Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space

Khadjiev, Djavvat; Ören, İdris

doi:10.2478/auom-2019-0018

Abstract

In this paper, for the orthogonal group G = O(2) and special orthogonal group G = O⁺(2) global G-invariants of plane paths and plane curves in two-dimensional Euclidean space E₂ are studied. Using complex numbers, a method to detect G-equivalences of plane paths in terms of the global G-invariants of a plane path is presented. General evident form of a plane path with the given G-invariants are obtained. For given two plane paths x(t) and y(t) with the common G-invariants, evident forms of all transformations g ∈ G, carrying x(t) to y(t), are obtained. Similar results have obtained for plane curves.

References

[1] Alcázar, J. G., Hermoso, Carlos, M., Muntingh, G., Detecting similarity of rational plane curves. J. Comput. Appl. Math 269 (2014), 113.10.1016/j.cam.2014.03.013
Search in Google Scholar Back to article
[2] Alexandrov, A. D., Reshetnyak, Yu. G., General Theory of Irregular Curves. Kluwer Acad. Publ., Dordrecht, 1989.10.1007/978-94-009-2591-5
Search in Google Scholar Back to article
[3] Aminov, Yu., Differential Geometry and Topology of Curves. CRC Press, New York, 2000.
Search in Google Scholar Back to article
[4] 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(7),(2007) 1-14.10.3103/S1066369X07070018
Search in Google Scholar Back to article
[5] Berger, M., Geometry I. Springer-Verlag, Berlin Heidelberg, 1987.
Search in Google Scholar Back to article
[6] Chou, K. S., Qu, C. Z., Integrable equations arising from motions of plane curves. Phys. D 162(1-2) (2002) 9-33.10.1016/S0167-2789(01)00364-5
Search in Google Scholar Back to article
[7] Chou, K. S., Qu, C. Z., Integrable equations and motions of plane curves. Symmetry in nonlinear mathematical physics, Part 1, 2 Kyiv, (2001) 281-290.10.1016/S0167-2789(01)00364-5
Search in Google Scholar Back to article
[8] Gibson, G. G., Elementary Geometry of Differentiable Curves. An Undergraduate Introduction. Cambridge University Press, 2001.10.1017/CBO9781139173377
Search in Google Scholar Back to article
[9] Guggenheimer, H., W., Differential Geometry. McGraw-Hill Book Company, Inc., New York, 1963.
Search in Google Scholar Back to article
[10] Hauer, M., Jüttler, B. Detecting affine equivalences of planar rational curves. EuroCG 2016, Lugano, Switzerland, March 30-April 1, 2016.
Search in Google Scholar Back to article
[11] Inoguchi, J., Attractive plane curves in differential geometry. Mathematical Progress in Expressive Image Synthesis III, (2016) 121-135.10.1007/978-981-10-1076-7_13
Search in Google Scholar Back to article
[12] Khadjiev, D., Application of the Invariant Theory to the Differential Geometry of Curves. Fan Publisher, Tashkent, 1988. [in Russian].
Search in Google Scholar Back to article
[13] Khadjiev, D., Complete systems of differential invariants of vector fields in a Euclidean space. Turk J. Math., 34, (2010) 543-560.10.3906/mat-0809-10
Search in Google Scholar Back to article
[14] Khadjiev, D., On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-Euclidean space. J Nonlinear Math Phys, 17, Supp 01 (2010) 49-70.10.1142/S1402925110000799
Search in Google Scholar Back to article
[15] Khadjiev D., Peksen Ö., The complete system of global differential and integral invariants of equiaffne curves. Diff. Geom. And Appl., 20 (2004) 168-175.10.1016/j.difgeo.2003.10.005
Search in Google Scholar Back to article
[16] Khadjiev, D., Ören, İ., Pekşen, Ö., Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry. Turk J. Math., 37 (2013) 80-94.10.3906/mat-1104-41
Search in Google Scholar Back to article
[17] Kreyszig, E., Introduction to Differential Geometry and Riemannian Geometry. Univ. of Toronto Press, 1968.10.3138/9781487589448
Search in Google Scholar Back to article
[18] Mozo-Fernandez, J., Munuera, C., Recognition of polynomial plane curves under affine transformations. AAECC 13 (2002) 121-136.10.1007/s002000200095
Search in Google Scholar Back to article
[19] O’Neill, B., Elementary Differential Geometry, Elsevier, 2006.10.1016/B978-0-12-088735-4.50011-0
Search in Google Scholar Back to article
[20] Ören, İ., Equivalence conditions of two Bézier curves in the Euclidean geometry. Iran J Sci Technol Trans Sci. DOI number: 10.1007/s40995-016-0129-1.(in press)10.1007/s40995-016-0129-1.(
Open DOI Search in Google Scholar Back to article
[21] Pekşen, Ö., Khadjiev, D., On invariants of null curves in the pseudo-Euclidean geometry. Diff. Geom. And Appl. 29 (2011) S183-187.10.1016/j.difgeo.2011.04.024
Search in Google Scholar Back to article
[22] Pekşen, Ö., Khadjiev, D., Ören, İ, Invariant parametrizations and complete systems of global invariants of curves in the pseudo-Euclidean geometry. Turk J. Math., 36 (2012) 147-160.10.3906/mat-0911-145
Search in Google Scholar Back to article
[23] Yaglom, I. M., Complex Numbers in Geometry. Academic Press, New York, 1968.
Search in Google Scholar Back to article
[24] Zwikker, C., The Advanced Geometry of Plane Curves and Their Applications. Dover Publications, INC., New York, 1963.
Search in Google Scholar Back to article

Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space

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