Bezier curves and surfaces based on modified Bernstein polynomials

11/20/2015
by   Khalid Khan, et al.
Universiti Putra Malaysia
0

In this paper, we use the blending functions of Bernstein polynomials with shifted knots for construction of Bezier curves and surfaces. We study the nature of degree elevation and degree reduction for Bezier Bernstein functions with shifted knots. Parametric curves are represented using these modified Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We get Bezier curve defined on [0, 1] when we set the parameter α=β to the value 0. We also present a de Casteljau algorithm to compute Bernstein Bezier curves and surfaces with shifted knots. The new curves have some properties similar to Bezier curves. Furthermore, some fundamental properties for Bernstein Bezier curves and surfaces are discussed.

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1 Introduction

It was S.N. Bernstein [1] in 1912, who first introduced his famous operators defined for any and for any function where denote the set of all continuous functions on which is equipped with sup-norm

(1.1)

and named it Bernstein polynomials to prove the Weierstrass theorem [9].

Bernstein showed that if , then where represents the uniform convergence. One can find a detailed monograph about the Bernstein polynomials in [10].

Later it was found that Bernstein polynomials possess many remarkable properties and has various applications in areas such as approximation theory [9], numerical analysis, computer-aided geometric design, and solutions of differential equations due to its fine properties of approximation [18].

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 [21] is the classical Bzier curve [2] constructed with the help of Bernstein basis functions. Other works related to different generalization of Bernstein polynomials and bezier curves and surfaces can be found in [3, 4, 5, 7, 8, 11, 12, 13, 14, 15, 16, 18, 19, 20]

In 1968 Stancu [22] showed that the polynomials

(1.2)

converge to continuous function uniformly in [0,1] for each real such that . The polynomials (1.2) are called as a Bernstein-Stancu polynomials.

In 2010, Gadjiev and Gorhanalizadeh [6] introduced the following construction of Bernstein-Stancu type polynomials with shifted knots:

(1.3)

where and are positive real numbers provided . It is clear that for , then polynomials (1.3) turn into the Bernstein-Stancu polynomials (1.2) and if then these polynomials turn into the classical Bernstein polynomials.

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 [7, 8, 17, 18].

The outline of this paper is as follow: Section introduces a modified Bernstein functions with shifted knots and their Properties. Section introduces degree elevation and degree reduction properties for these modified Bernstein functions. Section introduces a de Casteljau 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 with shifted knots are introduced and its properties that is inherited from the univariate case are being discussed.

In next section, we construct basis functions with shifted knots with the help of (1.3).

2 Bernstein functions with shifted knots

The Bernstein functions with shifted knots is defined as follows

(2.1)

where and are positive real numbers provided .

2.1 Properties of the Bernstein functions with shifted knots

Theorem 2.1

The Bernstein functions with shifted knots possess the following properties:

(1.) Non-negativity:

(2.)Partition of unity:


(3.) End-point interpolation property holds:


clearly both sided end point interpolation property holds.

(4.) Reducibility: when formula reduces to the classical Bernstein bases on .

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

Figure 1: ‘Cubic Bezier blending functions with shifted knots’
Figure 2: ‘Cubic Bezier blending functions’

Fig. 1 shows the modified Bernstein basis functions of degree with shifted knots for Here we can observe that sum of blending fuctions is always unity and also satisfies end point interpolation property. In case it turns out to be classical Bernstein basis on which is shown in Fig.

Apart from the basic properties above, the Bernstein functions with shifted knots also satisfy some recurrence relations, as for the classical Bernstein basis.

3 Degree elevation and reduction for Bernstein functions with shifted knots

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 (3.1),(3.2) and Theorem (3.1) are useful.

Degree elevation

(3.1)

and

(3.2)

Proof:

Consider

Similarly for

Theorem 3.1

Each Bernstein functions with shifted knots of degree n is a linear combination of two Bernstein functions with shifted knots of degree

(3.3)

where

and are positive real numbers satisfying .

Proof:

on using equation we can easily get

Theorem 3.2

Each Bernstein functions with shifted knots of degree n is a linear combination of two Bernstein functions with shifted knots of degree

(3.4)

where

and are positive real numbers satisfying .

Proof On using Pascal’s type relation i.e , we get

Theorem 3.3

The end-point property of derivative:

(3.5)
(3.6)

i.e. Bernstein-Bzier curves with shifted knots are tangent to fore-and-aft edges of its control polygon at end points.

Proof: Let

or


then on differentiating both hand side with respect to ‘t’, we have


Let

then


which implies


Now

and

Similarly after some computation, we have

3.1 Degree elevation for Bzier curves with shifted knots

Bzier curves with shifted knots 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.

where

(3.7)

The statement above can be derived from Theorem 3.1. When formula 3.7 reduce to the degree evaluation formula of the Bzier curves. If we let

denote the vector of control points of the initial B

zier 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:

where

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

3.2 de Casteljau algorithm:

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

Algorithm 1.

(3.8)

Then

(3.9)

It is clear that the results can be obtained from Theorem (3.2). When formula (3.8) and (3.9) recover the de Casteljau algorithms of classical Bzier curves. Let , then de Casteljau algorithm can be expressed as:

Algorithm 2.

(3.10)

where is a matrix and

4 Tensor product Bzier surfaces with shifted knots on

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

(4.1)

where and are modified Bernstein functions respectively. 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.

4.1 Properties

1. Geometric invariance and affine invariance property: Since

(4.2)

is an affine combination of its control points.

2. Convex hull property: is a convex combination of and lies in the convex hull of its control net.

3. Isoparametric curves property: The isoparametric curves and of a tensor product Bzier surface are respectively the Bzier curves with shifted knots of degree and degree namely,

The boundary curves of are evaluated by , , and .

4. Corner point interpolation property: The corner control net coincide with the four corners of the surface. Namely,

5. Reducibility: When formula (4.1) reduces to a classical tensor product Bzier patch.

4.2 Degree elevation and de Casteljau algorithm

Let be a tensor product Bzier surface with shifted knots of degree As an example, let us take obtaining the same surface as a surface of degree Hence we need to find new control points such that

(4.3)

Let

Then

(4.4)

which can be written in matrix form as

The de Casteljau algorithms are also easily extended to evaluate points on a Bzier surface. Given the control net

(4.5)

or

When one can directly use the algorithms above to get a point on the surface. When to get a point on the surface after applications of formula (4.5), we perform formula (3.10) for the intermediate point

Note: We get classical Bzier curves and surfaces for when we set the parameter

5 Future work

In near future, we will construct -analogue of Bzier curves and surfaces with shifted knots and we will also study de Casteljau algorithm and degree evaluation properties for curves and surfaces.

References