On Bernstein type polynomials and their applications
- Yilmaz Simsek^{1}Email author and
- Melih Gunay^{2}
https://doi.org/10.1186/s13662-015-0423-9
© Simsek and Gunay; licensee Springer. 2015
Received: 23 October 2014
Accepted: 24 February 2015
Published: 5 March 2015
Abstract
In this study we examine generating functions for the Bernstein type polynomials given in (Simsek in Fixed Point Theory Appl. 2013:80, 2013). We expand these generating functions using the parameters u and v. By applying these generating functions, we obtain some functional equations and partial differential equations. In addition, using these equations, we derive several identities and relations related to these polynomials. Finally, numerical values of these polynomials for selected cases are demonstrated with their plots.
Keywords
1 Introduction
With the advances in computer graphics and CAD, there has been renewed interest by researchers to study Bézier curves and surfaces [1, 2]. According to Goldman [3], free-form curves and surfaces are smooth shapes often describing man-made objects. For instance, the hood of a car, the hull of a ship, and the fuselage of an airplane are all examples of free-form shapes that differ from the typical surfaces as they can be described with a few parameters. On the other hand, free-form shapes such as the hood of car may not easily be described with a few parameters. Therefore, mathematical techniques for describing these surfaces focused on Bernstein polynomials and their various generalizations (cf. [1–13]).
Curves obtained by using Bernstein polynomials range from the design of new fonts to the creation of mechanical components and assemblies for large scale industrial design and manufacture. By using the Bernstein polynomials, one can easily find an explicit polynomial representation of a Bézier curves.
In addition to computer graphics, the Bernstein polynomials are also used in the approximation of functions, in statistics, in numerical analysis, in p-adic analysis, and in the solution of differential equations. Therefore, the goal of this paper is to develop a more flexible Bernstein type polynomial using its generating function and visualize the curves obtained with this function over a finite domain with set parameters.
The organization of the paper is as follows.
In Section 2, we give the definition, generating functions, and some properties of the Bernstein type basis functions with respect to u and v. In Section 3, we differentiate the generating function with respect to x and t and obtain select partial differential equations (PDEs). Using these equations, we derive a recurrence relation and derivative formula for Bernstein type basis functions. Finally curves are plotted using the Bernstein type basis function.
2 Properties of the Bernstein type basis functions
In this section, we give fundamental properties of the Bernstein basis functions and their generating functions. By using generating functions, we derive various functional equations and PDEs. Next, using these equations and PDEs, we obtain several identities related to the Bernstein type basis functions.
2.1 Generating functions
Recently the Bernstein polynomials have been defined and studied in many different ways, for example, by q-series, by complex functions, by p-adic Volkenborn integrals, and many algorithms (cf. [1, 2, 4–6]). Here by using an analytic function we construct generating functions for the Bernstein type basis functions related to nonnegative real parameters.
The Bernstein type basis functions \(S_{k}^{n}(x;b;u,v)\) are defined as follows.
Definition 1
Remark 1
Definition 2
We construct generating functions for the Bernstein type basis functions explicitly by the following theorem.
Theorem 1
Proof
2.2 Bernstein type polynomials
Remark 2
Remark 3
2.3 Sum of the Bernstein type basis functions
Theorem 2
Theorem 3
2.4 Alternating sum of the Bernstein type basis functions
Theorem 4
3 Differentiating the generating function
In this section, we give derivative of the Bernstein type basis functions. By differentiating the generating function in (2.3) with respect to x, we arrive at the following theorem.
Theorem 5
By using Theorem 5, we obtain the derivative of the Bernstein type basis functions by the following theorem.
Theorem 6
Remark 4
3.1 Recurrence relation
Here, by using higher order derivatives of the generating function with respect to t, we derive a partial differential equation. Using this equation, we shall give a recurrence relation for the Bernstein type basis functions. Here we use the same method as in [9].
Theorem 7
Using definition (2.2), (2.1), and Theorem 7, we obtain a recurrence relation for the Bernstein type basis functions by the following theorem.
Theorem 8
Theorem 9
4 Identities
In this section, we give a functional equation which is related to the generating function in (2.3). By using this functional equation, we derive two identities for the Bernstein type basis functions.
Theorem 10
Substituting \(m=1\) into (4.2), we obtain the following corollary:
Corollary 1
5 Simulation of the Bernstein type basis functions
Graphics of the Bernstein type polynomials are provided to visualize the shape of polynomials on finite domain. The effects of k, b, and n on the shape of the curve are demonstrated for the given range. These graphics may not only be used in Computer Aided Geometric Design (CAGD) but also in other areas (cf. [1–13]).
Declarations
Acknowledgements
The present investigation was supported by the Scientific Research Project Administration of Akdeniz University. We also would like to thank the referees for carefully reading and making valuable suggestions.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Farouki, RT, Goodman, TNT: On the optimal stability of the Bernstein basis. Math. Comput. 65, 1553-1566 (1996) View ArticleMATHMathSciNetGoogle Scholar
- Goldman, R: Identities for the univariate and bivariate Bernstein basis functions. In: Paeth, A (ed.) Graphics Gems, vol. V, pp. 149-169. Academic Press, San Diego (1995) View ArticleGoogle Scholar
- Goldman, R: An Integrated Introduction to Computer Graphics and Geometric Modeling. CRC Press, New York (2009) MATHGoogle Scholar
- Simsek, Y: Generating functions for the Bernstein type polynomials: a new approach to deriving identities and applications for these polynomials. Hacet. J. Math. Stat. 43(1), 1-14 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Simsek, Y, Acikgoz, M: A new generating function of (q-) Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal. 2010, Article ID 769095 (2010) View ArticleMathSciNetGoogle Scholar
- Simsek, Y: q-Beta polynomials and their applications. Appl. Math. Inf. Sci. 7(6), 2539-2547 (2013) View ArticleMathSciNetGoogle Scholar
- Simsek, Y: A new class of polynomials associated with Bernstein and beta polynomials. Math. Methods Appl. Sci. 37(5), 676-685 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Simsek, Y: Beta-type polynomials and their generating functions. Appl. Math. Comput. 254, 172-182 (2015) View ArticleMathSciNetGoogle Scholar
- Simsek, Y: Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions. Fixed Point Theory Appl. 2013, 80 (2013) View ArticleGoogle Scholar
- Goldman, R: Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling. Morgan Kaufmann, San Diego (2002) Google Scholar
- Lorentz, GG: Bernstein Polynomials. Chelsea, New York (1986) MATHGoogle Scholar
- Simsek, Y: Interpolation function of generalized q-Bernstein type polynomials and their application. In: Curves and Surfaces. Lecture Notes in Computer Science, vol. 6920, pp. 647-662. Springer, Berlin (2011) View ArticleGoogle Scholar
- Bernstein, SN: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Commun. Soc. Math. Kharkow (2) 13, 1-2 (1912-1913) Google Scholar