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Year 2020, Volume: 13 Issue: 2, 116 - 134, 15.10.2020

İdris Ören Djavvat Khadjiev

https://doi.org/10.36890/iejg.725297

Abstract

References

  • [1] Aminov, Yu.: Differential Geometry and Topology of Curves. CRC Press, New York, 2000.
  • [2] Aripov, R. G., Khadjiev (Khadzhiev) D.: The complete system of global differential and integral invariants of a curve in Euclidean geometry. Russian Mathematics (Iz VUZ), 51, No. 7, 1-14 (2007).
  • [3] Berger, M.: Geometry I. Springer-Verlag, Berlin Heidelberg, 1987.
  • [4] Gibson, C. G.: Elementary Geometry of Differentiable Curves. An Undergraduate Introduction, Cambridge University Press, 2001.
  • [5] Gray,A., Abbena, E. and Salamon,S.: Modern Differential Geometry of Curves and surfaces with Mathematica. Third edition. Studies in Advanced Mathematics. Chapman and Hall/CRC, Boca Raton, FL, 2006.
  • [6] Guggenheimer, H. W.: Differential Geometry. Dower Publicatiýns, INC., New York, 1977.
  • [7] Khadjiev, D.: Application of the Invariant Theory to the Differential Geometry of Curves. "Fan" Publisher, Tashkent, 1988. [in Russian].
  • [8] Khadjiev, D.: Complete systems of differential invariants of vector fields in a Euclidean space. Turkish J. Math. 34, 543-560 (2010).
  • [9] Khadjiev, D.: On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space. Journal of Nonlinear Mathematical Physics, 17, Supp 01, 49-70 (2010).
  • [10] Khadjiev D., Pekşen Ö.: The complete system of global differential and integral invariants of equiaffine curves. Diff. Geom. And Appl. 20, 168-175 (2004).
  • [11] Khadjiev, D., Ören, İ., Pekşen, Ö.: Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry. Turkish J. Math. 37 , 80-94 (2013).
  • [12] Khadjiev, D., Ören, İ., Pekşen, Ö.: Global invariants of path and curves for the group of all linear similarities in the two-dimensional Euclidean space. Int.J. Geo.Modern Phys,15(6), 1-28 (2018).
  • [13] Mundy, J.L., Zisserman, A.: Geometric Invariance in Computer Vision, Artificial Intelligence. MIT Press, Cambridge, 1992.
  • [14] Mundy, J.L., Zisserman, A., Forsyth, D.: Application of Invariance in Computer Vision. Lecture Notes in Computer Science, Springer, Berlin, 1992.
  • [15] Khadjiev, D., Ören, İ.: Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space. An. St. Univ. Ovidius Constanta, 27(2), 37-65 (2019).
  • [16] Montel, S., Ros, A.: Curves and Surfaces. American Mathematical Society, 2005.
  • [17] Pekşen, Ö., Khadjiev, D.: On invariants of null curves in the pseudo-Euclidean geometry. Diff. Geom. and Appl. 29, 183-187 (2011).
  • [18] Pekşen, Ö., Khadjiev, D., Ören, İ.: Invariant parametrizations and complete systems of global invariants of curves in the pseudo-Euclidean geometry. Turkish J. Math. 36, 147-160 (2012).
  • [19] Spivak, M.: Comprehensive Introduction to Differential Geometry. Vol. 2 Publish Or Perish, INC., Houston, Texas, 1999.
  • [20] Toponogov, V. A.: Differential Geometry of Curves and Surfaces. Birkhauser, Boston, 2006.

Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry

Year 2020, Volume: 13 Issue: 2, 116 - 134, 15.10.2020

İdris Ören Djavvat Khadjiev

https://doi.org/10.36890/iejg.725297

Abstract

This paper presents global differential invariants of curves and paths in the 2-dimensional Euclidean geometry for the groups of Euclidean transformations M(2) and special Euclidean transformations $M^{+}(2)$. For these groups, analogues of the fundamental theorem for Euclidean curves are obtained in terms of global differential invariants of a path and a curve. Moreover, for given two paths(or curves) with the common differential G-invariants, evident forms of all Euclidean transformations that maps one of the paths (or curves) to the other are found. .

Keywords

Path invariant Euclidean geometry

References

  • [1] Aminov, Yu.: Differential Geometry and Topology of Curves. CRC Press, New York, 2000.
  • [2] Aripov, R. G., Khadjiev (Khadzhiev) D.: The complete system of global differential and integral invariants of a curve in Euclidean geometry. Russian Mathematics (Iz VUZ), 51, No. 7, 1-14 (2007).
  • [3] Berger, M.: Geometry I. Springer-Verlag, Berlin Heidelberg, 1987.
  • [4] Gibson, C. G.: Elementary Geometry of Differentiable Curves. An Undergraduate Introduction, Cambridge University Press, 2001.
  • [5] Gray,A., Abbena, E. and Salamon,S.: Modern Differential Geometry of Curves and surfaces with Mathematica. Third edition. Studies in Advanced Mathematics. Chapman and Hall/CRC, Boca Raton, FL, 2006.
  • [6] Guggenheimer, H. W.: Differential Geometry. Dower Publicatiýns, INC., New York, 1977.
  • [7] Khadjiev, D.: Application of the Invariant Theory to the Differential Geometry of Curves. "Fan" Publisher, Tashkent, 1988. [in Russian].
  • [8] Khadjiev, D.: Complete systems of differential invariants of vector fields in a Euclidean space. Turkish J. Math. 34, 543-560 (2010).
  • [9] Khadjiev, D.: On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space. Journal of Nonlinear Mathematical Physics, 17, Supp 01, 49-70 (2010).
  • [10] Khadjiev D., Pekşen Ö.: The complete system of global differential and integral invariants of equiaffine curves. Diff. Geom. And Appl. 20, 168-175 (2004).
  • [11] Khadjiev, D., Ören, İ., Pekşen, Ö.: Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry. Turkish J. Math. 37 , 80-94 (2013).
  • [12] Khadjiev, D., Ören, İ., Pekşen, Ö.: Global invariants of path and curves for the group of all linear similarities in the two-dimensional Euclidean space. Int.J. Geo.Modern Phys,15(6), 1-28 (2018).
  • [13] Mundy, J.L., Zisserman, A.: Geometric Invariance in Computer Vision, Artificial Intelligence. MIT Press, Cambridge, 1992.
  • [14] Mundy, J.L., Zisserman, A., Forsyth, D.: Application of Invariance in Computer Vision. Lecture Notes in Computer Science, Springer, Berlin, 1992.
  • [15] Khadjiev, D., Ören, İ.: Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space. An. St. Univ. Ovidius Constanta, 27(2), 37-65 (2019).
  • [16] Montel, S., Ros, A.: Curves and Surfaces. American Mathematical Society, 2005.
  • [17] Pekşen, Ö., Khadjiev, D.: On invariants of null curves in the pseudo-Euclidean geometry. Diff. Geom. and Appl. 29, 183-187 (2011).
  • [18] Pekşen, Ö., Khadjiev, D., Ören, İ.: Invariant parametrizations and complete systems of global invariants of curves in the pseudo-Euclidean geometry. Turkish J. Math. 36, 147-160 (2012).
  • [19] Spivak, M.: Comprehensive Introduction to Differential Geometry. Vol. 2 Publish Or Perish, INC., Houston, Texas, 1999.
  • [20] Toponogov, V. A.: Differential Geometry of Curves and Surfaces. Birkhauser, Boston, 2006.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

İdris Ören 0000-0003-2716-3945

Djavvat Khadjiev This is me 0000-0001-7056-5662

Publication Date October 15, 2020
Acceptance Date September 15, 2020
Published in Issue Year 2020 Volume: 13 Issue: 2

Cite

APA Ören, İ., & Khadjiev, D. (2020). Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry. International Electronic Journal of Geometry, 13(2), 116-134. https://doi.org/10.36890/iejg.725297
AMA Ören İ, Khadjiev D. Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry. Int. Electron. J. Geom. October 2020;13(2):116-134. doi:10.36890/iejg.725297
Chicago Ören, İdris, and Djavvat Khadjiev. “Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry”. International Electronic Journal of Geometry 13, no. 2 (October 2020): 116-34. https://doi.org/10.36890/iejg.725297.
EndNote Ören İ, Khadjiev D (October 1, 2020) Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry. International Electronic Journal of Geometry 13 2 116–134.
IEEE İ. Ören and D. Khadjiev, “Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry”, Int. Electron. J. Geom., vol. 13, no. 2, pp. 116–134, 2020, doi: 10.36890/iejg.725297.
ISNAD Ören, İdris - Khadjiev, Djavvat. “Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry”. International Electronic Journal of Geometry 13/2 (October 2020), 116-134. https://doi.org/10.36890/iejg.725297.
JAMA Ören İ, Khadjiev D. Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry. Int. Electron. J. Geom. 2020;13:116–134.
MLA Ören, İdris and Djavvat Khadjiev. “Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry”. International Electronic Journal of Geometry, vol. 13, no. 2, 2020, pp. 116-34, doi:10.36890/iejg.725297.
Vancouver Ören İ, Khadjiev D. Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry. Int. Electron. J. Geom. 2020;13(2):116-34.
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