Research Article
Year 2023,
Volume: 72 Issue: 1, 137 - 158, 30.03.2023
Abstract
Let $E_{2}$ be the $2$-dimensional Euclidean space and $T$ be a set such that it has at least two elements. A mapping $\alpha : T\rightarrow E_{2}$ will be called a $T$-figure in $E_{2}$. Let $O(2, R)$ be the group of all orthogonal transformations of $E_{2}$. Put $SO(2, R)=\left\{ g\in O(2, R)|detg=1\right\}$, $MO(2, R)=\left\{F:E_{2}\rightarrow E_{2}\mid Fx=gx+b, g\in O(2,R), b\in E_{2}\right\}$,
$MSO(2, R)= \left\{F\in MO(2, R)|detg=1\right\}$.
The present paper is devoted to solutions of problems of $G$-equivalence of $T$-figures in $E_{2}$ for groups $G=O(2, R), SO(2, R)$, $MO(2, R)$, $MSO(2, R)$. Complete systems of $G$-invariants of $T$-figures in $E_{2}$ for these groups are obtained. Complete systems of relations between elements of the obtained complete systems of $G$-invariants are given for these groups.
Keywords
Supporting Institution
The Ministry of Innovative Development of the Republic of Uzbekistan and The Scientific and Technological Research Council of Turkey
Project Number
UT-OT-2020-2 and 119N643
Thanks
This work is supported by The Ministry of Innovative Development of the Republic of Uzbekistan (MID Uzbekistan) under Grant Number UT-OT-2020-2 and The Scientific and Technological Research Council of Turkey (T\"{U}B{\.I}TAK) under Grant Number 119N643.
References
- Aripov, R., Khadjiev, D., The complete system of global differential and integral invariants of a curve in Euclidean geometry, Izvestiya Vuzov, Ser. Mathematics, 542 (2007), 114, http://dx.doi.org/10.3103/S1066369X07070018.
- Berger, M., Geometry I, Springer-Verlag, Berlin, Heidelberg, 1987.
- Dieudonne, J. A. ,Carrell, J.B. , Invariant Theory, Academic Press, New-York, London, 1971.
- Greub, W. H. , Linear Algebra, Springer-Verlag, New York Inc., 1967.
- İncesu, M., Gürsoy, O., LS(2)-equivalence conditions of control points and application to planar Bezier curves, New Trends in Mathematical Sciences, 5(3) (2017), 70-84., http://dx.doi.org/10.20852/ntmsci.2017.186.
- Höfer, R., m-Point invariants of real geometries, Beitrage Algebra Geom., 40 (1999), 261-266.
- Ören, İ., Khadjiev, D., Pekşen, Ö., Identifications of paths and curves under the plane similarity transformations and their applications to mechanics, Journal of Geometry and Physics, 151 (2020), 1-17, 103619, https://doi.org/10.1016/j.geomphys.2020.103619.
- Khadjiev, D.,Application of the Invariant Theory to the Differential Geometry of Curves, Fan Publisher, Tashkent, 1988, [in Russian].
- Khadjiev, D., Pekşen, Ö., The complete system of global integral and differential invariants for equi-affine curves, Differential Geometry and its Applications, 20 (2004), 167-175, https://doi.org/10.1016/j.difgeo.2003.10.005.
- Khadjiev, D., Complete systems of differential invariants of vector fields in a Euclidean space, Turkish Journal of Mathematics, 34(2010), 543-559,https://doi.org/10.3906/mat-0809-10
- Khadjiev, D., On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space, Journal of Nonlinear Mathematical Physics, 17(1) (2010), 49-70, https://doi.org/10.1142/S1402925110000799.
- Khadjiev, D., Ören, İ., Pekşen, Ö., Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry, Turkish Journal of Mathematics, 37 (2013), 80-94, https://doi.org/10.3906/mat-1104-41.
- Khadjiev, D., Göksal, Y., Applications of hyperbolic numbers to the invariant theory in two-dimensional pseudo-Euclidean space, Adv. Appl. Clifford Algebras, 26 (2016) 645-668, https://doi.org/10.1007/s00006-015-0627-9
- Khadjiev, D., Ören, İ., Pekşen, Ö., 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, 15(6) (2018), 1850092, https://doi.org/10.1142/S0219887818500925.
- Khadjiev, D., Projective invariants of m-tuples in the one-dimensional projective space, Uzbek Mathematical Journal, 1 (2019) 61-73.
- Khadjiev, D., Ayupov, Sh., Beshimov, G., Complete systems of invariant of m-tuples for fundamental groups of the two-dimensional Euclidian space, Uzbek Mathematical Journal, 1 (2020), 57-84.
- Khadjiev, D., Bekbaev, U., Aripov, R., On equivalence of vector-valued maps, arXiv:2005.08707v1 [math GM] 13 May 2020.
- Khadjiev, D., Ayupov, Sh., Beshimov, G., Affine invariants of a parametric figure for fundamental groups of n-dimensional affine space, Uzbek Mathematical Journal, 65(4)(2021), 27-47.
- Mundy, J. L. , Zisserman, A., Forsyth , D.(Eds.), Applications of Invariance in Computer Vision, Springer-Verlag, Berlin, Heidelberg, New York, 1994.
- Mumford, D., Fogarty, J., Geometric Invariant Theory, Springer-Verlag, Berlin, Heidelberg, 1994.
- O’Rourke, J.,Computational Geometry in C , Cambridge University Press, 1997.
- Ören, İ., Equivalence conditions of two Bezier curves in the Euclidean geometry, Iranian Journal of Science and Technology, Transactions A: Science, 42(3) (2018), 1563-1577., http://dx.doi.org/10.1007/s40995-016-0129-1.
- Ören, İ., Invariants of m-vectors in Lorentzian geometry, International Electronic Journal of Geometry, 9(1)(2016), 38-44.
- Pekşen, Ö., Khadjiev, D., Invariants of curves in centro-affine geometry, J. Math. Kyoto Univ., 44(3)(2004), 603-613.
- Pekşen, Ö., Khadjiev, D., On invariants of null curves in the pseudo-Euclidean geometry, Differential Geometry and its Applications 29 (2011), 183-187, https://doi.org/10.1016/j.difgeo.2011.04.024.
- Pekşen, Ö., Khadjiev, D., Ören, İ., Invariant parametrizations and complete systems of global invariants of curves in the pseudo-euclidean geometry, Turkish Journal of Mathematics , 36 (2012), 147-160, http://dx.doi.org/10.3906/mat-0911-145.
- Reiss, T. H. ,Recognizing Planar Objects Using Invariant Image Features, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
- Sağıroğlu, Y., Khadjiev, D., Gözütok, U., Differential invariants of non-degenerate surfaces, Applications and Applied Mathematics, Special issue, 3 ( 2019), 35-57.
- Sibirskii, K. S., Introduction to the Algebraic Invariants of Differential Equations, Manchester University Press, New York, 1988.
- Springer, T. A., Invariant Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
- Weyl, H., The Classical Groups: Their Invariants and Representations, Princeton Univ. Press, Princeton, New Jersey, 1946.
Year 2023,
Volume: 72 Issue: 1, 137 - 158, 30.03.2023
Abstract
Project Number
UT-OT-2020-2 and 119N643
References
- Aripov, R., Khadjiev, D., The complete system of global differential and integral invariants of a curve in Euclidean geometry, Izvestiya Vuzov, Ser. Mathematics, 542 (2007), 114, http://dx.doi.org/10.3103/S1066369X07070018.
- Berger, M., Geometry I, Springer-Verlag, Berlin, Heidelberg, 1987.
- Dieudonne, J. A. ,Carrell, J.B. , Invariant Theory, Academic Press, New-York, London, 1971.
- Greub, W. H. , Linear Algebra, Springer-Verlag, New York Inc., 1967.
- İncesu, M., Gürsoy, O., LS(2)-equivalence conditions of control points and application to planar Bezier curves, New Trends in Mathematical Sciences, 5(3) (2017), 70-84., http://dx.doi.org/10.20852/ntmsci.2017.186.
- Höfer, R., m-Point invariants of real geometries, Beitrage Algebra Geom., 40 (1999), 261-266.
- Ören, İ., Khadjiev, D., Pekşen, Ö., Identifications of paths and curves under the plane similarity transformations and their applications to mechanics, Journal of Geometry and Physics, 151 (2020), 1-17, 103619, https://doi.org/10.1016/j.geomphys.2020.103619.
- Khadjiev, D.,Application of the Invariant Theory to the Differential Geometry of Curves, Fan Publisher, Tashkent, 1988, [in Russian].
- Khadjiev, D., Pekşen, Ö., The complete system of global integral and differential invariants for equi-affine curves, Differential Geometry and its Applications, 20 (2004), 167-175, https://doi.org/10.1016/j.difgeo.2003.10.005.
- Khadjiev, D., Complete systems of differential invariants of vector fields in a Euclidean space, Turkish Journal of Mathematics, 34(2010), 543-559,https://doi.org/10.3906/mat-0809-10
- Khadjiev, D., On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space, Journal of Nonlinear Mathematical Physics, 17(1) (2010), 49-70, https://doi.org/10.1142/S1402925110000799.
- Khadjiev, D., Ören, İ., Pekşen, Ö., Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry, Turkish Journal of Mathematics, 37 (2013), 80-94, https://doi.org/10.3906/mat-1104-41.
- Khadjiev, D., Göksal, Y., Applications of hyperbolic numbers to the invariant theory in two-dimensional pseudo-Euclidean space, Adv. Appl. Clifford Algebras, 26 (2016) 645-668, https://doi.org/10.1007/s00006-015-0627-9
- Khadjiev, D., Ören, İ., Pekşen, Ö., 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, 15(6) (2018), 1850092, https://doi.org/10.1142/S0219887818500925.
- Khadjiev, D., Projective invariants of m-tuples in the one-dimensional projective space, Uzbek Mathematical Journal, 1 (2019) 61-73.
- Khadjiev, D., Ayupov, Sh., Beshimov, G., Complete systems of invariant of m-tuples for fundamental groups of the two-dimensional Euclidian space, Uzbek Mathematical Journal, 1 (2020), 57-84.
- Khadjiev, D., Bekbaev, U., Aripov, R., On equivalence of vector-valued maps, arXiv:2005.08707v1 [math GM] 13 May 2020.
- Khadjiev, D., Ayupov, Sh., Beshimov, G., Affine invariants of a parametric figure for fundamental groups of n-dimensional affine space, Uzbek Mathematical Journal, 65(4)(2021), 27-47.
- Mundy, J. L. , Zisserman, A., Forsyth , D.(Eds.), Applications of Invariance in Computer Vision, Springer-Verlag, Berlin, Heidelberg, New York, 1994.
- Mumford, D., Fogarty, J., Geometric Invariant Theory, Springer-Verlag, Berlin, Heidelberg, 1994.
- O’Rourke, J.,Computational Geometry in C , Cambridge University Press, 1997.
- Ören, İ., Equivalence conditions of two Bezier curves in the Euclidean geometry, Iranian Journal of Science and Technology, Transactions A: Science, 42(3) (2018), 1563-1577., http://dx.doi.org/10.1007/s40995-016-0129-1.
- Ören, İ., Invariants of m-vectors in Lorentzian geometry, International Electronic Journal of Geometry, 9(1)(2016), 38-44.
- Pekşen, Ö., Khadjiev, D., Invariants of curves in centro-affine geometry, J. Math. Kyoto Univ., 44(3)(2004), 603-613.
- Pekşen, Ö., Khadjiev, D., On invariants of null curves in the pseudo-Euclidean geometry, Differential Geometry and its Applications 29 (2011), 183-187, https://doi.org/10.1016/j.difgeo.2011.04.024.
- Pekşen, Ö., Khadjiev, D., Ören, İ., Invariant parametrizations and complete systems of global invariants of curves in the pseudo-euclidean geometry, Turkish Journal of Mathematics , 36 (2012), 147-160, http://dx.doi.org/10.3906/mat-0911-145.
- Reiss, T. H. ,Recognizing Planar Objects Using Invariant Image Features, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
- Sağıroğlu, Y., Khadjiev, D., Gözütok, U., Differential invariants of non-degenerate surfaces, Applications and Applied Mathematics, Special issue, 3 ( 2019), 35-57.
- Sibirskii, K. S., Introduction to the Algebraic Invariants of Differential Equations, Manchester University Press, New York, 1988.
- Springer, T. A., Invariant Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
- Weyl, H., The Classical Groups: Their Invariants and Representations, Princeton Univ. Press, Princeton, New Jersey, 1946.
There are 31 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Articles |
| Authors | |
| Project Number | UT-OT-2020-2 and 119N643 |
| Publication Date | March 30, 2023 |
| Submission Date | October 3, 2021 |
| Acceptance Date | September 19, 2022 |
| Published in Issue | Year 2023 Volume: 72 Issue: 1 |
Cite
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.
