A de Casteljau Algorithm for Bernstein type Polynomials based on (p,q)-integers

by   Khalid Khan, et al.
Universiti Putra Malaysia

In this paper, a de Casteljau algorithm to compute (p,q)-Bernstein Bezier curves based on (p,q)-integers is introduced. We study the nature of degree elevation and degree reduction for (p,q)-Bezier Bernstein functions. The new curves have some properties similar to q-Bezier curves. Moreover, we construct the corresponding tensor product surfaces over the rectangular domain (u, v) ∈ [0, 1] × [0, 1] depending on four parameters. We also study the de Casteljau algorithm and degree evaluation properties of the surfaces for these generalization over the rectangular domain. Furthermore, some fundamental properties for (p,q)-Bernstein Bezier curves are discussed. We get q-Bezier curves and surfaces for (u, v) ∈ [0, 1] × [0, 1] when we set the parameter p1 = p2 = 1.



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1 Introduction

Recently, Mursaleen et al [9, 10] applied -calculus in approximation theory and introduced the first -analogue of Bernstein operators based on -integers. Motivated by the work of Mursaleen et al [9, 10], the idea of -calculus and its importance. We construct -Bzier curves and surfaces based on -integers which is further generalization of -Bzier curves and surfaces. For similar works based on -integers, one can refer [11, 12, 13, 15, 22, 25].

It was S.N. Bernstein [1] in 1912, who first introduced his famous operators defined for any and for any function


and named it Bernstein polynomials to prove the Weierstrass theorem [7]. Later it was found that Bernstein polynomials possess many remarkable properties and has various applications in areas such as approximation theory [7], numerical analysis, computer-aided geometric design, and solutions of differential equations due to its fine properties of approximation [19].

In computer aided geometric design (CAGD), Bernstein polynomials and its variants are used in order to preserve the shape of the curves or surfaces. One of the most important curve in CAGD [24] is the classical Bzier curve [2] constructed with the help of Bernstein basis functions.

In recent years, generalization of the Bzier curve with shape parameters has received continuous attention. Several authors were concerned with the problem of changing the shape of curves and surfaces, while keeping the control polygon unchanged and thus they generalized the Bzier curves in [5, 18, 19].

The rapid development of -calculus [23] has led to the discovery of new generalizations of Bernstein polynomials involving -integers [8, 16, 19] . The aim of these generalizations is to provide appropriate and powerful tools to application areas such as numerical analysis, computer-aided geometric design, and solutions of differential equations.

In 1987, Lupaş [8] introduced the first -analogue of Bernstein operator as follows


and investigated its approximating and shape-preserving properties.

In 1996, Phillips [17] proposed another -variant of the classical Bernstein operator, the so-called Phillips -Bernstein operator which attracted lots of investigations.


where defined for any and any function

The -variants of Bernstein polynomials provide one shape parameter for constructing free-form curves and surfaces, Phillips -Bernstein operator was applied well in this area.

In 2003, Oruk and Phillips [19] used the basis functions of Phillips -Bernstein operator for construction of -Bzier curves, which they call Phillips -Bzier curves, and studied the properties of degree reduction and elevation.

Thus with the development of -analogue of Bernstein operators and its variants, one natural question arises, how it can be used in order to preserve the shape of the curves or surfaces. In this way, it opens a new research direction which requires further investigations.

Before proceeding further, let us recall certain notations of -calculus .

The integers are defined by

The formula for -binomial expansion is as follow:

where -binomial coefficients are defined by

Details on -calculus can be found in [6, 9, 21].

The -Bernstein Operators introduced by Mursaleen et al is as follow:


Note when -Bernstein Operators given by (1.4) turns out to be -Bernstein Operators.

Also, we have

Again by some simple calculations and using the property of -integers, we get -analogue of Pascal’s relation as follow:


We apply -calculus and introduce first the -Bzier curves and surfaces based on -integers which is further generalization of -Bzier curves and surfaces, for example, [18, 19].

The outline of this paper is as follow: Section introduces a -analogue of the Bernstein functions and their Properties. Section introduces degree elevation and degree reduction properties for -analogue of the Bernstein functions. Section introduces a de Casteljau type algorithm for . In Section we define a tensor product patch based on algorithm and its geometric properties as well as a degree elevation technique are investigated. Furthermore tensor product of -Bzier surfaces on for -Bernstein polynomials are introduced and its properties that is inherited from the univariate case are being discussed.

2 -Bernstein functions

The -Bernstein functions is as follows



2.1 Properties of the -analogue of the Bernstein functions

Theorem 2.1

The -analogue of the Bernstein functions possess the following properties:

(1.) Non-negativity:

(2.) Partition of unity:

(3.)Both sided end-point property:


then both side end point interpolation property holds.

(3.) Reducibility: when formula reduces to the -Bernstein bases.

Proof: All these property can be deduced easily from equation (2.1).

Figure 1: ‘Cubic Bezier blending functions’
Figure 2: ‘Cubic Bezier blending functions’

Fig. 2 shows the -analogues of the Bernstein basis functions of degree with . Here we can observe that sum of blending fuctions is always unity and also end point interpolation property holds, when we put it turns out to be -Bernstein basis which is shown in Fig.

Apart from the basic properties above, the -analogue of the Bernstein functions also satisfy some recurrence relations, as for the classical Bernstein basis.

3 Degree elevation and reduction for -Bernstein functions

Technique of degree elevation has been used to increase the flexibility of a given curve. A degree elevation algorithm calculates a new set of control points by choosing a convex combination of the old set of control points which retains the old end points. For this purpose, the identities (3.3),(3.4) and Theorem (3.2) are useful.

Theorem 3.1

Each -Bernstein functions of degree n is a linear combination of two -Bernstein functions of degree


Proof: On using Pascal’s type relation based on -integers i.e we get


Similarly, if we use , we have

Degree elevation




By some simple calculation, we have

using this result, we get

similarly on considering,

finally we get

Theorem 3.2

Each -Bernstein function of degree is a linear combination of two -Bernstein functions of degree


Proof From equation we can easily get

For the blending functions are given by:

We observed that the both sided end point interpolation property and partition of unity property always holds in case of -Bernstein functions.

The de Casteljau algorithm describes how to subdivide a Bzier curve, when a Bzier curve is repeatedly subdivided, the collection of control polygons converge to the curve. Thus, the way of computing a Bzier curve is to simply subdivide it an appropriate number of times and compute the control polygons.

4 -Bernstein Bzier curves:

Let us define the -Bzier curves of degree n using the -analogues of the Bernstein functions as follows:


where and are control points. Joining up adjacent points to obtain a polygon which is called the control polygon of -Bézier curves.

4.1 Some basic properties of -Bzier curves.

Theorem 4.1

From the definition, we can derive some basic properties of -Bzier curves:

1. -Bzier curves have geometric and affine invariance.
2. -Bzier curves lie inside the convex hull of its control polygon.
3. The end-point interpolation property:
3. Reducibility: when formula 4.1 gives the -Bzier curves.

Proof. These properties of -Bzier curves can be easily deduced from corresponding properties of the -analogue of the Bernstein functions.

4.2 Degree elevation for -Bzier curves

-Bzier curves have a degree elevation algorithm that is similar to that possessed by the classical Bzier curves. Using the technique of degree elevation, we can increase the flexibility of a given curve.



The statement above can be derived using the identities Consider

We obtain

Now by shifting the limits, we have


is defined as the zero vector. Comparing coefficients on both side, we have


In general


When formula 4.4 reduce to the degree evaluation formula of the -Bzier curves. If we let denote the vector of control points of the initial -Bzier curve of degree and the vector of control points of the degree elevated -Bzier curve of degree then we can represent the degree elevation procedure as:


For any the vector of control points of the degree elevated -Bézier curves of degree is: As the control polygon converges to a -Bzier curve.

4.3 de Casteljau algorithm:

-Bzier curves of degree can be written as two kinds of linear combination of two -Bzier curves of degree and we can get the two selectable algorithms to evaluate -Bzier curves. The algorithms can be expressed as:

Algorithm 1.






It is clear that the results can be obtained from Theorem (LABEL:khalidthm). When formula (4.3) and (4.4) recover the de Casteljau algorithms of classical -Bzier curves. Let , then de Casteljau algorithm can be expressed as:

Algorithm 2.


where is a matrix and


5 Tensor product -Bernstein Bzier surfaces on

We define a two-parameter family of tensor product surfaces of degree as follow:


where and two real numbers are -analogue of Bernstein functions respectively with the parameter and We call the parameter surface tensor product -Bzier surface with degree We refer to the as the control points. By joining up adjacent points in the same row or column to obtain a net which is called the control net of tensor product -Bzier surface.

5.1 Properties

1. Geometric invariance and affine invariance property: Since