On the matrix representation of 5th order Bezier curve and derivatives in E$^{3}$

Abstract

Using the matrix representation form, the first, second, third, fourth, and fifth derivatives of 5th order Bezier curves are examined based on the control points in $E^3$. In addition to this, each derivative of 5th order Bezier curves is given by their control points. Further, a simple way has been given to find the control points of a Bezier curves and its derivatives by using matrix notations. An example has also been provided and the corresponding figures which are drawn by Geogebra v5 have been presented in the end.

Keywords

• Bezier curve
• cubic Bezier curve
• derivatives of Bezier curves
• matrix representation

References

1. Marsh, D., Applied Geometry for Computer Graphics and CAD, Springer Science and Business Media, 2006.
2. Derivatives of a Bezier Curve, https://pages.mtu.edu/ 126 shene/COURSES/cs3621/NOTES/spline /Bezier/bezier-der.html
3. Erkan, E., Yüce, S., Some notes on geometry of Bezier curves in Euclidean 4-space, Journal of Engineering Technology and Applied Sciences, 5(3) (2020), 93-101. https://doi.org/10.30931/jetas.837921
4. Taş, F., İlarslan, K., A new approach to design the ruled surface, International Journal of Geometric Methods in Modern Physics, 16(6) (2019), 1950093. https://doi.org/10.1142/S0219887819500932
5. Farin, G., Curves and Surfaces for Computer-Aided Geometric Design, Academic Press, 1996.
6. Hagen, H., Bezier-curves with curvature and torsion continuity, Rocky Mountain J. Math., 16(3) (1986), 629-638. https://doi.org/10.1216/RMJ-1986-16-3-629
7. Zhang, J. W. C., Jieqing, F., Bezier curves and surfaces, Graphical Models and Image Processing, 61(1) (1999), 2-15.
8. Incesu, M., LS (3)-equivalence conditions of control points and application to spatial Bezier curves and surfaces, AIMS Mathematics, 5(2) (2020) 1216-1246. https://doi.org/10.3934/math.2020084

9. Incesu, M., Gursoy, O., LS (2)-Equivalence conditions of control points and application to planar Bezier curves, New Trends in Mathematical Sciences, 5(3) (2017), 70-84. https://doi.org/10.20852/ntmsci.2017.186
10. Michael, S., Bezier Curves and Srfaces, Lecture 8, Floater Oslo, 2003.
11. Kılıçoğlu, Ş., Şenyurt, S., On the cubic Bezier curves in E3, Ordu University Journal of Science and Technology, 9(2) (2019) 83-97.
12. Kılıçoğlu, S., Şenyurt, S., On the involute of the cubic Bezier curve by using matrix representation in E3, European Journal of Pure and Applied Mathematics, 13 (2020), 216-226.https://doi.org/10.29020/nybg.ejpam.v13i2.3648