Primitivoids of curves in Minkowski plane

Yanlin Li; A. A. Abdel-Salam; M. Khalifa Saad; Yanlin Li; A. A. Abdel-Salam; M. Khalifa Saad

doi:10.3934/math.2023123

AIMS Mathematics

2023, Volume 8, Issue 1: 2386-2406. doi: 10.3934/math.2023123

Previous Article Next Article

Research article

Primitivoids of curves in Minkowski plane

1.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
2.
Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
3.
Department of Mathematics, Faculty of Science, NBU of Arar, Arar 1321, KSA
4.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, KSA

Received: 19 August 2022 Revised: 14 October 2022 Accepted: 19 October 2022 Published: 02 November 2022
MSC : 53A35, 53B30

In this work, we investigate the differential geometric characteristics of pedal and primitive curves in a Minkowski plane. A primitive is specified by the opposite structure for creating the pedal, and primitivoids are known as comparatives of the primitive of a plane curve. We inspect the relevance between primitivoids and pedals of plane curves that relate with symmetry properties. Furthermore, under the viewpoint of symmetry, we expand these notions to the frontal curves in the Minkowski plane. Then, we present the relationships and properties of the frontal curves in this category. Numerical examples are presented here in support of our main results.
- primitive,
- primitivoids,
- pedaloids,
- Minkowski plane
Citation: Yanlin Li, A. A. Abdel-Salam, M. Khalifa Saad. Primitivoids of curves in Minkowski plane[J]. AIMS Mathematics, 2023, 8(1): 2386-2406. doi: 10.3934/math.2023123

Related Papers:

Abstract

In this work, we investigate the differential geometric characteristics of pedal and primitive curves in a Minkowski plane. A primitive is specified by the opposite structure for creating the pedal, and primitivoids are known as comparatives of the primitive of a plane curve. We inspect the relevance between primitivoids and pedals of plane curves that relate with symmetry properties. Furthermore, under the viewpoint of symmetry, we expand these notions to the frontal curves in the Minkowski plane. Then, we present the relationships and properties of the frontal curves in this category. Numerical examples are presented here in support of our main results.

References

[1]	V. I. Arnold, Singularities of caustics and wave fronts, Dordrecht: Kluwer Academic Publishers, 1990.
[2]	V. I. Arnold, Encyclopedia of mathematical sciences, dynamical systems VIII, Berlin: Springer, 1989.
[3]	T. Nishimura, Normal forms for singularities of pedal curves produced by non-singular dual curve germs in $ S^{n} $, Geometriae Dedicata, 133 (2008), 59–66. https://doi.org/10.1007/s10711-008-9233-5 doi: 10.1007/s10711-008-9233-5
[4]	M. Božek, G. Foltán, On singularities of arbitrary order of pedal curves, Proc. Symp. Comput. Geom. SCG, 21 (2012), 22–27.
[5]	T. Fukunaga, M. Takahashi, Evolutes and involutes of frontals in the Euclidean plane, Demonstr. Math., 48 (2015), 147–166. https://doi.org/10.1515/dema-2015-0015 doi: 10.1515/dema-2015-0015
[6]	T. Fukunaga, M. Takahashi, Existence and uniqueness for Legendre curves, J. Geom., 104 (2013), 297–307. https://doi.org/10.1007/s00022-013-0162-6 doi: 10.1007/s00022-013-0162-6
[7]	T. Fukunaga, M. Takahashi, Evolutes of fronts in the Euclidean plane, J. Singul., 10 (2014), 92–107. https://doi.org/10.5427/jsing.2014.10f doi: 10.5427/jsing.2014.10f
[8]	Y. Li, D. Pei, Pedal curves of frontals in the Euclidean plane, Math. Method. Appl. Sci., 41 (2018), 1988–1997. https://doi.org/10.1002/mma.4724 doi: 10.1002/mma.4724
[9]	Y. Li, D. Pei, Pedal curves of fronts in the sphere, J. Nonlinear Sci. Appl., 9 (2016), 836–844.
[10]	G. A. ŞEKERCİ, Anti-pedals and primitives of curves in Minkowski plane, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 22 (2014), 92–99. https://doi.org/10.35414/akufemubid.1026512 doi: 10.35414/akufemubid.1026512
[11]	X. Zhao, D. Pei, Pedal curves of the mixed-type curves in the Lorentz-Minkowski plane, Mathematics, 9 (2021), 2852. https://doi.org/10.3390/math9222852 doi: 10.3390/math9222852
[12]	L. Verstraelen, On angles and Pseudo-angles in Minkowskian planes, Mathematics, 6 (2018), 52. https://doi.org/10.3390/math6040052 doi: 10.3390/math6040052
[13]	I. Castro, I. Castro-Infantes, J. Castro-Infantes, Curves in the Lorentz-Minkowski plane with curvature depending on their position, Open Math., 1 (2020), 749–770. https://doi.org/10.1515/math-2020-0043 doi: 10.1515/math-2020-0043
[14]	M. Choi, Y. H. Kim, Classification theorems of ruled surfaces in Minkowski three-space, Mathematics, 12 (2018), 318. https://doi.org/10.3390/math6120318 doi: 10.3390/math6120318
[15]	R. López, Ž. M. Šipuš, L. P. Gajčić, I. Protrka, Involutes of pseudo-null curves in Lorentz-Minkowski 3-space, Mathematics, 9 (2021), 1256. https://doi.org/10.3390/math9111256 doi: 10.3390/math9111256
[16]	S. Wei, Y. Wang, Gauss-bonnet theorems in the lorentzian heisenberg group and the lorentzian group of rigid motions of the Minkowski plane, Symmetry, 13 (2021), 173. https://doi.org/10.3390/sym13020173 doi: 10.3390/sym13020173
[17]	A. A. Abdel-Salam, M. Khalifa Saad, Classification of evolutoids and pedaloids in Minkowski space-time plane, WSEAS Trans. Math., 20 (2021), 97–105. https://doi.org/10.37394/23206.2021.20.10 doi: 10.37394/23206.2021.20.10
[18]	G. Şekerci, S. Izumiya, Evolutoids and pedaloids of Minkowski plane curves, Bull. Malays. Math. Sci. Soc., 44 (2021), 2813–2834. https://doi.org/10.1007/s40840-021-01091-1 doi: 10.1007/s40840-021-01091-1
[19]	Y. Li, Q. Sun, Evolutes of fronts in the Minkowski plane, Math. Med. Appl. Sci., 42 (2018), 1–11. https://doi.org/10.1002/mma.5402 doi: 10.1002/mma.5402
[20]	H. Yu, D. Pei, X. Cui, Evolutes of fronts on Euclidean 2-sphere, J. Nonlinear Sci. Appl., 8 (2015), 678–686.
[21]	S. Izumiya, N. Takeuchi, Primitivoids and inversions of plane curves, Beitr. Algebra Geom., 61 (2019), 317–334. https://doi.org/10.1007/s13366-019-00472-9 doi: 10.1007/s13366-019-00472-9
[22]	S. Izumiya, N. Takeuchi, Evolutoids and pedaloids of plane curves, Note Mat., 39 (2019), 13–23. https://doi.org/10.1285/i15900932v39n2p13 doi: 10.1285/i15900932v39n2p13
[23]	P. J. Giblin, J. P. Warder, Evolving evolutoids, Am. Math. Mon., 121 (2014), 871–889.
[24]	Y. Li, S. Şenyurt, A. Özduran, D. Canlı, The characterizations of parallel q-Equidistant ruled surfaces, Symmetry, 14 (2022), 1879. https://doi.org/10.3390/sym14091879 doi: 10.3390/sym14091879
[25]	Y. Li, F. Mofarreh, R. Abdel-Baky, Timelike circular surfaces and singularities in Minkowski 3-space, Symmetry, 14 (2022), 1914. https://doi.org/10.3390/sym14091914 doi: 10.3390/sym14091914
[26]	Y. Li, N. Alluhaibi, R. Abdel-Baky, One-parameter lorentzian dual spherical movements and invariants of the axodes, Symmetry, 14 (2022), 1930. https://doi.org/10.3390/sym14091930 doi: 10.3390/sym14091930
[27]	Y. Li, K. Eren, K. Ayvacı, S. Ersoy, Simultaneous characterizations of partner ruled surfaces using Flc frame, AIMS Math., 7 (2022), 20213–20229. https://doi.org/10.3934/math.20221106 doi: 10.3934/math.20221106
[28]	Y. Li, S. H. Nazra, R. Abdel-Baky, Singularity properties of timelike sweeping surface in Minkowski 3-space, Symmetry, 14 (2022), 1996. https://doi.org/10.3390/sym14101996 doi: 10.3390/sym14101996
[29]	Y. Li, R. Prasad, A. Haseeb, S. Kumar, S. Kumar, A study of clairaut semi-invariant riemannian maps from cosymplectic manifolds, Axioms, 11 (2022), 503. https://doi.org/10.3390/axioms11100503 doi: 10.3390/axioms11100503
[30]	Y. Li, M. Khatri, J. Singh, S. Chaubey, Improved Chen's inequalities for submanifolds of generalized Sasakian-space-forms, Axioms, 11, (2022), 324. https://doi.org/10.3390/axioms11070324
[31]	Y. Li, A. Uçum, K. İlarslan, Ç. Camcı, A new class of Bertrand curves in Euclidean 4-space, Symmetry, 14 (2022), 1191. https://doi.org/10.3390/sym14061191 doi: 10.3390/sym14061191
[32]	Y. Li, F. Mofarreh, R. Agrawal, A. Ali, Reilly-type inequality for the $\phi$-Laplace operator on semislant submanifolds of Sasakian space forms, J. Inequal. Appl., 1 (2022), 1–17.
[33]	Y. Li, F. Mofarreh, S. Dey, S. Roy, A. Ali, General relativistic space-time with $\eta_1$-Einstein metrics, Mathematics, 10 (2022), 2530.
[34]	Y. Li, A. Haseeb, M. Ali, LP-Kenmotsu manifolds admitting $\eta$-Ricci solitons and spacetime, J. Math., 2022 (2022), 6605127. https://doi.org/10.1155/2022/6605127
[35]	Y. Li, S. Mazlum, S. Senyurt, The Darboux trihedrons of timelike surfaces in the Lorentzian 3-space, Int. J. Geom. Methods Mod. Phys., 2022, 1–35. https://doi.org/10.1142/S0219887823500305
[36]	Y. Li, S. Mondal, S. Dey, A. Bhattacharyya, A. Ali, A study of conformal $\eta$-Einstein solitons on trans-Sasakian 3-manifold, J. Nonlinear Math. Phys., 2022 (2022), 1–27. https://doi.org/10.1007/s44198-022-00088-z doi: 10.1007/s44198-022-00088-z
[37]	Y. Li, K. Eren, K. Ayvacı, S. Ersoy, The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space, AIMS Math., 8 (2023), 2226–2239. https://doi.org/10.3934/math.2023115 doi: 10.3934/math.2023115
[38]	S. Gür, S. Şenyurt, L. Grilli, The Dual expression of parallel equidistant ruled surfaces in Euclidean 3-space, Symmetry, 14 (2022), 1062. https://doi.org/10.3390/sym14051062 doi: 10.3390/sym14051062
[39]	S. Şenyurt, S. Gür, Spacelike surface geometry, Int. J. Geom. Methods Mod. Phys., 14 (2022), 1750118. https://doi.org/10.1142/S0219887817501183
[40]	J. R. Sharma, S. Kumar, L. Jäntschi, On a class of optimal fourth order multiple root solvers without using derivatives, Symmetry, 11 (2019), 1452. https://doi.org/10.3390/sym11121452 doi: 10.3390/sym11121452
[41]	M. A. Tomescu, L. Jäntschi, D. I. Rotaru, Figures of graph partitioning by counting, sequence and layer matrices, Mathematics, 9 (2021), 1419. https://doi.org/10.3390/math9121419 doi: 10.3390/math9121419
[42]	D. M. Joita, M. A. Tomescu, D. Bàlint, L. Jäntschi, An application of the eigenproblem for biochemical similarity, Symmetry, 13 (2021), 1849. https://doi.org/10.3390/sym13101849 doi: 10.3390/sym13101849
[43]	L. Jäntschi, Introducing structural symmetry and asymmetry implications in development of recent pharmacy and medicine, Symmetry, 14 (2022), 1674. https://doi.org/10.3390/sym14081674 doi: 10.3390/sym14081674
[44]	L. Jäntschi, Binomial distributed data confidence interval calculation: formulas, algorithms and examples, Symmetry, 14 (2022), 1104. https://doi.org/10.3390/sym14061104 doi: 10.3390/sym14061104
[45]	L. Jäntschi, Formulas, Algorithms and examples for binomial distributed data confidence interval calculation: excess risk, relative risk and odds ratio, Mathematics, 9 (2021), 2506. https://doi.org/10.3390/math9192506 doi: 10.3390/math9192506
[46]	B. Donatella, L. Jäntschi, Comparison of molecular geometry optimization methods based on molecular descriptors, Mathematics, 9 (2021), 2855. https://doi.org/10.3390/math9222855 doi: 10.3390/math9222855
[47]	T. Mihaela, L. Jäntschi, R. Doina, Figures of graph partitioning by counting, sequence and layer matrices, Mathematics, 9 (2021), 1419. https://doi.org/10.3390/math9121419 doi: 10.3390/math9121419
[48]	S. Kumar, D. Kumar, J. R. Sharma, L. Jäntschi, A family of derivative free optimal fourth order methods for computing multiple roots, Symmetry, 12 (2020), 1969. https://doi.org/10.3390/sym12121969 doi: 10.3390/sym12121969
[49]	K. Deepak, R. Janak, L. Jäntschi, A novel family of efficient weighted-newton multiple root iterations, Symmetry, 12 (2020), 1494. https://doi.org/10.3390/sym12091494 doi: 10.3390/sym12091494
[50]	R. Janak, K. Sunil, L. Jäntschi, On derivative free multiple-root finders with optimal fourth order convergence, Mathematics, 8 (2020), 1091. https://doi.org/10.3390/math8071091 doi: 10.3390/math8071091
[51]	L. Jäntschi, Detecting extreme values with order statistics in samples from continuous distributions, Mathematics, 8 (2020), 216. https://doi.org/10.3390/math8020216 doi: 10.3390/math8020216
[52]	K. Deepak, R. Janak, L. Jäntschi, Convergence analysis and complex geometry of an efficient derivative-free iterative method, Mathematics, 7 (2019), 919.
[53]	L. Jäntschi, S. D. Bolboacă, Conformational study of $C_24$ cyclic polyyne clusters, Int. J. Quantum Chem., 118 (2018), 25614. https://doi.org/10.1002/qua.25614 doi: 10.1002/qua.25614

Reader Comments

Your name:*

Email:*
© 2023 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)