How to Find a Bezier Curve in $\mathbf{E}^{3}$

Year 2022, Volume: 5 Issue: 1, 12 - 24, 17.03.2022

Süleyman Şenyurt , Şeyda Kılıçoglu

https://doi.org/10.33434/cams.1021878

Cited By: 5

Abstract

"How to find any $n^{th}$ order B\'{e}zier curve if we know its first, second, and third derivatives?" Hence we have examined the way to find the B\'{e}zier curve based on the control points with matrix form, while derivatives are given in $\mathbf{E}^{3}$. Further, we examined the control points of a cubic B\'{e}zier curve with given derivatives as an example. In this study first we have examined how to find any $n^{th}$ order Bezier curve with known its first, second and third derivatives, which are inherently, the $\left( n-1\right) ^{th}$ order, the $\left(n-2\right) ^{th}$ and the $\left( n-3\right) ^{th}$ Bezier curves in respective order. There is a lot of the number of B\'{e}zier curves with known the derivatives with control points. Hence to find a B\'{e}zier curve we have to choose any control point of any derivation\. In this study we have chosen two special points which are the initial point $P_{0}$ and the endpoint $P_{n}$.

Keywords

Bezier curves, Cubic Bezier curves, Derivatives of Bezier curve

Project Number

yok

References

• [1] H. Hagen, Bezier-curves with curvature and torsion continuity, Rocky Mountain J. Math., 16(3), (1986), 629-638.
• [2] D. Marsh, Applied Geometry for Computer Graphics and CAD. Springer Science and Business Media., 2006.
• [3] G. Farin, Curves and Surfaces for Computer-Aided Geometric Design, Academic Press, 1996.
• [4] H. Zhang, F. Jieqing, Bezier Curves and Surfaces (2), State Key Lab of CAD&CG Zhejiang University, 2006.
• [5] S. Michael, Bezier Curves and Surfaces, Lecture 8, Floater Oslo Oct., 2003.
• [6] E. Erkan, S. Yüce, Serret-Frenet frame and curvatures of B´ezier curves, Mathematics, 6 (12) (2018), 321.
• [7] H. K. Samanci, S. Celik, M. Incesu, The Bishop frame of B´ezier curves, Life Sci. J, 12(6) 2015, 175-180.
• [8] H. K. Samanci, M. Incesu, Investigating a quadratic Bezier curve due to NCW and N-Bishop frames, Turk. J. Math. Compu. Sci., 12(2) (2020), 120-127.
• [9] Ş. Kılıçoğlu, S. Şenyurt, On the cubic bezier curves in E3, Ordu Uni. J. Sci. Techno., 9(2) (2019), 83-97.
• [10] Ş. Kılıçoğlu, S. Şenyurt, On the involute of the cubic B´ezier curve by using matrix representation in E3, European J. Pure App. Math., 13 (2020), 216-226.
• [11] A. Levent, B. Sahin, Cubic bezier-like transition curves with new basis function, Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 44(2) (2008) , 222-228.
• [12] Ş. Kılıçoğlu, S. Şenyurt, On the matrix representation of 5th order B´ezier curve and derivatives, Comm. Fac. Sci. Uni. Ankara Series A1 Math. Stat., in press 2021.
• [13] Ş. Kılıçoğlu, S. Şenyurt, On the Bertrand mate of a cubic B´ezier curve by using matrix representation in E3, 18th International Geometry Sym. 2021.
• [14] Ş. Kılıçoğlu, S. Şenyurt, On the Mannheim partner of a cubic Bezier curve in E3, 10th International Eurasian Conference on Mathematical Sciences and Applications, 2021.
• [15] A. Y. Ceylan, Curve couples of Bezier curves in Euclidean 2-space, Fundamental J. Math. App., 4(4) (2021), 245-250.
• [16] Ş. Kılıçoğlu, S. Şenyurt, On the matrix representation of Bezier curves and derivatives in E3, Sigma J. Engineering and Natural Sci., in Press 2021