Computation of symmetries of rational surfaces

Juan Gerardo Alcázar; Carlos Hermoso; Hüsnü Anıl Çoban; Uğur Gözütok; Juan Gerardo Alcázar; Carlos Hermoso; Hüsnü Anıl Çoban; Uğur Gözütok

doi:10.3934/era.2024282

Electronic Research Archive

2024, Volume 32, Issue 11: 6087-6108. doi: 10.3934/era.2024282

Previous Article Next Article

Research article Special Issues

Computation of symmetries of rational surfaces

1.
Departamento de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid, Spain
2.
Department of Mathematics, Karadeniz Technical University, Trabzon, Türkiye
3.
Department of Natural Sciences, National Defence University, Istanbul, Türkiye

Received: 11 September 2024 Revised: 31 October 2024 Accepted: 05 November 2024 Published: 12 November 2024

In this paper, we provided, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e., Gauss curvature and mean curvature. In practice, the algorithm works well for sparse parametrizations (e.g., toric surfaces) and PN surfaces. Additionally, we provided a specific, and symbolic, algorithm for computing the symmetries of ruled surfaces. This algorithm works extremely well in practice, since the problem is reduced to that of rational space curves, which can be efficiently solved by using existing methods. The algorithm for ruled surfaces is based on the fact, proven in the paper, that every symmetry of a rational surface must also be a symmetry of its line of striction, which is a rational space curve. The algorithms have been implemented in the computer algebra system Maple, and the implementations have been made public. Evidence of their performance is given in the paper.
- rational surfaces,
- ruled surfaces,
- symmetry detection,
- symbolic computation,
- differential invariants
Citation: Juan Gerardo Alcázar, Carlos Hermoso, Hüsnü Anıl Çoban, Uğur Gözütok. Computation of symmetries of rational surfaces[J]. Electronic Research Archive, 2024, 32(11): 6087-6108. doi: 10.3934/era.2024282

Related Papers:

Abstract

In this paper, we provided, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e., Gauss curvature and mean curvature. In practice, the algorithm works well for sparse parametrizations (e.g., toric surfaces) and PN surfaces. Additionally, we provided a specific, and symbolic, algorithm for computing the symmetries of ruled surfaces. This algorithm works extremely well in practice, since the problem is reduced to that of rational space curves, which can be efficiently solved by using existing methods. The algorithm for ruled surfaces is based on the fact, proven in the paper, that every symmetry of a rational surface must also be a symmetry of its line of striction, which is a rational space curve. The algorithms have been implemented in the computer algebra system Maple, and the implementations have been made public. Evidence of their performance is given in the paper.

References

[1]	J. G. Alcázar, M. Lávička, J. Vršek, Symmetries and similarities of planar algebraic curves using harmonic polynomials, J. Comput. Appl. Math., 357 (2019), 302–318. https://doi.org/10.1016/j.cam.2019.02.036 doi: 10.1016/j.cam.2019.02.036
[2]	P. Lebmair, J. Richter-Gebert, Rotations, translations and symmetry detection for complexified curves, Comput. Aided Geom. Des., 25 (2008), 707–719. https://doi.org/10.1016/j.cagd.2008.09.004 doi: 10.1016/j.cagd.2008.09.004
[3]	P. Lebmair, Feature Detection for Real Plane Algebraic Curves, Ph.D thesis, Universität München, 2009.
[4]	J. G. Alcázar, M. Lávička, J. Vršek, Computing symmetries of implicit algebraic surfaces, Comput. Aided Geom. Des., 104 (2023), 102221. https://doi.org/10.1016/j.cagd.2023.102221 doi: 10.1016/j.cagd.2023.102221
[5]	M. Bizzarri, M. Làvi$\breve{{\rm{c}}}$ka, J. Vr$\breve{{\rm{s}}}$ek, Computing projective equivalences of special algebraic varieties, J. Comput. Appl. Math., 367 (2020), 112438. https://doi.org/10.1016/j.cam.2019.112438 doi: 10.1016/j.cam.2019.112438
[6]	U. Gözütok, H. A. Çoban, Detecting isometries and symmetries of implicit algebraic surfaces, AIMS Math., 9 (2024), 4294–4308. https://doi.org/10.3934/math.2024212 doi: 10.3934/math.2024212
[7]	J. G. Alcázar, C. Hermoso, G. Muntingh, Symmetry detection of rational space curves from their curvature and torsion, Comput. Aided Geom. Des., 33 (2015), 51–65. https://doi.org/10.1016/j.cagd.2015.01.003 doi: 10.1016/j.cagd.2015.01.003
[8]	M. Hauer, B. Jüttler, Projective and affine symmetries and equivalences of rational curves in arbitrary dimension, J. Symb. Comput., 87 (2018), 68–86. https://doi.org/10.1016/j.jsc.2017.05.009 doi: 10.1016/j.jsc.2017.05.009
[9]	J. G. Alcázar, C. Hermoso, Involutions of polynomially parametrized surfaces, J. Comput. Appl. Math., 294 (2016), 23–38. https://doi.org/10.1016/j.cam.2015.08.002 doi: 10.1016/j.cam.2015.08.002
[10]	M. Hauer, B. Jüttler, J. Schicho, Projective and affine symmetries and equivalences of rational and polynomial surfaces, J. Comput. Appl. Math., 349 (2019), 424–437. https://doi.org/10.1016/j.cam.2018.06.026 doi: 10.1016/j.cam.2018.06.026
[11]	B. Jüttler, N. Lubbes, J. Schicho, Projective isomorphisms between rational surfaces, J. Algebra, 594 (2022), 571–596. https://doi.org/10.1016/j.jalgebra.2021.11.045 doi: 10.1016/j.jalgebra.2021.11.045
[12]	https://github.com/niels-lubbes/surface_equivalence
[13]	J. G. Alcázar, H. Dahl, G. Muntingh, Symmetries of canal surfaces and Dupin cyclides, Comput. Aided Geom. Des., 59 (2018), 68–85. https://doi.org/10.1016/j.cagd.2017.10.001 doi: 10.1016/j.cagd.2017.10.001
[14]	J. G. Alcázar, G. Muntingh, Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design, J. Comput. Appl. Math., 411 (2022), 114206. https://doi.org/10.1016/j.cam.2022.114206 doi: 10.1016/j.cam.2022.114206
[15]	J. G. Alcázar, E. Quintero, Affine equivalences, isometries and symmetries of ruled rational surfaces, J. Comput. Appl. Math., 364 (2020), 112339. https://doi.org/10.1016/j.cam.2019.07.004 doi: 10.1016/j.cam.2019.07.004
[16]	J. Kozak, M. Krajnc, V. Vitrih, A quaternion approach to polynomial PN surfaces, Comput. Aided Geom. Des., 47 (2016), 172–188. https://doi.org/10.1016/j.cagd.2016.05.007 doi: 10.1016/j.cagd.2016.05.007
[17]	https://www.ugurgozutok.com/academics/software
[18]	U. Gözütok, Source Code/Examples for the paper: Computation of symmetries of rational surfaces, Zenodo, 2024. https://doi.org/10.5281/zenodo.13970116
[19]	J. R. Sendra, F. Winkler, S. Pérez-Díaz, Rational Algebraic Curves—A Computer Algebra Approach, Springer, 2008. https://doi.org/10.1007/978-3-540-73725-4
[20]	M. P. Do Carmo, Differential Geometry of Curves and Surfaces, 2$^nd$ edition, Courier Dover Publications, 2016.
[21]	A. Gray, Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, 1999.
[22]	D. J. Struik, Lectures on Classical Differential Geometry, Dover Publications, 1998.
[23]	N. M. Patrikalakis, T. Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing, Springer, 2002. https://doi.org/10.1007/978-3-642-04074-0
[24]	G. Fischer, Plane Algebraic Curves, American Mathematical Society, 2001.
[25]	D. Pellis, M. Kilian, H. Pottmann, M. Pauly, Computational design of weingarten surfaces, ACM Trans. Graphics, 40 (2021), 1–11. https://doi.org/10.1145/3450626.3459939 doi: 10.1145/3450626.3459939
[26]	L. Y. Shen, S. Pérez-Díaz, Characterization of rational ruled surfaces, J. Symb. Comput., 63 (2014), 21–45. https://doi.org/10.1016/j.jsc.2013.11.003 doi: 10.1016/j.jsc.2013.11.003
[27]	J. G. Alcázar, R. Goldman, Detecting when an implicit equation or a rational parametrization defines a conical or cylindrical surface, or a surface of revolution, IEEE Trans. Visual Comput. Graphics, 23 (2017), 2550–2559. https://doi.org/10.1109/TVCG.2016.2625786 doi: 10.1109/TVCG.2016.2625786
[28]	Maple^TM, Maplesoft, a division of Waterloo Maple Inc. Waterloo, Ontario, 2021.
[29]	U. Gözütok, H. A. Çoban, Y. Sağiroğlu, J. G. Alcázar, A new method to detect projective equivalences and symmetries of $3D$ rational curves, J. Comput. Appl. Math., 419 (2023), 114782. https://doi.org/10.1016/j.cam.2022.114782 doi: 10.1016/j.cam.2022.114782

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)