Inversion transformations with respect to conics in hybrid number planes

İskender Öztürk; Hasan Çakır; Mustafa Özdemir; İskender Öztürk; Hasan Çakır; Mustafa Özdemir

doi:10.3934/math.2025651

AIMS Mathematics

2025, Volume 10, Issue 6: 14472-14487. doi: 10.3934/math.2025651

Previous Article Next Article

Research article

Inversion transformations with respect to conics in hybrid number planes

1.
Department of Mathematics, Antalya Science and Art Center, Ministry of National Education of the Republic of Turkey, Antalya, Turkey
2.
Department of Mathematics, Faculty of Arts and Sciences, Recep Tayyip Erdoğan University, Rize, Turkey
3.
Department of Mathematics, Faculty of Sciences, Akdeniz University, Antalya, Turkey

Received: 28 April 2025 Revised: 10 June 2025 Accepted: 17 June 2025 Published: 24 June 2025
MSC : 11R52, 30C20, 30G35, 51M15, 51N20

This paper presents a comprehensive study of geometric inversion with respect to central conics in hybrid number planes, which unify complex, hyperbolic, and dual numbers within a single algebraic structure. By employing the hybrid scalar product and the associated pseudo-Euclidean metric, the hybridian planes were classified as elliptic, hyperbolic, or parabolic. Explicit inversion formulas were derived for points, lines, and conics in each plane type. It was shown that lines passing through the inversion center remain invariant, while others transform into conics. Homothetic conics preserve their type under inversion, whereas non-homothetic conics yield cubic or quartic curves depending on their relation to the inversion center. These results extend classical inversion geometry into a unified hybrid setting, providing a new framework for geometric transformations in generalized number systems.
- inversion of conics,
- hybrid numbers,
- pseudo-Euclidean geometry,
- hypercomplex systems,
- geometric transformations
Citation: İskender Öztürk, Hasan Çakır, Mustafa Özdemir. Inversion transformations with respect to conics in hybrid number planes[J]. AIMS Mathematics, 2025, 10(6): 14472-14487. doi: 10.3934/math.2025651

Related Papers:

Abstract

This paper presents a comprehensive study of geometric inversion with respect to central conics in hybrid number planes, which unify complex, hyperbolic, and dual numbers within a single algebraic structure. By employing the hybrid scalar product and the associated pseudo-Euclidean metric, the hybridian planes were classified as elliptic, hyperbolic, or parabolic. Explicit inversion formulas were derived for points, lines, and conics in each plane type. It was shown that lines passing through the inversion center remain invariant, while others transform into conics. Homothetic conics preserve their type under inversion, whereas non-homothetic conics yield cubic or quartic curves depending on their relation to the inversion center. These results extend classical inversion geometry into a unified hybrid setting, providing a new framework for geometric transformations in generalized number systems.

References

[1]	I. M. Yaglom, Geometric transformations IV: circular transformations, Washington: Mathematical Association of America, 2014.
[2]	S. Ponnusamy, H. Silverman, Complex variables with applications, Boston: Birkhäuser, 2006. https://doi.org/10.1007/978-0-8176-4513-7
[3]	N. A. Childress, Inversion with respect to the central conics, Mathematics Magazine, 38 (1965), 147–149. https://doi.org/10.1080/0025570X.1965.11975615 doi: 10.1080/0025570X.1965.11975615
[4]	J. Hrdina, A. Návrat, P. Vašík, Geometric algebra for conics, Adv. Appl. Clifford Algebras, 28 (2018), 66. https://doi.org/10.1007/s00006-018-0879-2 doi: 10.1007/s00006-018-0879-2
[5]	D. Klawitter, Reflections in conics, quadrics and hyperquadrics via Clifford algebra, Beitr. Algebra Geom., 57 (2016), 221–242. https://doi.org/10.1007/s13366-014-0218-2 doi: 10.1007/s13366-014-0218-2
[6]	G. Casanova, Complement on elliptic and hyperbolic inversions, Adv. Appl. Clifford Algebras, 11 (2001), 293–295. https://doi.org/10.1007/BF03042319 doi: 10.1007/BF03042319
[7]	J. L. Ramírez, G. N. Rubiano, Elliptic inversion of two-dimensional objects: a graphical point of view with Mathematica, J. Geom., 3 (2014), 12–27.
[8]	G. Casanova, Conjugated of conics, Adv. Appl. Clifford Algebras, 18 (2008), 143–146. https://doi.org/10.1007/s00006-008-0069-8 doi: 10.1007/s00006-008-0069-8
[9]	G. Sobczyk, The hyperbolic number plane, The College Mathematics Journal, 26 (1995), 268–280. https://doi.org/10.1080/07468342.1995.11973712 doi: 10.1080/07468342.1995.11973712
[10]	W. Miller, R. Boehning, Gaussian, parabolic, and hyperbolic numbers, The Mathematics Teacher, 61 (1968), 377–382. https://doi.org/10.5951/MT.61.4.0377 doi: 10.5951/MT.61.4.0377
[11]	A. A. Harkin, J. B. Harkin, Geometry of generalized complex numbers, Mathematics Magazine, 77 (2004), 118–129. https://doi.org/10.1080/0025570X.2004.11953236 doi: 10.1080/0025570X.2004.11953236
[12]	V. V. Kisil, Geometry of Mobius transformations: Elliptic, parabolic and hyperbolic actions of sl2(r), Singapore: World Scientific, 2012. https://doi.org/10.1142/p835
[13]	V. V. Kisil, Erlangen programme at large 3.2 ladder operators in hypercomplex mechanics, Acta Polytech., 51 (2011), 44–53. https://doi.org/10.14311/1402 doi: 10.14311/1402
[14]	V. V. Kisil, Erlangen program at large: An overview, In: Advances in applied analysis, Basel: Birkhäuser, 2012, 1–94. https://doi.org/10.1007/978-3-0348-0417-2_1
[15]	V. V. Kisil, Induced representations and hypercomplex numbers, Adv. Appl. Clifford Algebras, 23 (2013), 417–440. https://doi.org/10.1007/s00006-012-0373-1 doi: 10.1007/s00006-012-0373-1
[16]	J. Rooney, On the three types of complex number and planar transformations, Environment and Planning B: Planning and Design, 5 (1978), 89–99. https://doi.org/10.1068/b050089 doi: 10.1068/b050089
[17]	J. Rooney, Generalised complex numbers in mechanics, In: Advances on theory and practice of robots and manipulators, Cham: Springer, 2014, 55–62. https://doi.org/10.1007/978-3-319-07058-2_7
[18]	M. Özdemir, Introduction to hybrid numbers, Adv. Appl. Clifford Algebras, 28 (2018), 11. https://doi.org/10.1007/s00006-018-0833-3 doi: 10.1007/s00006-018-0833-3
[19]	İ. Öztürk, M. Özdemir, Similarity of hybrid numbers, Math. Method. Appl. Sci., 43 (2020), 8867–8881. https://doi.org/10.1002/mma.6580 doi: 10.1002/mma.6580
[20]	İ. Öztürk, M. Özdemir, Elliptical rotations with hybrid numbers, Indian J. Pure Appl. Math., 55 (2024), 23–39. https://doi.org/10.1007/s13226-022-00343-5 doi: 10.1007/s13226-022-00343-5
[21]	H. Çakır, M. Özdemir, Hybrid number matrices, Filomat, 37 (2023), 9215–9227. https://doi.org/10.2298/FIL2327215C doi: 10.2298/FIL2327215C
[22]	H. Çakır, Consimilarity of hybrid number matrices and hybrid number matrix equations ${\mathbf A}\widetilde{{\mathbf X}}-{\bf{XB}} = {\mathbf C}$, AIMS Mathematics, 10 (2025), 8220–8234. https://doi.org/10.3934/math.2025378 doi: 10.3934/math.2025378
[23]	S. Ulrych, Relativistic quantum physics with hyperbolic numbers, Phys. Lett. B, 625 (2005), 313–323. https://doi.org/10.1016/j.physletb.2005.08.072 doi: 10.1016/j.physletb.2005.08.072
[24]	R. C. Nunes, Erlangen's program for space-time through space-time geometric algebra induced by the R vector characteristic of the ring of hybrid numbers Z, (2021), arXiv: 2106.11106. https://doi.org/10.48550/arXiv.2106.11106
[25]	F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti, The mathematics of Minkowski space-time: with an introduction to commutative hypercomplex numbers, Basel: Birkhäuser, 2008. https://doi.org/10.1007/978-3-7643-8614-6
[26]	F. Catoni, R. Cannata, V. Catoni, P. Zampetti, Hyperbolic trigonometry in two-dimensional space-time geometry, Italian Physical Society, 118 (2003), 475–492. https://doi.org/10.1393/ncb/i2003-10012-9 doi: 10.1393/ncb/i2003-10012-9
[27]	H. A. Diao, X. X. Fei, H. Y. Liu, K. Yang, Visibility, invisibility, and unique recovery of inverse electromagnetic problems with conical singularities, Inverse Probl. Imag., 18 (2024), 541–570. https://doi.org/10.3934/ipi.2023043 doi: 10.3934/ipi.2023043
[28]	H. A. Diao, H. Y. Liu, L. Y. Tao, Stable determination of an impedance obstacle by a single far-field measurement, Inverse Probl., 40 (2024), 055005. https://doi.org/10.1088/1361-6420/ad3087 doi: 10.1088/1361-6420/ad3087
[29]	H. A. Diao, R. X. Tang, H. Y. Liu, J. X. Tang, Unique determination by a single far-field measurement for an inverse elastic problem, Inverse Probl. Imag., 18 (2024), 1405–1430. https://doi.org/10.3934/ipi.2024020 doi: 10.3934/ipi.2024020
[30]	B. O'neill, Semi-Riemannian geometry with applications to relativity, New York: Academic Press, 1983.
[31]	D. G. Zill, P. D. Shanahan, Complex analysis: A first course in complex analysis with applications, Burlington: Jones & Bartlett Learning, 2009.
[32]	H. Simsek, M. Özdemir, Generating hyperbolical rotation matrix for a given hyperboloid, Linear Algebra Appl., 496 (2016), 221–245. https://doi.org/10.1016/j.laa.2016.01.038 doi: 10.1016/j.laa.2016.01.038
[33]	P. S. Modenov, A. S. Parkhomenko, Euclidean and affine transformations: geometric transformations, New York: Academic Press, 2014.
[34]	J. L. Ramírez, Inversions in an ellipse, Forum Geometricorum, 14 (2014), 107–115.

Reader Comments

Your name:*

Email:*
© 2025 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)