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http://dx.doi.org/10.5831/HMJ.2021.43.1.88

ON THE GEOMETRY OF RATIONAL BÉZIER CURVES

Ceylan, Ayse Yilmaz (Department of Mathematics, Akdeniz University)
Turhan, Tunahan (Department of Mathematics and Science Education, Suleyman Demirel University)
Tukel, Gozde Ozkan (Isparta University of Applied Sciences)

Publication Information

Honam Mathematical Journal / v.43, no.1, 2021 , pp. 88-99 More about this Journal

Abstract

The purpose of this paper is to assign a movable frame to an arbitrary point of a rational Bézier curve on the 2-sphere S2 in Euclidean 3-space R3 to provide a better understanding of the geometry of the curve. Especially, we obtain the formula of geodesic curvature for a quadratic rational Bézier curve that allows a curve to be characterized on the surface. Moreover, we give some important results and relations for the Darboux frame and geodesic curvature of a such curve. Then, in specific case, given characterizations for the quadratic rational Bézier curve are illustrated on a unit 2-sphere.

Keywords

Darboux frame field; Geodesic curvature; Rational $B{\acute{e}}zier$ curve; 2-sphere;

Citations & Related Records

Reference

1	G. Brunnet and P. E. Crouch, Elastic curves on the sphere, Advances in Computational Mathematics, 2(1) (1994), 23-40. DOI
2	M. P. Do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, NJ. 1976.
3	G. E. Farin, Curves and surfaces for CAGD:. a practical guide, Morgan Kaufmann Publishers., 2002.
4	E. Erkan and S. Yuce, Serret-Frenet frame and curvatures of Bezier curves, Mathematics 6(12) (2018), 321. DOI
5	M. S. Floater, Derivatives of rational Bezier curves, Computer Aided Geometric Design 9(3) (1993), 161-174. DOI
6	D. Marsh, Applied geometry for computer graphics and CAD., Springer 2006.
7	B. O'Neill, Elementary differential geometry, Elsevier 2006.
8	F. C. Park and B. Ravani, Bezier curves on Riemannian manifolds and Lie groups with kinematics applications, Journal of Mechanical Design 117 (1995), 36-40. DOI
9	K. Shomake, Animating rotations with quaternion curves, ACM SIGGRAPH Computer Graphics 19 (1985), 245-254. DOI
10	T. Popiel and L. Noakes, C² spherical Bezier splines, Computer aided geometric design 23(3) (2006), 261-275. DOI
11	B. Tantay and F. Tas, The Curvature of a Bezier Control Polyline, Mathematical and Computational Applications 16(2) (2011), 350-358. DOI
12	A. Yilmaz Ceylan, T. Turhan and G. O. Tukel, Darboux frame fields of rational Bezier curves on the two dimensional sphere, BILTEK CONFERENCE-III Current Studies on Science, Technology Social Sciences Haziran 19-20, Adana, Turkey, (2020), 243-248.
13	G. Ozkan and A. Yucesan, Generalized relaxed elastic line on an oriented surface, Ukrainian Journal of Mathematics 64(8) (2012), 1121-1131.