-resultants, in: Proceedings Geometric Modeling and Processing 2000. Theory and Applications, IEEE, 2000, pp. 310–318.Google Scholar
8. Chionh et al., 2002 Chionh E.-W., Zhang M., Goldman R.N., Fast computation of the Bezout and Dixon resultant matrices, J. Symb. Comput. 33 (1) (2002) 13–29.Google Scholar
9. Cox, 2001 Cox D., Equations of parametric curves and surfaces via syzygies, Contemp. Math. 286 (2001) 1–20.Google Scholar
10. Cox, 2003 Cox D., Curves, surfaces, and syzygies, Contemp. Math. 334 (2003) 131–150.Google Scholar
11. Cox et al., 2000 Cox D., Goldman R., Zhang M., On the validity of implicitization by moving quadrics for rational surfaces with no base points, J. Symb. Comput. 29 (3) (2000) 419–440.Google Scholar
12. D.A. Cox et al., 2000 Cox D.A., Little J., O'Shea D., Ideals, Varieties, and Algorithms, Springer, 2000.Google Scholar
13. Cox et al., 2006 Cox D.A., Little J., O'Shea D., Using Algebraic Geometry, vol. 185, Springer Science & Business Media, 2006.Google Scholar
14. Dixon, 1908 Dixon A.L., The eliminant of three quantics in two independent variables, Proc. Lond. Math. Soc. (2) 7 (1908) 49–69.Google Scholar
15. Jia et al., 2022a Jia X., Chen F., Yao S., Singularity computation for rational parametric surfaces using moving planes, ACM Trans. Graph. (TOG) 42 (1) (2022) 1–14.Google Scholar
16. Jia et al., 2022b Jia X., Li K., Cheng J., Computing the intersection of two rational surfaces using matrix representations, Comput. Aided Des. 150 (2022).Google Scholar
17. Lai and Chen, 2017 Lai Y., Chen F., An improved algorithm for constructing moving quadrics from moving planes, J. Syst. Sci. Complex. 30 (6) (2017) 1483–1506.Google Scholar
18. Lai and Chen, 2016 Lai Y., Chen F., Implicitizing rational surfaces using moving quadrics constructed from moving planes, J. Symb. Comput. 77 (Nov.-Dec. 2016) 127–161.Google Scholar
19. Lai et al., 2019 Lai Y., Chen F., Shi X., Implicitizing rational surfaces without base points by moving planes and moving quadrics, Comput. Aided Geom. Des. 70 (2019) 1–15.Google Scholar
20. Lai et al., 2022 Lai Y., Chen F., Shi X., Gao Y., A constructive approach to implicitizing rational surfaces with LCI base points by moving planes and moving quadrics, Comput. Aided Geom. Des. 92 (2022).Google Scholar
21. Sederberg and Chen, 1995 Sederberg T.W., Chen F., Implicitization using moving curves and surfaces, in: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, Association for Computing Machinery, New York, NY, USA, 1995, pp. 301–308.Google Scholar
22. Sederberg et al., 1984 Sederberg T.W., Anderson D.C., Goldman R.N., Implicit representation of parametric curves and surfaces, Comput. Vis. Graph. Image Process. 28 (1) (1984) 72–84.Google Scholar
23. Shen and Goldman, 2018 Shen L.-Y., Goldman R., Combining complementary methods for implicitizing rational tensor product surfaces, Comput. Aided Des. 104 (2018) 100–112.Google ScholarDigital Library
24. Shen and Goldman, 2017 Shen L.Y., Goldman R., Implicitizing rational tensor product surfaces using the resultant of three moving planes, ACM Trans. Graph. 36 (5) (2017) 1–14.Google Scholar

Index Terms

1. Efficient computation of moving planes for rational parametric surfaces with base points using Dixon resultants

   1. Computing methodologies

      1. Symbolic and algebraic manipulation

         1. Symbolic and algebraic algorithms

            1. Algebraic algorithms

            2. Linear algebra algorithms

   2. Mathematics of computing

      1. Mathematical analysis

         1. Numerical analysis

            1. Computations on matrices

   3. Theory of computation

      1. Randomness, geometry and discrete structures

         1. Computational geometry

• Using moving planes to implicitize rational surfaces generated from a planar curve and a space curve

  A rational surface(1) S ( s , t ) = ( A ( s ) a ( t ) , B ( s ) b ( t ) , C ( s ) c ( t ) , C ( s ) d ( t ) ) can be generated from a rational planer curve P * ( s ) = ( A ( s ) , B ( s ) , C ( s ) ) and a rational space curve P ( t ) = ( a ( t ) , b ( ...

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• Implicitizing rational surfaces without base points by moving planes and moving quadrics

  Highlights

  • A tensor product rational surface of bidegree ( m , n ) without base points can always be implicitized by moving quadrics together with moving planes.

  Abstract

  It was proven by Cox, Goldman and Zhang that a tensor product rational surface without base points can be implicitized by moving quadrics whenever the rational surface doesn't contain low degree moving planes following it. However, ...

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• Implicitizing Rational Tensor Product Surfaces Using the Resultant of Three Moving Planes

  Implicitizing rational surfaces is a fundamental computational task in Computer Graphics and Computer Aided Design. Ray tracing, collision detection, and solid modeling all benefit from implicitization procedures for rational surfaces. The univariate ...

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