Computing affine equivalences and symmetries of trigonometric curves in arbitrary dimension

Abstract

We present a new and efficient algorithm to compute affine equivalences and symmetries between two trigonometric curves in an arbitrary dimension. The algorithm benefits from the power of invariance and polynomial gcd and factoring without solving any system of equations. The algorithm is implemented in MAPLE, and extensive experimentations demonstrating the efficiency of the method are given.

Keywords

• affine equivalence
• trigonometric curve
• rational curve

References

1. [1] J.G. Alcázar, U. Gözütok, H.A. Çoban, and C. Hermoso, Detecting affine equivalences between implicit planar algebraic curves, Acta Appl. Math. 182, 2, 2022.
2. [2] J.G. Alcázar, C. Hermoso, and G. Muntingh, Detecting similarity of rational plane curves, J. Comput. Appl. Math. 269, 1–13, 2014.
3. [3] J.G. Alcázar, C. Hermoso, and G. Muntingh, Detecting symmetries of rational plane and space curves, Comput. Aided Geom. Des. 31, 199–209, 2014.
4. [4] J.G. Alcázar and E. Quintero, Affine equivalences of trigonometric curves, Acta Appl. Math. 170, 691–708, 2020.
5. [5] A. Berner, M. Bokeloh, M.Wand, A. Schilling, and H.P. Seidel, A graph-based approach to symmetry detection, in: Symposium on Volume and Point Based Graphics, 1–8, 2008.
6. [6] M. Bizzarri, M. Lávička, and J. Vršek, Computing projective equivalences of special algebraic varieties, J Comput. Appl. Math. 367, 112438, 2020.
7. [7] M. Bizzarri, M. Lávička, and J. Vršek, Symmetries of discrete curves and point clouds via trigonometric interpolation, J Comput. Appl. Math. 408, 114124, 2022.
8. [8] M. Bokeloh, A. Berner, M. Wand, H.P. Seidel, and A. Schilling, Symmetry detection using line features, Comput. Graph Forum 28 (2), 697–706, 2009.

9. [9] M. Boutin, Numerically invariant signature curves, Int. J. Comput. Vis. 40 (3), 235– 248, 2000.
10. [10] P. Brass and C. Knauer, Testing congruence and symmetry for general 3-dimensional objects, Comput. Geom. 27, 3–11, 2004.
11. [11] U. Gözütok, H.A. Çoban, Y. Sarolu, and J.G. Alcázar, A new method to detect projective equivalences and symmetries of rational 3D curves, J. Comput. Appl. Math. 419, 114782, 2023.
12. [12] https://www.ugurgozutok.com/academics/software
13. [13] M. Hauer and B. Jüttler, Projective and affine symmetries and equivalences of rational curves in arbitrary dimension, J. Symb. Comput. 87, 68–86, 2018.
14. [14] M. Hauer, B. Jüttler, and J. Schicho, Projective and affine symmetries and equivalences of rational and polynomial surfaces, J. Comput. Appl. Math. 349, 424–437, 2018.
15. [15] H. Hong, Implicitization of curves parametrized by generalized trigonometric polynomials, in: Proceedings of Applied Algebra, Algebraic Algorithms and Error Correcting Codes (AAECC-11), 285–296, 1995.
16. [16] H. Hong and J. Schicho, Algorithms for trigonometric curves (simplification, implicitization, parameterization), J. Symb. Comput. 26 (3), 279–300, 1998.
17. [17] E.A. Hook, Multiple points on Lissajous’s curves in two and three dimensions, Ann. Math., Second Series, 4 (2), 67–88, 1903.
18. [18] Z. Huang and F.S. Cohen, Affine-invariant B-spline moments for curve matching, IEEE Trans. Image Process 5 (10), 1473–1480, 1996.
19. [19] J.D. Lawrence, A catalog of special plane curves, New York, Dover, pp. 178-179 and 181-183, 1972.
20. [20] P. Lebmeir and J. Richter-Gebert, Rotations, translations and symmetry detection for complexified curves, Comput. Aided Geom. Des. 25, 707–719, 2008.
21. [21] G. Loy and J. Eklundh, Detecting symmetry and symmetric constellations of features, in: Proceedings ECCV 2006, 9th European Conference on Computer Vision, 508–521, 2006.
22. [22] Maple™, Maplesoft, a Division of Waterloo Maple Inc., Waterloo, Ontario, 2021.
23. [23] N.J. Mitra, L.J. Guibas, and M. Pauly, Partial and approximate symmetry detection for 3D geometry, ACM Trans. Graph 25 (3), 560–568, 2006.
24. [24] K. Palmer, T. Ridgway, O. Al-Rawi, I. Johnson, and M. Poullis, Lissajous Figures: An Engineering Tool for Root Cause Analysis of Individual CasesA Preliminary Concept, J Extra Corpor. Technol. 43 (3), 153–156, 2011.
25. [25] R. Sulanke, The fundamental theorem for curves in the n-dimensional Euclidean space, 2020, http://www-irm.mathematik.hu-berlin.de/~sulanke/ diffgeo/euklid/ECTh.pdf.
26. [26] C. Sun and J. Sherrah, 3D symmetry detection using the extended Gaussian image, IEEE Trans. Pattern Anal. Mach. Intell. 19, 164–168, 1997.
27. [27] H. Yalçn, M. Ünel, and W. Wolowich, Implicitization of parametric curves by matrix annihilation, Int. J. Comput. Vis. 54, 105–115, 2003.