Global Invariants of Objects in two-Dimensional Minkowski Space

Ren, DRS; Khadjiev, Djavvat; Peken, Mer; Beshimov, Gayrat

doi:10.1080/1726037X.2022.2120271

Open Access Research Article

Global Invariants of Objects in two-Dimensional Minkowski Space

^*DRS RenCorresponding authororen@ktu.edu.trDepartment of MathematicsFaculty of ScienceKaradenz Technical UniversityTrabzon, 61080, TrkyeView full profile → , Djavvat Khadjievkhdjavvat@gmail.comV.I. Romanovsky Institute of Mathemetics, Uzbekistan Academy of SciencesNational University of Uzbekistan Named After Mirzo UlugbekTashkent, UzbekistanView full profile → , Mer Pekenpeksen@ktu.edu.trDepartment of MathematicsFaculty of ScienceKaradenz Technical UniversityTrabzon, 61080, TrkyeView full profile → , Gayrat Beshimovgayratbeshimov@gmail.comNational University of Uzbekistan Named After Mirzo UlugbekTashkent, UzbekistanView full profile →

* Corresponding author · click or hover a name for details

pp. 249–273Vol. 20Issue 2July 2022DOI: 10.1080/1726037X.2022.2120271XML

Received:: 11 Apr 2022
Accepted:: 10 Aug 2022
Published Online:: 13 Nov 2022
Article type:: Research Article
Language:: EN
Article no.:: 1726037X.2022.2120271
Pages:: 249–273

Abstract

In this paper, ”the concept of T-object” is defined in two dimensional Minkowski space. By this concept, the complete systems of invariants of m-point, paths and vector fields are investigated as a single theory. Using this approach, we obtain the complete systems of invariants of T-object and the equivalence conditions of two T-objects are given in terms of their global invariants.

Keywords

Minkowski spaceInvariantObject

Subject Classifications

14L2415A6315A72

References

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R.Hőfer, m-point invariants of real geometries , Beitrage Algebra Geom ., Vol. 40 , 261 - 266 ( 1999 ).
E.Stasiak, Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1 , Publ.Math. Debrecen , Vol. 57 , Number 1-2 , 55 - 69 ( 2000 ).
Őren, On invariants of m-vectors in Lorentzian Geometry , International Electronic Journal of Geometry , Vol. 9 , Number 1 , 38 - 44 ( 2016 ). doi: 10.36890/iejg.591885
Őren, Equivalence conditions of two Bézier Curves in the Euclidean geometry, Iranian Journal of Science and Technology , Transactions A: Science (ISTT ), Vol. 42 , Number 3 , 1563 - 1577 ( 2018 ).
Őren, The equivalence problem of vectors in the two-dimensional Minkowski spacetime and its applications to Bézier curves , J. math. Comput.Sci ., Vol. 6 , Number 1 , 1 - 21 ( 2016 ).
Őren, M.ncesu, Recognition of complex polynomial Bezier curves under similarity transformations , Commun.Fac.Sci.Univ.Ank.Ser . A1 Math. Stat., Vol. 69 , Number 2 , 1377 - 1388 ( 2020 ).
D.Khadjiev, Y.Gőksal, Applications of hyperbolic numbers to the invariant theory in twodimensional pseudo-Euclidean space , Adv. Appl. Clifford Algebras , Vol. 26 , 645 - 668 ( 2016 ). doi: 10.1007/s00006-015-0627-9
D.Khadjiev, S.Ayupov, G.Beshimov, Complete systems of invariant of m-tuples for fundamental groups of the two-dimensional Euclidean space , Uzbek Mathematical Journal , Vol. 1 , 57 - 84 ( 2020 ). doi: 10.29229/uzmj.2020-1-6
D.Khadjiev, U.Bekbaev, R.Aripov, On equivalence of vector value maps, https://arxiv.org/abs/2005.08707 , (2020).
D.Khadjiev, Projective invariants of m-tuples in the one-dimensional projective space , Uzbek Mathematical Journal , Vol. 1 , 60 - 72 ( 2019 ).
H. E.Bez, Generalized invariant-geometry conditions for the rational Bzier paths , International Journal of Computer Mathematics , Vol. 87 , Number 12 , 2722 - 2732 ( 2010 ). doi: 10.1080/00207160902745333
H. A.oban, Complete system of invariants of vectors for isometry group in n-dimensional unitary space , Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. , Vol. 68 , Number 1 , 362 - 370 ( 2019 ).
R.Aripov, D.Khadjiev, The complete system of global differential and integral invariants of a curve in Euclidean geometry , Russian Mathematics , Vol. 51 , 1 - 14 ( 2007 ). doi: 10.3103/S1066369X07070018
D.Khadjiev, Őren, Ő. Peken, Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry , Turkish Journal of Mathematics , Vol. 37 , 80 - 94 ( 2013 ).
Ő.Peken, D.Khadjiev, On invariants of curves in centro-affine geometry , J. Math. Kyoto Univ.(JMKYAZ ) Vol. 44 , Number 3 , 603 - 613 ( 2004 ).
Ő.Peken, D.Khadjiev, Őren, Invariant parametrizations and complete systems of global invariants of curves in the pseudo-euclidean geometry , Turkish Journal of Mathematics , Vol. 36 , 147 - 160 ( 2012 ).
Őren, D.Khadjiev, Recognition of paths and curves in the 2-dimensional Euclidean geometry , International Electronic Journal of Geometry , Vol. 13 , Number 2 , 116 - 1348 ( 2020 ).
Őren, D.Khadjiev, Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space , An. St. Univ. Ovidius Constanta , Vol. 27 , Number 2 , 37 - 65 ( 2019 ).
Őren, D.Khadjiev, Recognition of plane paths and plane curves under linear pseudo-similarity transformations , Journal of Geometry , Vol. 111 , Number 3 , 1 - 24 ( 2020 ). doi: 10.1007/s00022-020-00551-6
Őren, D.Khadjiev, Ő.Peken, Identifications of paths and curves under the plane similarity transformations and their applications to mechanics , Journal of Geometry and Physics , Vol. 151 , 103619 , 1 - 17 ( 2020 ).
D.Khadjiev, Őren, Ő.Peken, Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space , International Journal of Geometric Methods in Modern Physics , Vol. 15 , Number 6 , 1850092 ( 2018 ). doi: 10.1142/S0219887818500925
K.K.Muminov, Equivalence of Curves with Respect to the Action of the Symplectic Group , Russian Mathematics (Iz. VUZ ),Vol. 53 , Number 6 , 2428 ( 2009 ).
V.Chilin, K.Muminov, Differential invariants of curves in Galilean geometry , AIP Conference Proceedings , 23 - 65 ( 2021 ).
H.Weyl, The Classical Groups: Their Invariants and Representations , Princeton Univ. Press , Princeton, New Jersey, 1946 .
C.Zhang, S.Du, J.Liu, Y.Li, J.Xue, Y.Liu, Robust iterative closest point algorithm with bounded rotation angle for 2D registration , Neurocomputing ,Vol. 195 , 172 - 180 ( 2016 ). doi: 10.1016/j.neucom.2015.06.107
P.J.Besl, Neil D.McKay, A method for registration of 3-D shapes , IEEE Transactions on Pattern Analysis and Machine Intelligence , Vol. 14 , Number 2 , 239 - 256 ( 1992 ). doi: 10.1109/34.121791
S.Du, N.Zheng, S.Ying, J.Liu, Affine iterative closest point algorithm for point set registration , Pattern Recognition Letters , Vol. 31 , 791 - 799 ( 2010 ). doi: 10.1016/j.patrec.2010.01.020
H.Chen, X.Zhang, S.Du, Z.Wu, and N.Zheng, A Correntropy-based Affine Iterative Closest Point Algorithm for Robust Point Set Registration , IEEE/CAA J. Autom. Sin . Vol. 6 , Number 4 , 981 - 991 ( 2019 ). doi: 10.1109/JAS.2019.1911579
I.Weiss, Geometric invariants and object recognition , J. Math. Imaging Vision , Vol. 10 , Number 3 , 201 - 231 ( 1993 ).
J. G.Alcázar, Hermoso, M.Carlos, G.Muntingh, Detecting similarity of rational plane curves , J. Comput. Appl. Math . , Vol. 269 , 113 ( 2014 ). doi: 10.1016/j.cam.2014.03.013
J. G.Alcázar, C.Hermoso, G.Muntingh, Similarity detection of rational space curves , J. Symbolic Comput . , Vol. 85 , 424 ( 2018 ). doi: 10.1016/j.jsc.2017.07.001
J. G.Alcázar, GemaM.Diaz-Toca, C.Hermoso On the Problem of Detecting When Two Implicit Plane Algebraic Curves Are Similar , Internat. J. Algebra Comput , Vol. 29 , Number 5 , 775 - 793 ( 2019 ). doi: 10.1142/S0218196719500279
M.Hauer, B.Jüttler, Detecting affine equivalences of planar rational curves, EuroCG 2016, Lugano, Switzerland, March 30-April 1, 2016.
J.Sánchez Reyes, Detecting symmetries in polynomial Bézier curves , J. Comput. Appl. Math . , Vol. 288 , 274283 ( 2015 ). doi: 10.1016/j.cam.2015.04.025
J.Mozo-Fernandez and C.Munuera, Recognition of Polynomial Plane Curves Under Affine Transformations , AAECC 13 , 121 - 136 ( 2002 ). doi: 10.1007/s002000200095
M.İncesu, O.Gürsoy, LS(2)-Equivalence conditions of control points and application to planar Bézier curves , New Trends in Mathematical Science , Vol. 3 , 70 - 84 ( 2017 ). doi: 10.20852/ntmsci.2017.186
[38] K. S.Sibirskii, Introduction to the Algebraic Invariants of Differential Equations , Manchester University Press , New York, 1988 .
D.Khadjiev, On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space , J. Nonlinear Math. Phys . , Vol. 17 , Number 1 , 49 - 70 ( 2010 ).

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[1] [1] M.Berger, Geometry I , Springer-Verlag , Berlin, Heidelberg, 1987 .

[2] R.Hőfer, m-point invariants of real geometries , Beitrage Algebra Geom ., Vol. 40 , 261 - 266 ( 1999 ).

[3] E.Stasiak, Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1 , Publ.Math. Debrecen , Vol. 57 , Number 1-2 , 55 - 69 ( 2000 ).

[4] Őren, On invariants of m-vectors in Lorentzian Geometry , International Electronic Journal of Geometry , Vol. 9 , Number 1 , 38 - 44 ( 2016 ). doi: 10.36890/iejg.591885

[5] Őren, Equivalence conditions of two Bézier Curves in the Euclidean geometry, Iranian Journal of Science and Technology , Transactions A: Science (ISTT ), Vol. 42 , Number 3 , 1563 - 1577 ( 2018 ).

[6] Őren, The equivalence problem of vectors in the two-dimensional Minkowski spacetime and its applications to Bézier curves , J. math. Comput.Sci ., Vol. 6 , Number 1 , 1 - 21 ( 2016 ).

[7] Őren, M.ncesu, Recognition of complex polynomial Bezier curves under similarity transformations , Commun.Fac.Sci.Univ.Ank.Ser . A1 Math. Stat., Vol. 69 , Number 2 , 1377 - 1388 ( 2020 ).

[8] D.Khadjiev, Y.Gőksal, Applications of hyperbolic numbers to the invariant theory in twodimensional pseudo-Euclidean space , Adv. Appl. Clifford Algebras , Vol. 26 , 645 - 668 ( 2016 ). doi: 10.1007/s00006-015-0627-9

[9] D.Khadjiev, S.Ayupov, G.Beshimov, Complete systems of invariant of m-tuples for fundamental groups of the two-dimensional Euclidean space , Uzbek Mathematical Journal , Vol. 1 , 57 - 84 ( 2020 ). doi: 10.29229/uzmj.2020-1-6

[10] D.Khadjiev, U.Bekbaev, R.Aripov, On equivalence of vector value maps, https://arxiv.org/abs/2005.08707 , (2020).

[11] D.Khadjiev, Projective invariants of m-tuples in the one-dimensional projective space , Uzbek Mathematical Journal , Vol. 1 , 60 - 72 ( 2019 ).

[12] H. E.Bez, Generalized invariant-geometry conditions for the rational Bzier paths , International Journal of Computer Mathematics , Vol. 87 , Number 12 , 2722 - 2732 ( 2010 ). doi: 10.1080/00207160902745333

[13] H. A.oban, Complete system of invariants of vectors for isometry group in n-dimensional unitary space , Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. , Vol. 68 , Number 1 , 362 - 370 ( 2019 ).

[14] R.Aripov, D.Khadjiev, The complete system of global differential and integral invariants of a curve in Euclidean geometry , Russian Mathematics , Vol. 51 , 1 - 14 ( 2007 ). doi: 10.3103/S1066369X07070018

[15] D.Khadjiev, Őren, Ő. Peken, Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry , Turkish Journal of Mathematics , Vol. 37 , 80 - 94 ( 2013 ).

[16] Ő.Peken, D.Khadjiev, On invariants of curves in centro-affine geometry , J. Math. Kyoto Univ.(JMKYAZ ) Vol. 44 , Number 3 , 603 - 613 ( 2004 ).

[17] Ő.Peken, D.Khadjiev, Őren, Invariant parametrizations and complete systems of global invariants of curves in the pseudo-euclidean geometry , Turkish Journal of Mathematics , Vol. 36 , 147 - 160 ( 2012 ).

[18] Őren, D.Khadjiev, Recognition of paths and curves in the 2-dimensional Euclidean geometry , International Electronic Journal of Geometry , Vol. 13 , Number 2 , 116 - 1348 ( 2020 ).

[19] Őren, D.Khadjiev, Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space , An. St. Univ. Ovidius Constanta , Vol. 27 , Number 2 , 37 - 65 ( 2019 ).

[20] Őren, D.Khadjiev, Recognition of plane paths and plane curves under linear pseudo-similarity transformations , Journal of Geometry , Vol. 111 , Number 3 , 1 - 24 ( 2020 ). doi: 10.1007/s00022-020-00551-6

[21] Őren, D.Khadjiev, Ő.Peken, Identifications of paths and curves under the plane similarity transformations and their applications to mechanics , Journal of Geometry and Physics , Vol. 151 , 103619 , 1 - 17 ( 2020 ).

[22] D.Khadjiev, Őren, Ő.Peken, Global invariants of paths and curves for the group of all linear similarities in the two-dimensional Euclidean space , International Journal of Geometric Methods in Modern Physics , Vol. 15 , Number 6 , 1850092 ( 2018 ). doi: 10.1142/S0219887818500925

[23] K.K.Muminov, Equivalence of Curves with Respect to the Action of the Symplectic Group , Russian Mathematics (Iz. VUZ ),Vol. 53 , Number 6 , 2428 ( 2009 ).

[24] V.Chilin, K.Muminov, Differential invariants of curves in Galilean geometry , AIP Conference Proceedings , 23 - 65 ( 2021 ).

[25] H.Weyl, The Classical Groups: Their Invariants and Representations , Princeton Univ. Press , Princeton, New Jersey, 1946 .

[26] C.Zhang, S.Du, J.Liu, Y.Li, J.Xue, Y.Liu, Robust iterative closest point algorithm with bounded rotation angle for 2D registration , Neurocomputing ,Vol. 195 , 172 - 180 ( 2016 ). doi: 10.1016/j.neucom.2015.06.107

[27] P.J.Besl, Neil D.McKay, A method for registration of 3-D shapes , IEEE Transactions on Pattern Analysis and Machine Intelligence , Vol. 14 , Number 2 , 239 - 256 ( 1992 ). doi: 10.1109/34.121791

[28] S.Du, N.Zheng, S.Ying, J.Liu, Affine iterative closest point algorithm for point set registration , Pattern Recognition Letters , Vol. 31 , 791 - 799 ( 2010 ). doi: 10.1016/j.patrec.2010.01.020

[29] H.Chen, X.Zhang, S.Du, Z.Wu, and N.Zheng, A Correntropy-based Affine Iterative Closest Point Algorithm for Robust Point Set Registration , IEEE/CAA J. Autom. Sin . Vol. 6 , Number 4 , 981 - 991 ( 2019 ). doi: 10.1109/JAS.2019.1911579

[30] I.Weiss, Geometric invariants and object recognition , J. Math. Imaging Vision , Vol. 10 , Number 3 , 201 - 231 ( 1993 ).

[31] J. G.Alcázar, Hermoso, M.Carlos, G.Muntingh, Detecting similarity of rational plane curves , J. Comput. Appl. Math . , Vol. 269 , 113 ( 2014 ). doi: 10.1016/j.cam.2014.03.013

[32] J. G.Alcázar, C.Hermoso, G.Muntingh, Similarity detection of rational space curves , J. Symbolic Comput . , Vol. 85 , 424 ( 2018 ). doi: 10.1016/j.jsc.2017.07.001

[33] J. G.Alcázar, GemaM.Diaz-Toca, C.Hermoso On the Problem of Detecting When Two Implicit Plane Algebraic Curves Are Similar , Internat. J. Algebra Comput , Vol. 29 , Number 5 , 775 - 793 ( 2019 ). doi: 10.1142/S0218196719500279

[34] M.Hauer, B.Jüttler, Detecting affine equivalences of planar rational curves, EuroCG 2016, Lugano, Switzerland, March 30-April 1, 2016.

[35] J.Sánchez Reyes, Detecting symmetries in polynomial Bézier curves , J. Comput. Appl. Math . , Vol. 288 , 274283 ( 2015 ). doi: 10.1016/j.cam.2015.04.025

[36] J.Mozo-Fernandez and C.Munuera, Recognition of Polynomial Plane Curves Under Affine Transformations , AAECC 13 , 121 - 136 ( 2002 ). doi: 10.1007/s002000200095

[37] M.İncesu, O.Gürsoy, LS(2)-Equivalence conditions of control points and application to planar Bézier curves , New Trends in Mathematical Science , Vol. 3 , 70 - 84 ( 2017 ). doi: 10.20852/ntmsci.2017.186

[38] [38] K. S.Sibirskii, Introduction to the Algebraic Invariants of Differential Equations , Manchester University Press , New York, 1988 .

[39] D.Khadjiev, On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space , J. Nonlinear Math. Phys . , Vol. 17 , Number 1 , 49 - 70 ( 2010 ).

Journal of Dynamical Systems and Geometric Theories

WoS JIF 2026 : 0.5 (Q3)

Global Invariants of Objects in two-Dimensional Minkowski Space

Abstract

Keywords

Subject Classifications

References