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:
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).
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
- [1] S. N. Bernstein, Constructive proof of Weierstrass approximation theorem, Comm. Kharkov Math. Soc. (1912)
- [2] P.E. Bzier, Numerical Control-Mathematics and applications, John Wiley and Sons, London, 1972.
- [3] Cetin Disibuyuk and Halil Oruc, Tensor Product -Bernstein Polynomials, BIT Numerical Mathematics, Springer 48 (2008) 689-700.
- [4] Cetin Disibuyuk. ”Tensor Product −Bernstein Bzier Patches”, Lecture Notes in Computer Science, 2009.
- [5] Rida T. Farouki, V. T. Rajan, Algorithms for polynomials in Bernstein form, Computer Aided Geometric Design, Volume 5, Issue 1, June 1988 .
- [6] A.D. Gadjiev, A.M. Ghorbanalizadeh, Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables, Appl. Math. Comput. 216 (3) (2010) 890-901.
- [7] Khalid Khan, D.K. Lobiyal, Adem Kilicman, A de Casteljau Algorithm for Bernstein type Polynomials based on -integers, arXiv 1507.04110.
- [8] Khalid Khan, D.K. Lobiyal, Bezier curves based on Lupas -analogue of Bernstein polynomials in CAGD, arXiv:1505.01810.
- [9] P. P. Korovkin, Linear operators and approximation theory, Hindustan Publishing Corporation, Delhi, 1960.
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