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

1 Introduction

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

It was S.N. Bernstein [1] in 1912, who first introduced his famous operators $B_{n} :$ $C [0, 1] \to C [0, 1]$ defined for any $n \in N$ and for any function $f \in C [0, 1]$

B_{n} (f; x) = n \sum k = 0 (\begin{matrix} n k \end{matrix}) x^{k} (1 - x)^{n - k} f (\frac{k}{n}), x \in [0, 1] .

(1.1)

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 B $´ e$ zier curve [2] constructed with the help of Bernstein basis functions.

In recent years, generalization of the B $´ e$ zier 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 B $´ e$ zier curves in [5, 18, 19].

The rapid development of $q$ -calculus [23] has led to the discovery of new generalizations of Bernstein polynomials involving $q$ -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 $q$ -analogue of Bernstein operator as follows

L_{n, q} (f; x) = n \sum k = 0 \frac{f (\frac{[k]_{q}}{[n]_{q}}) {[\begin{matrix} n k \end{matrix}]}_{q} q^{\frac{k (k - 1)}{2}} x^{k} (1 - x)^{n - k}}{n \prod j = 1 {(1 - x) + q^{j - 1} x}},

(1.2)

and investigated its approximating and shape-preserving properties.

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

B_{n, q} (f; x) = n \sum k = 0 {[\begin{matrix} n k \end{matrix}]}_{q} x^{k} n - k - 1 \prod s = 0 (1 - q^{s} x) f (\frac{[k]_{q}}{[n]_{q}}), x \in [0, 1]

(1.3)

where $B_{n, q} :$ $C [0, 1] \to C [0, 1]$ defined for any $n \in N$ and any function $f \in C [0, 1] .$

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

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

Thus with the development of $(p, q)$ -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 $(p, q)$ -calculus .

The $(p, q)$ integers $[n]_{p, q}$ are defined by

[n]_{p, q} = \frac{p^{n} - q^{n}}{p - q}, n = 0, 1, 2, \dots, p > q > 0.

The formula for $(p, q)$ -binomial expansion is as follow:

(a x + b y)_{p, q}^{n} := n \sum k = 0 p^{\frac{(n - k) (n - k - 1)}{2}} q^{\frac{k (k - 1)}{2}} {[\begin{matrix} n k \end{matrix}]}_{p, q} a^{n - k} b^{k} x^{n - k} y^{k},

(x + y)_{p, q}^{n} = (x + y) (p x + q y) (p^{2} x + q^{2} y) \dots (p^{n - 1} x + q^{n - 1} y),

(1 - x)_{p, q}^{n} = (1 - x) (p - q x) (p^{2} - q^{2} x) \dots (p^{n - 1} - q^{n - 1} x),

where $(p, q)$ -binomial coefficients are defined by

{[\begin{matrix} n k \end{matrix}]}_{p, q} = \frac{[n]_{p, q}!}{[k]_{p, q}! [n - k]_{p, q}!} .

Details on $(p, q)$ -calculus can be found in [6, 9, 21].

The $(p, q)$ -Bernstein Operators introduced by Mursaleen et al is as follow:

(1.4)

Note when $p = 1,$ $(p, q)$ -Bernstein Operators given by (1.4) turns out to be $q$ -Bernstein Operators.

Also, we have

	$(1 - x)_{p, q}^{n}$	$= n - 1 \prod s = 0 (p^{s} - q^{s} x) = (1 - x) (p - q x) (p^{2} - q^{2} x) . . . (p^{n - 1} - q^{n - 1} x)$
		$= n \sum k = 0 {(- 1)}^{k} p^{\frac{(n - k) (n - k - 1)}{2}} q^{\frac{k (k - 1)}{2}} {[\begin{matrix} n k \end{matrix}]}_{p, q} x^{k}$

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

{[\begin{matrix} n k \end{matrix}]}_{p, q} = q^{n - j} {[\begin{matrix} n - 1 k - 1 \end{matrix}]}_{p, q} + p^{j} {[\begin{matrix} n - 1 k \end{matrix}]}_{p, q}

(1.5)

{[\begin{matrix} n k \end{matrix}]}_{p, q} = p^{n - j} {[\begin{matrix} n - 1 k - 1 \end{matrix}]}_{p, q} + q^{j} {[\begin{matrix} n - 1 k \end{matrix}]}_{p, q}

(1.6)

We apply $(p, q)$ -calculus and introduce first the $(p, q)$ -B $´ e$ zier curves and surfaces based on $(p, q)$ -integers which is further generalization of $q$ -B $´ e$ zier curves and surfaces, for example, [18, 19].

The outline of this paper is as follow: Section $2$ introduces a $(p, q)$ -analogue of the Bernstein functions $B_{p, q}^{k, n}$ and their Properties. Section $3$ introduces degree elevation and degree reduction properties for $(p, q)$ -analogue of the Bernstein functions. Section $4$ introduces a de Casteljau type algorithm for $B_{p, q}^{k, n}$ . In Section $5$ we define a tensor product patch based on algorithm $1$ and its geometric properties as well as a degree elevation technique are investigated. Furthermore tensor product of $(p, q)$ -B $´ e$ zier surfaces on $[0, 1] \times [0, 1]$ for $(p, q)$ -Bernstein polynomials are introduced and its properties that is inherited from the univariate case are being discussed.

2 $(p, q)$ -Bernstein functions

The $(p, q)$ -Bernstein functions is as follows

B_{p, q}^{k, n} (t) = \frac{1}{p^{\frac{n (n - 1)}{2}}} {[\begin{matrix} n k \end{matrix}]}_{p, q} p^{\frac{k (k - 1)}{2}} t^{k} (1 - t)_{p, q}^{n - k}, t \in [0, 1]

(2.1)

where

(1 - t)_{p, q}^{n - k} = n - k - 1 \prod s = 0 (p^{s} - q^{s} t)

2.1 Properties of the $(p, q)$ -analogue of the Bernstein functions

Theorem 2.1

The $(p, q)$ -analogue of the Bernstein functions possess the following properties:

(1.) Non-negativity: $B_{p, q}^{k, n} (t) \geq 0$ $k = 0, 1, . . ., n, t \in [0, 1] .$

(2.) Partition of unity:

n \sum k = 0 B_{p, q}^{k, n} (t) = 1, for every t \in [0, 1] .

(3.)Both sided end-point property:

B_{p, q}^{k, n} (0) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1, if k = 0 0, k \neq 0 \end{matrix}

B_{p, q}^{k, n} (1) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1, % if k = n 0, k \neq n \end{matrix}

when $p = 1,$

then both side end point interpolation property holds.

(3.) Reducibility: when $p = 1,$ formula $(2.1)$ reduces to the $q$ -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 $(p, q)$ -analogues of the Bernstein basis functions of degree $3$ with $q = 0.5, p = .8$ . Here we can observe that sum of blending fuctions is always unity and also end point interpolation property holds, when we put $p = 1,$ it turns out to be $q$ -Bernstein basis which is shown in Fig. $??? .$

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

3 Degree elevation and reduction for $(p, q)$ -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 $(p, q)$ -Bernstein functions of degree n is a linear combination of two $(p, q)$ -Bernstein functions of degree $n - 1 :$

B_{p, q}^{k, n} (t) = q^{n - k} p^{k - 1} t B_{p, q}^{k - 1, n - 1} (t) + (p^{n - 1} - p^{k} q^{n - k - 1} t) B_{p, q}^{k, n - 1} (t)

(3.1)

B_{p, q}^{k, n} (t) = p^{n - 1} t B_{p, q}^{k - 1, n - 1} (t) + (q^{k} p^{n - k - 1} - q^{n - 1} t) B_{p, q}^{k, n - 1} (t)

(3.2)

Proof: On using Pascal’s type relation based on $(p, q)$ -integers i.e $(???),$ we get

	$= \frac{1}{p^{\frac{n (n - 1)}{2}}} q^{n - k} {[\begin{matrix} n - 1 k - 1 \end{matrix}]}_{p, q} p^{\frac{k (k - 1)}{2}} t^{k} (1 - t)_{p, q}^{n - k} + \frac{1}{p^{\frac{n (n - 1)}{2}}} p^{k} {[\begin{matrix} n - 1 k \end{matrix}]}_{p, q} p^{\frac{k (k - 1)}{2}} t^{k} (1 - t)_{p, q}^{n - k}$

	$+ \frac{1}{p^{\frac{n (n - 1)}{2}}} p^{k} (p^{n - k - 1} - q^{n - k - 1} t) {[\begin{matrix} n - 1 k \end{matrix}]}_{p, q} p^{\frac{k (k - 1)}{2}} t^{k} (1 - t)_{p, q}^{n - k - 1}$
	$= q^{n - k} p^{k - 1} t B_{p, q}^{k - 1, n - 1} (t) + (p^{n - 1} - p^{k} q^{n - k - 1} t) B_{p, q}^{k, n - 1} (t)$

Thus

B_{p, q}^{k, n} (t) = q^{n - k} p^{k - 1} t B_{p, q}^{k - 1, n - 1} (t) + (p^{n - 1} - p^{k} q^{n - k - 1} t) B_{p, q}^{k, n - 1} (t)

Similarly, if we use $(???)$ , we have

B_{p, q}^{k, n} (t) = p^{n - 1} t B_{p, q}^{k - 1, n - 1} (t) + (q^{k} p^{n - k - 1} - q^{n - 1} t) B_{p, q}^{k, n - 1} (t)

Degree elevation

q^{n - k} p^{k} t B_{p, q}^{k, n} (t) = (1 - \frac{p^{k + 1} {[n - k]}_{p, q}}{{[n + 1]}_{p, q}}) B_{p, q}^{k + 1, n + 1} (t)

(3.3)

(p^{n} - p^{k} q^{n - k} t) B_{p, q}^{k, n} (t) = (\frac{p^{k} {[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) B_{p, q}^{k, n + 1} (t)

(3.4)

Proof:

	$q^{n - k} p^{k} t B_{p, q}^{k, n} (t)$	$= \frac{1}{p^{\frac{n (n - 1)}{2}}} q^{n - k} p^{k} t ({[\begin{matrix} n k \end{matrix}]}_{p, q} p^{\frac{k (k - 1)}{2}} t^{k} (1 - t)_{p, q}^{n - k})$

		$= q^{n - k} \frac{{[k + 1]}_{p, q}}{{[n + 1]}_{p, q}} B_{p, q}^{k + 1, n + 1} (t)$

By some simple calculation, we have

q^{n - k} \frac{{[k + 1]}_{p, q}}{{[n + 1]}_{p, q}} = 1 - \frac{p^{k + 1} {[n - k]}_{p, q}}{} {[n + 1]}_{p, q},

using this result, we get

q^{n - k} p^{k} t B_{p, q}^{k, n} (t) = (1 - \frac{p^{k + 1} {[n - k]}_{p, q}}{{[n + 1]}_{p, q}}) B_{p, q}^{k + 1, n + 1} (t),

similarly on considering,

	$(p^{n} - p^{k} q^{n - k} t) B_{p, q}^{k, n} (t)$	$= (p^{n} - p^{k} q^{n - k} t) (\frac{1}{p^{\frac{n (n - 1)}{2}}} {[\begin{matrix} n k \end{matrix}]}_{p, q} p^{\frac{k (k - 1)}{2}} t^{k} (1 - t)_{p, q}^{n - k})$
		$= \frac{(p^{n} - p^{k} q^{n - k} t)}{(p^{n - k} - q^{n - k} t)} \frac{1}{p^{\frac{n (n - 1)}{2}}} \frac{{[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}} ({[\begin{matrix} n + 1 k \end{matrix}]}_{p, q} p^{\frac{k (k - 1)}{2}} t^{k} (1 - t)_{p, q}^{n + 1 - k})$

finally we get

(p^{n} - p^{k} q^{n - k} t) B_{p, q}^{k, n} (t) = (\frac{p^{k} {[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) B_{p, q}^{k, n + 1} (t)

Theorem 3.2

Each $(p, q)$ -Bernstein function of degree $n$ is a linear combination of two $(p, q)$ -Bernstein functions of degree $n + 1.$

(3.5)

Proof From equation $???, ???$ we can easily get

For $n = 3,$ the blending functions are given by:

B_{p, q}^{0, 3} = \frac{1}{p^{3}} {[\begin{matrix} 30 \end{matrix}]}_{p, q} (1 - t) (p - q t) (p^{2} - q^{2} t)

B_{p, q}^{1, 3} = \frac{1}{p^{3}} {[\begin{matrix} 31 \end{matrix}]}_{p, q} t (1 - t) (p - q t)

B_{p, q}^{2, 3} = \frac{1}{p^{3}} {[\begin{matrix} 32 \end{matrix}]}_{p, q} p t^{2} (1 - t)

B_{p, q}^{3, 3} = {[\begin{matrix} 33 \end{matrix}]}_{p, q} t^{3}

We observed that the both sided end point interpolation property and partition of unity property always holds in case of $(p, q)$ -Bernstein functions.

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

4 $(p, q)$ -Bernstein B $´ e$ zier curves:

Let us define the $(p, q)$ -B $´ e$ zier curves of degree n using the $(p, q)$ -analogues of the Bernstein functions as follows:

P (t; p, q) = n \sum i = 0 P_{i} B_{p, q}^{i, n} (t)

(4.1)

where $P_{i} \in R^{3}$ $(i = 0, 1, . . ., n)$ and $p > q > 0.$ $P_{i}$ are control points. Joining up adjacent points $P_{i},$ $i = 0, 1, 2, . . ., n$ to obtain a polygon which is called the control polygon of $(p, q)$ -Bézier curves.

4.1 Some basic properties of $(p, q)$ -B $´ e$ zier curves.

Theorem 4.1

From the definition, we can derive some basic properties of $(p, q)$ -B $´ e$ zier curves:

1. $(p, q)$ -B $´ e$ zier curves have geometric and affine invariance.
2. $(p, q)$ -B $´ e$ zier curves lie inside the convex hull of its control polygon.
3. The end-point interpolation property: $P (0; p, q) = P_{0}, P (1; p, q) = P_{n} .$
3. Reducibility: when $p = 1,$ formula 4.1 gives the $q$ -B $´ e$ zier curves.

Proof. These properties of $(p, q)$ -B $´ e$ zier curves can be easily deduced from corresponding properties of the $(p, q)$ -analogue of the Bernstein functions.

4.2 Degree elevation for $(p, q)$ -B $´ e$ zier curves

$(p, q)$ -B $´ e$ zier curves have a degree elevation algorithm that is similar to that possessed by the classical B $´ e$ zier curves. Using the technique of degree elevation, we can increase the flexibility of a given curve.

P (t; p, q) = n \sum k = 0 P_{k} B_{p, q}^{k, n} (t)

P (t; p, q) = n + 1 \sum k = 0 P_{k}^{*} B_{p, q}^{i + 1, n} (t),

where

P^{*} = p^{- n} (1 - \frac{p^{k} {[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) P_{k - 1} + p^{k - n} (\frac{{[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) P_{k}

(4.2)

The statement above can be derived using the identities $(???) and (???) .$ Consider

p^{n} P (t; p, q) = (p^{n} - p^{k} q^{n - k} t) P (t; p, q) + p^{k} q^{n - k} t P (t; p, q) .

We obtain

p^{n} P (t; p, q) = n \sum k = 0 (p^{k} \frac{{[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) P_{k}^{0} B_{p, q}^{k, n + 1} (t) + n \sum k = 0 (1 - \frac{p^{k + 1} {[n - k]}_{p, q}}{{[n + 1]}_{p, q}}) P_{k}^{0} B_{p, q}^{k + 1, n + 1} (t)

Now by shifting the limits, we have

p^{n} P (t; p, q) = n + 1 \sum k = 0 (p^{k} \frac{{[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) P_{k}^{0} B_{p, q}^{k, n + 1} (t) + n + 1 \sum k = 0 (1 - \frac{p^{k} {[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) P_{k - 1}^{0} B_{p, q}^{k, n + 1} (t)

where $P_{- 1}^{0}$

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

p^{n} P_{k}^{1} = (p^{k} \frac{{[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) P_{k}^{0} + (1 - \frac{p^{k} {[n + 1 - k]}_{p, q}}{{[n + 1]}_{p, q}}) P_{k - 1}^{0}

In general

where $r = 1, . . . . . n and i = 0, 1, 2, . . ., n + r$

When $p = 1,$ formula 4.4 reduce to the degree evaluation formula of the $q$ -B $´ e$ zier curves. If we let $P = (P_{0}, P_{1}, . . ., P_{n})^{T}$ denote the vector of control points of the initial $(p, q)$ -B $´ e$ zier curve of degree $n,$ and $P^{(1)} = (P_{0}^{*}, P_{1}^{*}, . . ., P_{n + 1}^{*})$ the vector of control points of the degree elevated $(p, q)$ -B $´ e$ zier curve of degree $n + 1,$ then we can represent the degree elevation procedure as:

P^{(1)} = T_{n + 1} P,

where

T_{n + 1} = \frac{1}{[n + 1]_{p, q}} {⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{[n + 1]_{p, q}}{p^{n}} & 0 & \dots & 0 & 0 \frac{([n + 1]_{p, q} - p [1]_{p, q})}{p^{n}} & \frac{[1]_{p, q}}{p^{n - 2}} & \dots & 0 & 0 ⋮ & ⋮ & ⋱ & ⋮ & ⋮ 0 & \dots & \frac{([n + 1]_{p, q} - p^{2} [n - 1]_{p, q})}{p^{- n}} & \frac{[n - 1]_{p, q}}{p^{n - 4}} & 0 0 & 0 & \dots & 0 & \frac{[n + 1]_{p, q}}{p^{n}} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦}_{(n + 2) \times (n + 1)}

For any $l \in N,$ the vector of control points of the degree elevated $(p, q)$ -Bézier curves of degree $n + l$ is: $P^{(l)} = T_{n + l} T_{n + 2} . . . . . . . . T_{n + 1} P .$ As $l ⟶ \infty,$ the control polygon $P^{(l)}$ converges to a $(p, q)$ -B $´ e$ zier curve.

4.3 de Casteljau algorithm:

$(p, q)$ -B $´ e$ zier curves of degree $n$ can be written as two kinds of linear combination of two $(p, q)$ -B $´ e$ zier curves of degree $n - 1,$ and we can get the two selectable algorithms to evaluate $(p, q)$ -B $´ e$ zier curves. The algorithms can be expressed as:

Algorithm 1.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} P_{i}^{0} (t; p, q) \equiv P_{i}^{0} \equiv P_{i} i = 0, 1, 2......, n P_{i}^{r} (t; p, q) = p^{r - 1} t P_{i + 1}^{r - 1} (t; p, q) + (q^{k} p^{r - i - 1} - q^{r - 1} t) P_{i}^{r - 1} (t; p, q) r = 1, . . ., n, i = 0, 1, 2......, n - r ., \end{matrix}

(4.3)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} P_{i}^{0} (t; p, q) \equiv P_{i}^{0} \equiv P_{i} i = 0, 1, 2......, n P_{i}^{r} (t; p, q) = p^{i} q^{r - k - 1} t P_{i + 1}^{r - 1} (t; p, q) + (p^{r - 1} - p^{k} q^{r - i - 1} t) P_{i}^{r - 1} (t; p, q) r = 1, . . ., n, i = 0, 1, 2......, n - r ., \end{matrix}

(4.4)

Then

P (t; p, q) = n - 1 \sum i = 0 P_{i}^{1} (t; p, q) = . . . = \sum P_{i}^{r} (t; p, q) B_{p, q}^{i, n - r} (t) = . . . = P_{0}^{n} (t; p, q)

(4.5)

It is clear that the results can be obtained from Theorem (LABEL:khalidthm). When $p = 1,$ formula (4.3) and (4.4) recover the de Casteljau algorithms of classical $q$ -B $´ e$ zier curves. Let $P^{0} = (P_{0}, P_{1}, . . ., P_{n})^{T}$ , $P^{r} = (P_{0}^{r}, P_{1}^{r}, . . . ., P_{n - r}^{r})^{T},$ then de Casteljau algorithm can be expressed as:

Algorithm 2.

P^{r} (t; p, q) = M_{r} (t; p, q) . . . . M_{2} (t; p, q) M_{1} (t; p, q) P^{0}

(4.6)

where $M_{r} (t; p, q)$ is a $(n - r + 1) \times (n - r + 2)$ matrix and

M_{r} (t; p, q) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} (p^{r - 1} - q^{r - 1} t) & p^{r - 1} t & \dots & 0 & 0 0 & (q p^{r - 2} - q^{r - 1} t) & p^{r - 1} t & 0 & 0 ⋮ & ⋮ & ⋱ & ⋮ & ⋮ 0 & \dots & (q^{n - r - 1} p^{2 r - n - 2} - q^{r - 1} t) & p^{r - 1} t & 0 0 & 0 & \dots & (q^{n - r} p^{2 r - n - 1} - q^{r - 1} t) & p^{r - 1} t \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

M_{r} (t; p, q) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} (p^{r - 1} - q^{r - 1} t) & q^{r - 1} t & \dots & 0 & 0 0 & (p^{r - 1} - p q^{r - 2} t) & p q^{r - 2} t & 0 & 0 ⋮ & ⋮ & ⋱ & ⋮ & ⋮ 0 & \dots & 0 0 & 0 & \dots & (p^{r - 1} - p^{n - r} q^{- n - 1} t) & p^{n - r} q^{- n - 1} t \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

5 Tensor product $(p, q)$ -Bernstein B $´ e$ zier surfaces on $[0, 1] \times [0, 1]$

We define a two-parameter family $P (u, v)$ of tensor product surfaces of degree $m \times n$ as follow:

P (u, v) = m \sum i = 0 n \sum j = 0 P_{i, j} B_{p_{1}, q_{1}}^{i, m} (u) B_{p_{2}, q_{2}}^{j, n} (v), (u, v) \in [0, 1] \times [0, 1],

(5.1)

where $P_{i, j} \in R^{3} (i = 0, 1, . . ., m, j = 0, 1, . . ., n)$ and two real numbers $p_{1} > q_{1} > 0, p_{2} > q_{2} > 0,$ $b_{p_{1}, q_{1}}^{i, m} (u), b_{p_{2}, q_{2}}^{j, n} (v)$ are $(p, q)$ -analogue of Bernstein functions respectively with the parameter $p_{1}, q_{1}$ and $p_{2}, q_{2} .$ We call the parameter surface tensor product $(p, q)$ -B $´ e$ zier surface with degree $m \times n .$ We refer to the $P_{i, j}$ 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 $(p, q)$ -B $´ e$ zier surface.

5.1 Properties

1. Geometric invariance and affine invariance property: Since

m \sum i = 0 n \sum j = 0 B_{p_{1}, q_{1}}^{i, m} (u) B_{p_{2}, q_{2}}^{j, n}

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

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Bézier curves based on Lupaş (p,q)-analogue of Bernstein polynomials in CAGD

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Supersingular Curves With Small Non-integer Endomorphisms

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1 Introduction

2 $(p, q)$ -Bernstein functions

2.1 Properties of the $(p, q)$ -analogue of the Bernstein functions

Theorem 2.1

3 Degree elevation and reduction for $(p, q)$ -Bernstein functions

Theorem 3.1

Theorem 3.2

4 $(p, q)$ -Bernstein B $´ e$ zier curves:

4.1 Some basic properties of $(p, q)$ -B $´ e$ zier curves.

Theorem 4.1

4.2 Degree elevation for $(p, q)$ -B $´ e$ zier curves

4.3 de Casteljau algorithm:

5 Tensor product $(p, q)$ -Bernstein B $´ e$ zier surfaces on $[0, 1] \times [0, 1]$

5.1 Properties

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

Authors

Related Research

Bézier curves based on Lupaş (p,q)-analogue of Bernstein polynomials in CAGD

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Analyzing Midpoint Subdivision

Algorithms and identities for (p,q)-Bézier curves via (p,q)-Blossom

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This week in AI

1 Introduction

2 (p,q)-Bernstein functions

2.1 Properties of the (p,q)-analogue of the Bernstein functions

Theorem 2.1

3 Degree elevation and reduction for (p,q)-Bernstein functions

Theorem 3.1

Theorem 3.2

4 (p,q)-Bernstein B´ezier curves:

4.1 Some basic properties of (p,q)-B´ezier curves.

Theorem 4.1

4.2 Degree elevation for (p,q)-B´ezier curves

4.3 de Casteljau algorithm:

5 Tensor product (p,q)-Bernstein B´ezier surfaces on [0,1]×[0,1]

5.1 Properties

2 $(p, q)$ -Bernstein functions

2.1 Properties of the $(p, q)$ -analogue of the Bernstein functions

3 Degree elevation and reduction for $(p, q)$ -Bernstein functions

4 $(p, q)$ -Bernstein B $´ e$ zier curves:

4.1 Some basic properties of $(p, q)$ -B $´ e$ zier curves.

4.2 Degree elevation for $(p, q)$ -B $´ e$ zier curves

5 Tensor product $(p, q)$ -Bernstein B $´ e$ zier surfaces on $[0, 1] \times [0, 1]$