A RATIONAL PARAMETRIZATION OF BÉZIER LIKE CURVES
Keywords:
rational Bernstein functions, functions approximation, Bézier curves, de Casteljau algorithm.DOI:
https://doi.org/10.17654/0972111822003Abstract
In this paper, we construct a family of Bernstein functions using a class of rational parametrizations. The new family of rational Bernstein basis on an index $\alpha \in(-\infty, 0) \cup(1,+\infty)$, and for a given degree $k \in \mathbb{N}^*$, these basis functions are rationals with a numerator and a denominator each of polynomials of degree $k$. All of the classical properties as positivity, partition of unity hold for these rational Bernstein bases. They constitute approximation basis functions for spaces of continuous functions. The Bézier curves obtained satisfy the classical properties. We have the classical computational algorithms like the de Casteljau algorithm and the algorithm of subdivision with the similar accuracy. Given a degree k and a control polygon points, all of these algorithms converge to the same Bézier curve as the classical case. That means the Bézier curve is independent of the index $\alpha$. The classical polynomial Bernstein basis seems to be an asymptotic case of our new class of rational Bernstein basis.
Received: August 10, 2021
Accepted: December 3, 2021
References
Carl de Boor, On calculating with B-splines, J. Approx. Theory 6 (1972), 50-62.
M. G. Cox, The numerical evaluation of B-spline, J. Inst. Math. Appl. 10 (1972), 134-149.
D. F. Rogers, An Introduction to NURBS with Historical Perspective, Ed. Morgan Kaufmann Publishers, 2001.
S. Biswas and B. C. Lovell, Bezier and Spline in Image Processing and Machine Vision, Ed. Springer, 2008.
A. Chandrasekharan, D. Grobe and R. Drechsler, Design Automation Techniques for Approximation Circuits-verification, Synthesis and Test, Ed. Springer, 2019.
P. E. Ceruzzi, A History of Modern Computing, Ed. The MIT Press, 2003.
David Solomon, Curves and Surfaces for Computer Graphics, Ed. Springer Sciences + Business Media Inc., 2006.
Donald Knuth, The Metafont Book, Ed. Addison-Wesley, 1986.
Duncan Marsh, Applied Geometry for Computer Graphics and CAD, Ed. Springer-Verlag, 2005.
V. Gupta, T. M. Rassia, P. N. Agrawal and A. M. Acu, Recent Advances in Constructive Approximation Theory, Ed. Springer, 2018.
B. Jüttler and R. Piene, Geometric Modeling and Algebraic Geometry, Ed. Springer, 2008.
A. Kunoth, T. Lyche, G. Sangalli and S. Serra-Capizzano, Splines and PDEs-From Approximation Theory and Numerical Linear Algebra, Ed. Springer, 2018.
E. Lengyel, Mathematics for 3D Game Programming and Computer Graphics, Ed. Charles River Media Inc., 2003.
G. E. Randriambelosoa, On a family of rational polynomials for Bezier curves and surfaces, Communication privéeà l’Université d’Antananarivo, Madagascar, 2012.
I. J. Schoenberg, Contribution to the Problem of Approximation of Equidistant Data by Analytical Function, I. J. Schoenberg selected papers edited by Carl de Boor Ed. Springer, 1973.
K. S. Surana, Numerical Methods and Methods of Approximation in Science and Engineering, Ed. CRC Press, 2018.
K. J. Versprille, Computer-aided design applications of the rational B-spline approximation form, Ph.D. Thesis in System and Information Science, Syracuse University, 1975.
G. Wahba, Spline Models for Observational Data, Ed. SIAM, 1990.
Allaoui Mohamed, Jamal Adetola, Koffi Wilfrid Houédanou and Aurélien Goudjo, A new class of curves of rational B-spline type, Preprint submitted to Mathematical Methods in the Applied Sciences, 2021, 38 pp. https://doi.org/10.22541/au.163253831.19883484/v1.
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