İDRİS ÖREN

On the Control Invariants of Planar Bezier Curves for the Groups ´ M(2) and S M(2)

Let G = M(2) be the group generated by all orthogonal transformations and translations of the 2-dimensional Euclidean space E2 or G = S M(2) be the subgroup of M(2) generated by rotations and translations ofE2. In this paper, global G-invariants of plane Bezier curves in ´ E2 are introduced. Using complex numbers and theglobal G-invariants of a plane Bezier curves, for given two plane B ´ ezier curves ´ x(t) and y(t), evident forms of alltransformations g ∈ G, carrying x(t) to y(t), are obtained. Similar results are given for plane polynomial curves.

PDF

___

[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(7) (2007), 1-14 .

[3] Berger, M., Geometry I, Springer-Verlag, Berlin Heidelberg, 1987. [4] Gibson, C. G., Elementary Geometry of Differentiable Curves, 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 Publ, INC., New York, 1977.

[7] 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.

[8] Khadjiev, D., Complete systems of differential invariants of vector fields in a Euclidean space, Turk J. Math., 34(2010), 543-560.

[9] Khadjiev, D., Ören, İ., Peks¸en, 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] 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.

[11] Khadjiev D., Peks¸en O., ¨ The complete system of global differential and integral invariants of equiaffine curves, Diff. Geom. And Appl., 20 (2004) 168-175.

[12] Marsh D, Applied geometry for computer graphics and CAD, Springer-Verlag, London,1999.

[13] Montel, S., Ros, A., Curves and Surfaces, American Mathematical Society, 2005.

[14] O’Neill, B., Elementary Differential Geometry, Elsevier,Academic Press, Amsterdam, 2006.

[15] Ören, İ., Equivalence conditions of two B´e zier curves in the Euclidean geometry, Iran J Sci Technol Trans Sci., 42 (2018),1563-1577. 1, 4, 4,

[16] Peks¸en, O., 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.

[17] Spivak, M., Comprehensive Introduction to Differential Geometry, Publish Or Perish, INC., Houston, Texas, 1999.