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Year 2018, Volume: 10, 74 - 81, 29.12.2018

Abstract

References

  • Aminov, Yu., Differential Geometry and Topology of Curves, CRC Press, New York, 2000.
  • Aripov, R. G., Khadjiev (Khadzhiev) D., The complete system of global di_erential and integral invariants of a curve in Euclidean geometry, Russian Mathematics (Iz VUZ), 51(7) (2007), 1-14 .
  • Berger, M., Geometry I, Springer-Verlag, Berlin Heidelberg, 1987.
  • Gibson, C.G., Elementary Geometry of Differentiable Curves, Cambridge University Press, 2001.
  • Gray, A., Abbena, E., Salamon,S., Modern Di_erential Geometry of Curves and surfaces with Mathematica, Third edition. Studies in Advanced Mathematics. Chapman and Hall/CRC, Boca Raton, FL, 2006.
  • Guggenheimer, H. W., Differential Geometry, Dower Publ, INC., New York, 1977.
  • Khadjiev D., Pekşen Ö ., The complete system of global differential and integral invariants of equiaffine curves, Dif. Geom. And Appl., 20(2004) 168-175.
  • 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.
  • Khadjiev, D., Complete systems of di_erential invariants of vector fields in a Euclidean space, Turk J. Math., 34, (2010) 543-560.
  • Khadjiev, D., On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space, Journal of Nonlinear Mathematical Physics, 17 (2010) 49-70.
  • 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),(2018),1-28.
  • Marsh D, Applied geometry for computer graphics and CAD, Springer-Verlag, London,1999.
  • Montel, S., Ros, A., Curves and Surfaces, American Mathematical Society, 2005.
  • O’Neill, B., Elementary Differential Geometry, Elsevier, Academic Press, Amsterdam, 2006.
  • Ören, İ., Equivalence conditions of two Bezier curves in the Euclidean geometry, Iran J Sci Technol Trans Sci., 42 (2018),1563-1577.
  • 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.
  • Spivak, M., Comprehensive Introduction to Differential Geometry, Publish Or Perish, INC., Houston, Texas, 1999.

On the control invariants of planar Bezier curves for the groups M(2) and SM(2)

Year 2018, Volume: 10, 74 - 81, 29.12.2018

Abstract

Let G=M(2) be the group generated by all orthogonal transformations and translations of the 2-dimensional Euclidean space E2 or G=SM(2) be the subgroup of M(2) generated by rotations and translations of E2. In this paper, global G-invariants of plane Bezier  curves in E2 are introduced. Using complex numbers and the global G-invariants of a plane B  curves,  for given two plane B  curves x(t) and y(t), evident forms of all transformations g\in G, carrying x(t) to y(t),  are obtained. Similar results are given for plane polynomial curves.

	

References

  • Aminov, Yu., Differential Geometry and Topology of Curves, CRC Press, New York, 2000.
  • Aripov, R. G., Khadjiev (Khadzhiev) D., The complete system of global di_erential and integral invariants of a curve in Euclidean geometry, Russian Mathematics (Iz VUZ), 51(7) (2007), 1-14 .
  • Berger, M., Geometry I, Springer-Verlag, Berlin Heidelberg, 1987.
  • Gibson, C.G., Elementary Geometry of Differentiable Curves, Cambridge University Press, 2001.
  • Gray, A., Abbena, E., Salamon,S., Modern Di_erential Geometry of Curves and surfaces with Mathematica, Third edition. Studies in Advanced Mathematics. Chapman and Hall/CRC, Boca Raton, FL, 2006.
  • Guggenheimer, H. W., Differential Geometry, Dower Publ, INC., New York, 1977.
  • Khadjiev D., Pekşen Ö ., The complete system of global differential and integral invariants of equiaffine curves, Dif. Geom. And Appl., 20(2004) 168-175.
  • 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.
  • Khadjiev, D., Complete systems of di_erential invariants of vector fields in a Euclidean space, Turk J. Math., 34, (2010) 543-560.
  • Khadjiev, D., On invariants of immersions of an n-dimensional manifold in an n-dimensional pseudo-euclidean space, Journal of Nonlinear Mathematical Physics, 17 (2010) 49-70.
  • 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),(2018),1-28.
  • Marsh D, Applied geometry for computer graphics and CAD, Springer-Verlag, London,1999.
  • Montel, S., Ros, A., Curves and Surfaces, American Mathematical Society, 2005.
  • O’Neill, B., Elementary Differential Geometry, Elsevier, Academic Press, Amsterdam, 2006.
  • Ören, İ., Equivalence conditions of two Bezier curves in the Euclidean geometry, Iran J Sci Technol Trans Sci., 42 (2018),1563-1577.
  • 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.
  • Spivak, M., Comprehensive Introduction to Differential Geometry, Publish Or Perish, INC., Houston, Texas, 1999.
There are 17 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

İdris Ören

Publication Date December 29, 2018
Published in Issue Year 2018 Volume: 10

Cite

APA Ören, İ. (2018). On the control invariants of planar Bezier curves for the groups M(2) and SM(2). Turkish Journal of Mathematics and Computer Science, 10, 74-81.
AMA Ören İ. On the control invariants of planar Bezier curves for the groups M(2) and SM(2). TJMCS. December 2018;10:74-81.
Chicago Ören, İdris. “On the Control Invariants of Planar Bezier Curves for the Groups M(2) and SM(2)”. Turkish Journal of Mathematics and Computer Science 10, December (December 2018): 74-81.
EndNote Ören İ (December 1, 2018) On the control invariants of planar Bezier curves for the groups M(2) and SM(2). Turkish Journal of Mathematics and Computer Science 10 74–81.
IEEE İ. Ören, “On the control invariants of planar Bezier curves for the groups M(2) and SM(2)”, TJMCS, vol. 10, pp. 74–81, 2018.
ISNAD Ören, İdris. “On the Control Invariants of Planar Bezier Curves for the Groups M(2) and SM(2)”. Turkish Journal of Mathematics and Computer Science 10 (December 2018), 74-81.
JAMA Ören İ. On the control invariants of planar Bezier curves for the groups M(2) and SM(2). TJMCS. 2018;10:74–81.
MLA Ören, İdris. “On the Control Invariants of Planar Bezier Curves for the Groups M(2) and SM(2)”. Turkish Journal of Mathematics and Computer Science, vol. 10, 2018, pp. 74-81.
Vancouver Ören İ. On the control invariants of planar Bezier curves for the groups M(2) and SM(2). TJMCS. 2018;10:74-81.