Approximation of conic sections by weighted Lupaş post-quantum Bézier curves

Asif Khan; Mohammad Iliyas; Khalid Khan; Mohammad Mursaleen

doi:10.1515/dema-2022-0016

Article Open Access

Approximation of conic sections by weighted Lupaş post-quantum Bézier curves

Asif Khan , Mohammad Iliyas , Khalid Khan and Mohammad Mursaleen

Published/Copyright: August 3, 2022

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From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

This paper deals with weighted Lupaş post-quantum Bernstein blending functions and Bézier curves constructed with the help of bases via ( p , q ) -integers. These blending functions form normalized totally positive bases. Due to the rational nature of weighted Lupaş post-quantum Bézier curves and positive weights, they help in investigating from geometric point of view. Their degree elevation properties and de Casteljau algorithm have been studied. It has been shown that quadratic weighted Lupaş post-quantum Bézier curves can represent conic sections in two-dimensional plane. Graphical analysis has been presented to discuss geometric interpretation of weight and conic section representation by weighted Lupaş post-quantum Bézier curves. This new generalized weighted Lupaş post-quantum Bézier curve provides better approximation and flexibility to a particular control point as well as control polygon due to extra parameter p and q in comparison to classical rational Bézier curves, Lupaş q -Bézier curves and weighted Lupaş q -Bézier curves.

Keywords: post-quantum integers; weighted Lupaş post-quantum Bézier curve; rational Bézier curve; normalized totally positive basis; shape parameter; conic sections

MSC 2010: 65D17; 41A10; 41A25; 41A36

1 Introduction and preliminaries

Computer aided geometric design (CAGD) deals with the mathematical description of shape and computational aspects of geometric objects. It emphasizes on mathematics which is compatible with computers in shape designing. Study of parametric curves and surfaces is basically based on control points and control nets, respectively. CAGD has applications in industrial as well as applied mathematics. The tools which are used in CAGD are related to Computer Science and Mathematics. It has applications in other areas such as Computer graphics, Approximation theory, Numerical Analysis, Data Structures, Computer Algebra etc. Due to its mathematical nature, CAGD has been used in other engineering fields along with Computer Science.

To mimic the shape of curves and surfaces better, one needs parameters which can provide flexibility. In this sequel, applications of post-quantum calculus play an important role in CAGD.

In 1912, Bernstein [1] introduced the famous Bernstein polynomial (operator) for any bounded function f : [ 0 , 1 ] → R as follows:

(1.1) B μ ( f ; v ) = ∑ l = 0 μ b μ , l ( v ) f l μ , v ∈ [ 0 , 1 ] .

It was proved that the sequence of operators B μ : C [ 0 , 1 ] → C [ 0 , 1 ] for any μ ∈ N and f ∈ C [ 0 , 1 ] converges uniformly to f on [0,1], where

(1.2) b μ , l ( v ) = μ l v l ( 1 − v ) μ − l ( l = 0 , 1 , 2 , … , μ ) .

In CAGD, basis (1.2) of these Bernstein polynomials (1.1) has a great role in preserving the shape of the curves or surfaces. Graphic design programs such as Adobe’s illustrator and photoshop inkspaces utilize Bernstein polynomials to form what are known as Bézier curves.

Preserving the shape of the parametric curve or surface depends on basis functions used in constructing the curves. Totally positive basis, helps in investigating nice shape preserving properties by virtue of totally positive matrices and its variation diminishing properties. A parametric curve is constructed with blending basis functions which form a totally positive system which mimics the shape or design of its control polygon [2,3].

More details about total positivity of basis functions and the shape of the curves can be found in [4,5,6, 7,8].

The rapid development of quantum calculus or q -calculus [9] has given birth to new generalizations of Bernstein polynomials based on q -integers [7,10].

In 1987, Lupaş [10] proposed the first q -Bernstein operators (rational) for μ ∈ N and l ∈ { 0 , 1 , 2 , … , μ } as follows:

(1.3) L μ , q ( f ; v ) = ∑ l = 0 μ f [ l ] q [ μ ] q μ l q q l ( l − 1 ) 2 v l ( 1 − v ) μ − l ∏ j = 1 μ { ( 1 − v ) + q j − 1 v }

and studied its approximating and shape-preserving properties due to extra parameter q .

In 1996, Phillips introduced another positive linear q -operator (polynomials) B μ , q : C [ 0 , 1 ] → C [ 0 , 1 ] for μ ∈ N and l ∈ { 0 , 1 , 2 , … , μ } to study approximation properties and shape preserving properties [11].

(1.4) B μ , q ( f ; v ) = ∑ l = 0 μ μ l q v l ∏ s = 0 μ − l − 1 ( 1 − q s v ) f [ l ] q [ μ ] q , v ∈ [ 0 , 1 ] , f ∈ C [ 0 , 1 ] .

Later on, in 2003, Oruç and Phillips [7] utilized the basis functions of Phillips q -Bernstein operator (1.4) for construction of Phillips q -Bézier curves and applied the concept of total positivity to investigate the degree elevation, degree reduction and shape preserving properties of the curve.

In 2016, Han et al. [12] introduced weighted Lupaş q -Bézier curves by attaching positive weights to Lupaş q -Bézier curves to represent conic sections exactly and studied also shape preserving properties of this curve in view of normalized totally positive (NTP) basis.

Extension of quantum calculus is post-quantum calculus. Recently, in the field of approximation theory [13] and CAGD [8], ( p , q ) -calculus has been emerging as a new area. To mimic the shape of curves and surfaces better one needs parameters which can provide flexibility. In this sequel, applications of post-quantum calculus play an important role in CAGD, see [8].

First of all, Mursaleen et al. [13] initiated ( p , q ) -calculus in approximation theory and presented post-quantum Bernstein operators using post-quantum integers for 0 < q < p ≤ 1 and f ∈ C [ 0 , 1 ] as follows:

(1.5) B μ , p , q ( f ; v ) = 1 p μ ( μ − 1 ) 2 ∑ l = 0 μ μ l p , q p l ( l − 1 ) 2 v l ∏ s = 0 μ − l − 1 ( p s − q s v ) f [ l ] p , q p l − μ [ μ ] p , q , v ∈ [ 0 , 1 ] .

It is easy to note that putting p = 1 in post-quantum-Bernstein operators (1.5) reduces to Phillips q -Bernstein operators (1.4). They also introduced and studied approximation properties based on post-quantum analogue of Bleimann-Butzer-Hahn operators in [26]. This extra parameter p has some advantages [8,14]. For other relevant works, one can refer to [2,5,6,15,16, 17,18,19, 20,21,22, 23,24,25, 26,27].

Before going into our main results we give some basic concepts related to our work about post-quantum calculus.

The post-quantum integers [ μ ] p , q for any p > 0 and q > 0 are defined by

[ μ ] p , q = p μ − 1 + p μ − 2 q + p μ − 3 q 2 + … + p q μ − 2 + q μ − 1 = p μ − q μ p − q , if p ≠ q ≠ 1 , μ p μ − 1 , if p = q ≠ 1 , [ μ ] q , if p = 1 , μ , if p = q = 1 ,

where [ μ ] q is the q -integers and μ = 0 , 1 , 2 , … .

The post-quantum-binomial expansion is given by the following expressions:

( a u + b v ) p , q μ ≔ ∑ l = 0 μ p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 μ l p , q a μ − l b l u μ − l v l , ( u + v ) p , q μ = ( u + v ) ( p u + q v ) ( p 2 u + q 2 v ) ⋯ ( p μ − 1 u + q μ − 1 v ) ,

where post-quantum-binomial coefficients are defined as

μ l p , q = [ μ ] p , q ! [ l ] p , q ! [ μ − l ] p , q ! .

For more details on post-quantum calculus, see [13,28].

Post-quantum analogues of Euler’s identity and Pascal’s relation are as follows:

( 1 − v ) p , q μ = ∏ s = 0 μ − 1 ( p s − q s v ) = ( 1 − v ) ( p − q v ) ( p 2 − q 2 v ) … ( p μ − 1 − q μ − 1 v ) = ∑ l = 0 μ ( − 1 ) l p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 μ l p , q v l ,

(1.6) μ l p , q = q μ − l μ − 1 l − 1 p , q + p l μ − 1 l p , q ,

(1.7) μ l p , q = p μ − l μ − 1 l − 1 p , q + q l μ − 1 l p , q .

Recently, Khalid and Lobiyal [8] introduced Lupaş post-quantum analogue of Bernstein polynomial and used their basis blending functions for constructing Lupaş post-quantum (rational)-Bézier curves and surfaces based on post-quantum integers for all p > 0 and q > 0 , which is further generalization of Lupaş q -Bézier curves and surfaces [6,7].

Motivated from the above discussion, we utilize the concepts of weighted Lupaş q -Bézier curves [12] and post-quantum calculus. We construct weighted Lupaş post-quantum Bernstein blending functions by adjoining positive weights to basis of Lupaş post-quantum analogue of Bernstein operators [8]. These weighted Lupaş post-quantum Bernstein blending functions are used to construct weighted Lupaş ( p , q ) -Bézier curves based on two parameters p and q which are rational in nature. In view of NTP bases, shape-preserving properties of the new curves are studied.

The paper is arranged as follows. We define weighted Lupaş post-quantum Bernstein blending functions over [ 0 , 1 ] in Section 2 and prove that they form basis of a vector space of rational functions with the same denominator denoted by R μ , which are NTP also. In Section 3, with the help of weighted Lupaş post-quantum Bernstein blending functions, we construct weighted Lupaş post-quantum Bézier curves and discuss their fundamental properties, end point property of derivative, degree elevation and de Casteljau algorithms. Section 4 deals with the geometric interpretation of weight and representation of conic sections by quadratic weighted Lupaş post-quantum Bézier curves with the help of graphical demonstration. In Section 5, we give some important remarks and another way of representation of weighted Lupaş post-quantum Bézier curves. The effects of weights and post-quantum integers on the shape of the curves and surfaces have been discussed.

2 Weighted Lupaş post-quantum Bernstein basis

If we add positive weights to Lupaş ( p , q )-Bernstein blending functions, then we obtain weighted Lupaş post-quantum Bernstein functions.

Definition 2.1

For any given real numbers p > 0 and q > 0 , v ∈ [ 0 , 1 ] and any given real numbers w l > 0 ( l = 0 , 1 , 2 , … , μ ), weighted Lupaş post-quantum Bernstein basis functions of degree μ are defined as

(2.1) s μ , l ( v ; p , q ) = w l γ μ , l ( v ; p , q ) ∑ l = 0 μ w l γ μ , l ( v ; p , q ) , l = 0 , 1 , 2 , … , μ ,

where

γ μ , l ( v ; p , q ) = μ l p , q p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 v l ( 1 − v ) μ − l , l = 0 , 1 , 2 , … , μ .

Figure 1(a) shows weighted Lupaş post-quantum cubic Bézier Bernstein blending functions for q = 1.5 , p = 4.5 and weights w 1 = 1 , w 2 = 2 , w 3 = 4.5 , w 4 = 6 . Figure 1(b) shows weighted Lupaş post-quantum cubic Bézier Bernstein blending functions for q = 25 , p = 13 and weights w 1 = 1 , w 2 = 2 , w 3 = 4 , w 4 = 3 .

Figure 1

Weighted Lupaş post-quantum cubic Bézier Bernstein blending functions. (a) q = 1.5 , p = 4.5 . (b) q = 25 , p = 13 .

By using properties of Lupaş post-quantum Bernstein function [8], some properties of weighted Lupaş post-quantum Bernstein blending functions are as follows:

Non-negative: s μ , l ( v ; p , q ) ≥ 0 , l = 0 , 1 , 2 , … , μ , v ∈ [ 0 , 1 ] .
Partition of unity: ∑ l = 0 μ s μ , l ( v ; p , q ) = 1 , v ∈ [ 0 , 1 ] .
End-point property:
s μ , l ( 0 ; p , q ) = 1 , if l = 0 , 0 , l ≠ 0 , s μ , l ( 1 ; p , q ) = 1 , if l = μ , 0 , l ≠ μ .
Post-quantum inverse symmetry: When w l = w μ − l , for l = 0 , 1 , 2 , … , μ , we have
s μ , μ − l ( v ; p , q ) = s μ , μ − l v ; 1 q , 1 p = s μ , l 1 − v ; 1 p , 1 q .
.
Reducibility: If each weight w l = w ≠ 0 : ( for all l = 0 , 1 , 2 , … , μ ) , weighted Lupaş post-quantum Bernstein blending function (2.1) turns into Lupaş post-quantum analogue of Bernstein function [8]; if we put p = 1 , formula (2.1) converts into weighted Lupaş q Bernstein functions [12] and when p = q = 1 , (2.1) changes into classical rational Bernstein functions.

Remark 2.2

Let A μ ( v ; p , q ) = ∑ l = 0 μ w l γ μ , l ( v ; p , q ) , i.e., denominator of weighted Lupaş post-quantum Bernstein function of degree μ and b μ , l ( v ) = μ l v l ( 1 − v ) μ − l be the classical Bernstein basis functions of degree μ , after simplification of the following equation w l γ μ , l ( v ; p , q ) = α μ , l b μ , l ( v ) . We obtain

α μ , l = w l μ l p , q μ l p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 , l = 0 , 1 , 2 , … , μ .

The weighted Lupaş post-quantum Bernstein function of degree μ can also be represented as

s μ , l ( v : p , q ) = α μ , l b μ , l ( v ) A μ ( v ; p , q ) , l = 0 , 1 , 2 , … , μ .

Let us consider vector space P μ of all polynomials of degree less than or equal to μ + 1 . Observe that classical Bernstein basis functions b μ , 0 ( v ) , b μ , 1 ( v ) , … , b μ , μ ( v ) span μ + 1 dimensional space P μ . In the similar fashion, it is easy to note that weighted Lupaş post-quantum Bernstein functions form the basis of vector space R μ of rational functions of dimension μ + 1 with the common denominator A μ ( v ; p , q ) , i.e.,

span { s μ , l ( v : p , q ) ∣ l = 0 , 1 , 2 , … , μ } = P ( v ) A μ ( v ; p , q ) ∣ P ( v ) ∈ P μ = R μ .

2.1 Total positivity of weighted Lupaş post-quantum Bernstein functions

Let us recall some basics from [12,29] about totally positive matrix, TP basis and totally positive function sequences.

Definition 2.3

A matrix M with real entries is called totally positive (strictly totally positive), if all minors of M are nonnegative (positive), i.e,

M l 1 , l 2 , ⋯ , l r j 1 , j 2 , ⋯ , j r = det m l 1 , j 1 … m l 1 , j r … … … m l r , j 1 … m l r , j r ≥ 0 ( > 0 ) ,

for all l 1 < l 2 < ⋯ < l r and j 1 < j 2 < ⋯ < j r .

Instead of examining all minors of matrices, Karlin [30] pointed out a criterion to check those minors which are created by consecutive columns and rows to judge strictly totally positivity of real matrix M . In other words, a matrix with real entries M is called strictly totally positive if

M l , l + 1 , l + 2 , ⋯ , l + r j , j + 1 , j + 2 , ⋯ , j + r > 0 ,

for all l , j and r . For more details about totally positive matrices, see [31,32].

Definition 2.4

If for any given points 0 < v 0 < v 1 < v 2 < ⋯ < v μ on interval I , the collocation matrix ( f j ( v l ) ) l , j = 0 μ constructed by sequence of real-valued functions ( f 0 ( v ) , f 1 ( v ) , … , f μ ( v ) ) is totally positive. This sequence of real-valued functions ( f 0 ( v ) , f 1 ( v ) , … , f μ ( v ) ) is called totally positive on an interval I . If these totally positive functions ( f 0 ( v ) , f 1 ( v ) , … , f μ ( v ) ) are linearly independent, then they are known as TP basis; the TP basis satisfying ∑ l = 0 μ f l ( v ) = 1 , are known as NTP basis.

In the following theorem, we discuss about total positivity of weighted Lupaş post-quantum Bernstein basis functions.

Theorem 2.1

For the rational function space R μ , weighted Lupaş post-quantum analogue of Bernstein basis functions s μ , 0 ( v ; p , q ) , s μ , 1 ( v ; p , q ) , … , s μ , μ ( v ; p , q ) form a normalized basis, which is totally positive for all p > 0 , q > 0 and v ∈ [ 0 , 1 ] .

Proof

For any points 0 ≤ v 0 < v 1 < ⋯ < v μ ≤ 1 , let M μ be the collocation matrix s μ , j ( v l ; p , q ) l , j = 0 μ , that is,

M μ = s μ , 0 ( v 0 ; p , q ) s μ , 1 ( v 0 ; p , q ) ⋯ s μ , μ ( v 0 ; p , q ) s μ , 0 ( v 1 ; p , q ) s μ , 1 ( v 1 ; p , q ) ⋯ s μ , μ ( v 1 ; p , q ) ⋮ ⋮ ⋮ ⋮ s μ , 0 ( v μ ; p , q ) s μ , 1 ( v μ ; p , q ) ⋯ s μ , μ ( v μ ; p , q ) .

To show s μ , 0 ( v ; p , q ) , s μ , 1 ( v ; p , q ) , … , s μ , μ ( v ; p , q ) are TP basis, we need to show that M μ is totally positive matrix. We use induction on μ to prove M μ is totally positive matrix for any μ ∈ N . For μ = 1 , M 1 = s μ , 0 ( v 0 ; p , q ) s μ , 1 ( v 0 ; p , q ) s μ , 0 ( v 1 ; p , q ) s μ , 1 ( v 1 ; p , q ) , det ( M 1 ) = w 0 w 0 ( 1 − v 0 ) + w 1 v 0 ⋅ w 1 w 0 ( 1 − v 1 ) + w 1 v 1 ⋅ ( v 1 − v 0 ) ≥ 0 .

Observe that all the elements of M 1 are nonnegative and det ( M 1 ) ≥ 0 . Hence, the result holds for μ = 1 .

Suppose that matrix M μ is totally positive for some μ ≥ 2 , i.e., all its minors and elements are nonnegative. We prove the matrix M μ + 1 is totally positive. Note that elements and the minors of order k ( 2 ≤ k ≤ μ ) of the matrix M μ + 1 are nonnegative. We only need to show det ( M μ + 1 ) ≥ 0 . det ( M μ + 1 ) = η 1 ⋅ η 2 ⋅ η 3 ⋅ det ( D ) , where η 1 = ∏ j = 0 μ + 1 w j ∑ l = 0 μ + 1 w l γ μ + 1 , l ( v ; p , q ) ≥ 0 ,

η 2 = μ + 1 0 p , q ⋅ μ + 1 1 p , q ⋅ … ⋅ μ + 1 μ + 1 p , q ≥ 0 ,

η 3 = ( p q ) 0 ⋅ ( p q ) 1 ⋅ … ⋅ ( p q ) μ ( μ + 1 ) 2 ≥ 0 ,

and

D = ( 1 − v 0 ) μ + 1 v 0 ( 1 − v 0 ) μ ⋯ v 0 μ + 1 ( 1 − v 1 ) μ + 1 v 1 ( 1 − v 1 ) μ ⋯ v 1 μ + 1 ⋮ ⋮ ⋮ ⋮ ( 1 − v μ + 1 ) μ + 1 v μ + 1 ( 1 − v μ + 1 ) μ ⋯ v μ + 1 μ + 1 .

Note that matrix D here denote the collocation matrix of basis functions ( 1 − v ) μ + 1 , v ( 1 − v ) μ , … , v μ + 1 about any points 0 ≤ v 0 < v 1 < v 2 < ⋯ < v μ < v μ + 1 ≤ 1 , which is totally positive also, this implies that det ( D ) ≥ 0 . Thus, det ( M μ + 1 ) ≥ 0 , this implies that matrix M μ + 1 becomes totally positive matrix also. Finally, for every μ ∈ N the collocation matrix M μ is totally positive matrix. Since the basis s μ , l ( v ; p , q ) satisfies partition of unity, ⇒ ∑ l = 0 μ s μ , l ( v ; p , q ) = 1 , this means weighted Lupaş post-quantum Bernstein basis functions s μ , 0 ( v ; p , q ) , s μ , 1 ( v ; p , q ) , … , s μ , μ ( v ; p , q ) are the basis of the vector space R μ of rational functions, which are NTP too.□

Remark 2.5

From the reducibility of weighted Lupaş post-quantum Bernstein functions of degree μ and Theorem 2.1, Lupaş post-quantum ( p , q ) Bernstein blending functions come out to be NTP basis. Also the Bézier curve constructed using these NTP basis functions will have variation diminishing property [8,12,33]. Apart from this, these curves preserve convexity and monotonicity. When all the weights w l = w ≠ 0 : ( l = 0 , 1 , 2 , … , μ ) ; weighted Lupaş post-quantum analogue of Bernstein functions of degree μ turns out to be Lupaş post-quantum Bernstein functions of degree μ . When p = 1 , weighted Lupaş post-quantum analogue of Bernstein functions of degree μ is reduced to weighted Lupaş quantum analogue of Bernstein functions of degree μ [12].

Now we discuss weighted Lupaş post-quantum Bézier curves and their shape preserving properties in the following section.

3 Weighted Lupaş post-quantum Bézier curves

Weighted Lupaş post-quantum Bézier curve of degree μ constructed with the help of weighted Lupaş post-quantum Bernstein blending basis function is defined as follows:

ℛ ( v ; p , q ) = ∑ l = 0 μ P l s μ , l ( v ; p , q ) , 0 ≤ v ≤ 1 ,

where P l ∈ R 3 ( l = 0 , 1 , 2 , … , μ ) are control points, p > 0 and q > 0 , and real positive numbers w 0 , w 1 , … , w μ are the weights.

3.1 Properties

Some properties of Lupaş post-quantum Bézier curve [8] which have been inherited by weighted Lupaş post-quantum Bézier curve are as follows:

Weighted Lupaş post-quantum Bézier curves satisfy geometrical invariance and affine invariance property.
Weighted Lupaş post-quantum Bézier curves lie in Convex hull of its control polygon.
Weighted Lupaş post-quantum Bézier curves interpolate the end points, that is, ℛ ( 0 ; p , q ) = P 0 , ℛ ( 1 ; p , q ) = P μ .
Post-quantum inverse symmetry: When w l = w μ − l and when we reverse the order of control points, weighted Lupaş post-quantum Bézier curve will coincide with weighted Lupaş post-quantum Bézier curve with the parameters p and q replaced by 1 p and 1 q , respectively.
Reducibility: when all the weights w l = w ≠ 0 ( l = 0 , 1 , 2 , … , μ ) , weighted Lupaş post-quantum Bézier curve turns into Lupaş post-quantum Bézier curve [8]; when p = 1 weighted Lupaş post-quantum Bézier curve is reduced to weighted Lupaş q -Bézier curve [12], and when q = 1 , it changes into classical rational-Bézier curve.

Theorem 3.1

The derivative at end points and its properties:

ℛ ′ ( 0 : p , q ) = [ μ ] p , q w 1 ( P 1 − P 0 ) w 0 p μ − 1 ,

ℛ ′ ( 1 : p , q ) = [ μ ] p , q w μ − 1 ( P μ − P μ − 1 ) w μ q μ − 1 .

Proof

ℛ ( v ; p , q ) = ∑ l = 0 μ P l s μ , l ( v ; p , q ) = ∑ l = 0 μ w l γ μ , l ( v ; p , q ) P l ∑ l = 0 μ w l γ μ , l ( v ; p , q ) = Q ( v ; p , q ) W ( v ; p , q ) , where γ μ , l ( v ; p , q ) = μ l p , q p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 v l ( 1 − v ) μ − l , l = 0 , 1 , 2 , … , μ ,

γ μ , l ( 0 ; p , q ) = p μ ( μ − 1 ) 2 , if l = 0 , 0 , l ≠ 0 , γ μ , l ( 1 ; p , q ) = q μ ( μ − 1 ) 2 , if l = μ , 0 , l ≠ μ .

Since

ℛ ( v ; p , q ) W ( v ; p , q ) = Q ( v ; p , q ) .

Differentiating the above expression with respect to “ v ,” we have

ℛ ′ ( v ; p , q ) W ( v ; p , q ) + ℛ ( v ; p , q ) W ′ ( v ; p , q ) = Q ′ ( v ; p , q ) .

γ μ , l ′ ( v ; p , q ) = [ μ ] p , q [ l ] p , q μ − 1 l − 1 p , q p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 l v l − 1 ( 1 − v ) μ − l − [ μ ] p , q [ μ − l ] p , q μ − 1 l p , q p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 ( μ − l ) v l ( 1 − v ) μ − l − 1 = [ μ ] p , q [ l ] p , q q l − 1 l γ μ − 1 , l − 1 ( v ; p , q ) − [ μ ] p , q [ μ − l ] p , q p μ − l − 1 ( μ − l ) γ μ − 1 , l ( v ; p , q ) = c μ , l γ μ − 1 , l − 1 ( v ; p , q ) − d μ , μ − l γ μ − 1 , l ( v ; p , q ) ,

where

c μ , l = [ μ ] p , q [ l ] p , q q l − 1 l , d μ , μ − l = [ μ ] p , q [ μ − l ] p , q p μ − l − 1 ( μ − l ) .

Q ′ ( v ; p , q ) = ∑ l = 0 μ w l γ μ , l ′ ( v ; p , q ) P l = ∑ l = 0 μ − 1 ( w l + 1 C μ , l + 1 P l + 1 − w l d μ , μ − l P l ) γ μ − 1 , l ( v ; p , q ) , W ′ ( v ; p , q ) = ∑ l = 0 μ w l γ μ , l ′ ( v ; p , q ) = ∑ l = 0 μ − 1 ( w l + 1 C μ , l + 1 − w l d μ , μ − l ) γ μ − 1 , l ( v ; p , q ) .

Thus, we obtain our results

ℛ ′ ( 0 : p , q ) = [ μ ] p , q w 1 ( P 1 − P 0 ) w 0 p μ − 1 , ℛ ′ ( 1 : p , q ) = [ μ ] p , q w μ − 1 ( P μ − P μ − 1 ) w μ q μ − 1 . □

Theorem 3.2

Weighted Lupaş post-quantum Bézier curve will have no more intersection with the line as the line have with control polygon, thus weighted Lupaş post-quantum Bézier curves have variation diminishing property. Also, weighted Lupaş post-quantum Bézier curves preserve the monotonicity and convexity.

3.2 Degree elevation of weighted Lupaş post-quantum Bézier curves

One can increase the flexibility as well as local control by using the technique of degree elevation.

Let

ℛ ( v ; p , q ) = ∑ l = 0 μ P l s μ , l ( v ; p , q ) = ∑ l = 0 μ w l γ μ , l ( v ; p , q ) P l ∑ l = 0 μ w l γ μ , l ( v ; p , q ) , 0 ≤ v ≤ 1 ,

where

γ μ , l ( v ; p , q ) = μ l p , q p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 v l ( 1 − v ) μ − l , l = 0 , 1 , 2 , … , μ ,

then

ℛ ( v ; p , q ) = ∑ l = 0 μ + 1 w l ∗ γ μ + 1 , l ( v ; p , q ) P l ∗ ∑ l = 0 μ + 1 w l ∗ γ μ + 1 , l ( v ; p , q ) , l = 0 , 1 , 2 , … , μ + 1 ,

where

w l ∗ = 1 − p l [ μ + 1 − l ] p , q [ μ + 1 ] p , q w l − 1 + p l [ μ + 1 − l ] p , q [ μ + 1 ] p , q w l , P l ∗ = 1 − p l [ μ + 1 − l ] p , q [ μ + 1 ] p , q w l − 1 P l − 1 + p l [ μ + 1 − l ] p , q [ μ + 1 ] p , q w l P l / w l ∗ .

3.3 de Casteljau algorithm for weighted Lupaş post-quantum Bézier curves

Weighted Lupaş post-quantum Bézier curve of degree μ can be expressed as linear combination of two kinds of weighted Lupaş post-quantum Bézier curve of degree μ − 1 . The de Casteljau algorithm is written as follows:

Algorithm 1.

(3.1) w l 0 ( v ; p , q ) ≡ w l 0 ≡ w l l = 0 , 1 , 2 … … , μ , w l r ( v ; p , q ) = q μ − r v w l + 1 r − 1 ( v ; p , q ) + p μ − r ( 1 − v ) w l r − 1 ( v ; p , q ) , r = 1 , … , μ , l = 0 , 1 , 2 … … , μ − r ,

(3.2) P l 0 ( v ; p , q ) ≡ P l 0 ≡ P l l = 0 , 1 , 2 … … , μ , P l r ( v ; p , q ) = ( q μ − r v w l + 1 r − 1 ( v ; p , q ) P l + 1 r − 1 ( v ; p , q ) + p μ − r ( 1 − v ) w l r − 1 ( v ; p , q ) P l r − 1 ( v ; p , q ) ) / w l r ( v ; p , q ) , r = 1 , … , μ , l = 0 , 1 , 2 … … , μ − r ,

(3.3) w l 0 ( v ; p , q ) ≡ w l 0 ≡ w l l = 0 , 1 , 2 … … , μ , w l r ( v ; p , q ) = p μ − l − r q l v w l + 1 r − 1 ( v ; p , q ) + p μ − l − r q l ( 1 − v ) w l r − 1 ( v ; p , q ) , r = 1 , … , μ , l = 0 , 1 , 2 … … , μ − r ,

P l 0 ( v ; p , q ) ≡ P l 0 ≡ P l l = 0 , 1 , 2 … … , μ , P l r ( v ; p , q ) = ( p μ − l − r q l v w l + 1 r − 1 ( v ; p , q ) P l + 1 r − 1 ( v ; p , q ) + p μ − l − r q l ( 1 − v ) w l r − 1 ( v ; p , q ) P l r − 1 ( v ; p , q ) ) / w l r ( v ; p , q ) , r = 1 , … , μ , l = 0 , 1 , 2 … … , μ − r , (3.4)

then

ℛ ( v ; p , q ) = ∑ l = 0 μ − 1 w l 1 γ μ − 1 , l ( v ; p , q ) P l 1 ∑ l = 0 μ w l 1 γ μ − 1 , l ( v ; p , q ) = ⋯ = ℛ ( v ; p , q ) = ∑ l = 0 μ − k w l k γ μ − k , l ( v ; p , q ) P l k ∑ l = 0 μ − k w l k γ μ − k , l ( v ; p , q ) = ⋯ = P μ 0 ( v ; p , q ) .

4 Weights and conic section

Note that for constructing weighted Lupaş post-quantum Bézier curve using NTP basis, weights w 0 and w 1 must be positive and for l = 1 , 2 , … , μ − 1 , weights w l ≥ 0 are allowed, which results in interesting curve shapes, for more details see [21,34]. For fixed p > 0 and q > 0 , w 0 , w μ > 0 , and w l ∈ [ 0 , ∞ ) ( l = 1 , 2 , … , μ − 1 ) , if w l increases, the point ℛ ( v ; p , q ) moves toward the control point P l . If w l decreases, the point ℛ ( v ; p , q ) goes away from the control point P l . In mathematical language, the above theory can be expressed as

lim w l → + ∞ ℛ ( v ; p , q ) = P 0 , v = 0 , P l , v ∈ ( 0 , 1 ) , P μ , v = 1 .

One can note from Figure 2 that on increasing particular weights say w 2 in our case, the curve moves closer to the control point P 2 . For all above figures all other weights are fixed except w 2 . We have taken w 2 = 0.9 for figure (a), w 2 = 7 for figure (b), w 2 = 19 for figure (c) and w 2 = 45 for figure (d).

4.1 Weights (geometric interpretation)

Suppose we have given control polygon made by μ + 1 vectors P l ∈ R 3 ( l = 0 , 1 , 2 , … , μ ) . For any real numbers p > 0 and q > 0 , let us choose μ fixed weights w 0 , w 1 , … , w l − 1 , w l + 1 , … , w μ such that w 0 , w μ > 0 and remaining weights w j ∈ [ 0 , ∞ ) . Then the following points are defined for a fixed v ∈ ( 0 , 1 ) :

S ≔ ℛ ( v ; p , q ; w l = 0 ) , M ≕ ℛ ( v ; p , q ; w l = 1 ) , S l ≔ ℛ ( v ; p , q ; w l arbitrary ) .

Since

S l ≔ ℛ ( v ; p , q ; w l arbitrary ) = ∑ j = 0 μ w j P j γ μ , j ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) = ∑ j ≠ l μ w j P j γ μ , j ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) + w l P l γ μ , l ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) , S ≔ ℛ ( v ; p , q ; 0 ) = ∑ j ≠ l μ w j P j γ μ , j ( v ; p , q ) ∑ j ≠ l μ w j γ μ , j ( v ; p , q ) ,

and

M ≕ ℛ ( v ; p , q ; 1 ) = ∑ j ≠ l μ w j P j γ μ , j ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) + P l γ μ , j ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) .

Finally, it is easy to see S l and M can be expressed as:

S l = ( 1 − λ ) S + λ P l , M = ( 1 − ν ) S + ν P l ,

where

λ = w l γ μ , l ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) , 1 − λ = ∑ j ≠ l μ w l γ μ , j ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) , ν = γ μ , j ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) , 1 − ν = ∑ j ≠ l μ w j γ μ , j ( v ; p , q ) ∑ j = 0 μ w j γ μ , j ( v ; p , q ) .

This mean line passing through S and P l contains the points M and S l on it. Moreover,

∣ M P l ∣ ∣ S M ∣ : ∣ S l P l ∣ ∣ S S l ∣ = 1 − ν ν : 1 − λ λ = ∑ j ≠ l μ w j γ μ , j ( v ; p , q ) γ μ , j ( v ; p , q ) : ∑ j ≠ l μ w j γ μ , j ( v ; p , q ) w l γ μ , j ( v ; p , q ) = w l .

The above relation is the cross ratio of the points P l , S , M , S l in this order. Observe that as S l moves toward P l , λ goes to 1 and thus w l approaches to infinity for a fixed v ∈ ( 0 , 1 ) as λ is constant. To understand the above theory, see Figure 5 in [12].

4.2 Conic sections

Weighted Lupaş post-quantum Bézier curve ℛ ( v ; p , q ) of degree 2 is called quadratic weighted Lupaş post-quantum Bézier curve.

ℛ ( v ; p , q ) = ∑ l = 0 2 P l r 2 , l ( v ; p , q ) = ∑ l = 0 2 w l a 2 , l ( v ; p , q ) P l ∑ l = 0 2 w l a 2 , l ( v ; p , q ) .

We can represent conic sections by quadratic weighted Lupaş post-quantum Bézier curve, different kinds of conic sections are determined by the denominator A 2 ( v ; p , q ) [35],

A 2 ( v ; p , q ) = ∑ l = 0 2 w l γ 2 , l ( v ; p , q ) = ( 1 − v ) 2 w 0 + [ 2 ] p , q v ( 1 − v ) w 1 + q v 2 w 2 = ( w 0 − [ 2 ] p , q w 1 + q w 2 ) v 2 + ( [ 2 ] p , q w 1 − 2 w 0 ) v + w 0 .

The roots of the above equations are

v 1 , 2 = ( 2 w 0 − [ 2 ] p , q w 1 ) + − w 1 [ 2 ] p , q 2 − 4 q k 2 ( w 0 − [ 2 ] p , q w 1 + q w 2 ) ,

where k = w 0 w 2 w 1 2 , k determines different shapes of conic sections. k is called the conic shape factor. Choosing w 0 = w 2 = 1 . Then if w 1 = 1 , ℛ ( v ; p , q ) becomes parabola.

Assuming w 1 ≠ 1 , it is to note that

If k > [ 2 ] p , q 2 4 q , then ℛ ( v ; p , q ) will not have solutions in R , this means there is no point at infinity which lies on the curve, it gives an ellipse;
If k = [ 2 ] p , q 2 4 q , then ℛ ( v ; p , q ) will have one solution in R , this means there is one point at infinity which lies on the curve, it determines a parabola;
If k < [ 2 ] p , q 2 4 q , then ℛ ( v ; p , q ) will have two roots, this means there will be two points at infinity which lies on the curve, hence it determines a hyperbola.

Writing the above conditions in terms of w 1 gives us

w 1 2 < 4 q [ 2 ] p , q 2 ⇒ ℛ ( v ; p , q ) is an ellipse;
w 1 2 = 4 q [ 2 ] p , q 2 ⇒ ℛ ( v ; p , q ) is a parabola;
w 1 2 > 4 q [ 2 ] p , q 2 ⇒ ℛ ( v ; p , q ) is a hyperbola.

4.3 Shape effects

The parameters p , q and weights w l : ( l = 0 , 1 , 2 , … , μ ) are shape parameters for weighted Lupaş post-quantum Bézier curves. These extra parameters give more freedom and modeling flexibility in controlling the curve shape. Weighted Lupaş post-quantum Bézier curves for different values of p , q and weights have been demonstrated in Figures 2, 3, and 4.

$Figure 2 The effect on weighted Lupaş ( p , q ) \left(p,q) -Bézier curves by changing weight w 2 {w}_{2} . (a) w = [ 1 , 10 , 0.9 , 40 , 50 ] w=\left[1,10,0.9,40,50] . (b) w = [ 1 , 10 , 7 , 40 , 50 ] w=\left[1,10,7,40,50] . (c) w = [ 1 , 10 , 19 , 40 , 50 ] w=\left[1,10,19,40,50] . (d) w = [ 1 , 10 , 45 , 40 , 50 ] w=\left[1,10,45,40,50] .$

Figure 2

The effect on weighted Lupaş ( p , q ) -Bézier curves by changing weight w 2 . (a) w = [ 1 , 10 , 0.9 , 40 , 50 ] . (b) w = [ 1 , 10 , 7 , 40 , 50 ] . (c) w = [ 1 , 10 , 19 , 40 , 50 ] . (d) w = [ 1 , 10 , 45 , 40 , 50 ] .

$Figure 3 Conic sections produced by different w 1 {w}_{1} . (a) Line, (b) ellipse, (c) parabola, (d) hyperbola.$

Figure 3

Conic sections produced by different w 1 . (a) Line, (b) ellipse, (c) parabola, (d) hyperbola.

$Figure 4 Comparison of weighted Lupaş ( p , q ) \left(p,q) -Bézier curves for weights w = [ 1 , 2 , 4 , 4 , 4 , 1 ] w=\left[1,2,4,4,4,1] . (a) Weighted Lupaş ( p , q ) \left(p,q) -Bézier curve, (b) weighted Lupaş q q -Bézier curve.$

Figure 4

Comparison of weighted Lupaş ( p , q ) -Bézier curves for weights w = [ 1 , 2 , 4 , 4 , 4 , 1 ] . (a) Weighted Lupaş ( p , q ) -Bézier curve, (b) weighted Lupaş q -Bézier curve.

5 Some remarks

It is to note that when p = 1 , weighted Lupaş post-quantum Bézier curve changes into weighted Lupaş q -Bézier curve and γ μ , l ( v ; p , q ) are equal to γ μ , l ( v ; q ) = μ l q q l ( l − 1 ) 2 v l ( 1 − v ) μ − l , l = 0 , 1 , 2 , … , μ . When we put q = 1 , the curve converts into classical rational Bézier curve; and γ μ , l ( v ; q ) changes into classical Bernstein polynomials b μ , l ( v ) = μ l v l ( 1 − v ) μ − l , l = 0 , 1 , 2 , … , μ .

If we take p , q ∈ ( 0 , 1 ) ∪ ( 1 , ∞ ) , the situation is totally different. Throughout this section, p , q ≠ 1 is assumed.

5.1 Remark 1

Weighted Lupaş post-quantum Bézier curve is rational in nature with different positive weights. However, we can convert in rational Bernstein-Bézier form as

(5.1) ℛ ( t ) = ∑ l = 0 μ w ¯ l P l b μ , l ( v ) ∑ l = 0 μ w ¯ l b μ , l ( v ) , 0 ≤ v ≤ 1 ,

with same control points P l . But weight w ¯ l linked to control point P l is related by the following relation:

(5.2) w ¯ l = w l δ l , δ l ≔ δ l ( p , q ) = μ l p , q μ l p ( μ − l ) ( μ − l − 1 ) 2 q l ( l − 1 ) 2 , l = 0 , 1 , 2 , … , μ .

Rational Bernstein-Bézier representation is one of the most important representations and has been widely applied in the field of CAGD [36]. One can obtain the properties derived for weighted Lupaş post-quantum Bézier curve using the relation (5.1) and (5.2). The first-order derivative of ℛ ( v ; p , q ) at the endpoint v = 0 is given by using formulas (5.1) and (5.2)

ℛ ′ ( 0 ) = μ w ¯ 1 w ¯ 0 ( P 1 − P 0 ) = μ w 1 p 1 − μ w 0 μ 1 p , q μ 1 = [ μ ] p , q w 1 ( P 1 − P 0 ) w 0 p μ − 1 .

It is the same result which is already proved in Theorem 3.1.

Rational curves in CAGD are generally used to study it as the projection of higher-dimensional polynomial curves. Homogenous form of a weighted Lupaş post-quantum Bézier curve ℛ ¯ ( v ; p , q ) = ∑ l = 0 μ ( w l P l , w l ) γ μ , l ( v ; p , q ) .

It is to note that in case p , q ≠ 1 , the property partition of unity need not be followed by blending functions γ μ , l ( v ; p , q ) . Due to this the curve ℛ ¯ ( v ; p , q ) may not lie inside the convex hull of ( w l P l , w l ) . On the other hand, it will be better to transform ℛ ( v ; p , q ) into its rational Bernstein-Bézier form (5.1). Its homogeneous form given by ℛ ¯ ( v ) = ∑ l = 0 μ ( w ¯ l P l , w ¯ l ) b μ , l ( v ) has the convex hull property.

5.2 Remark 2

Since in relation (5.2), ℛ ( v ; p , q ) depends jointly on the contribution by weights w l , parameters p and q . On fixing all the weights w l , p and q act as shape parameter of weighted Lupaş post-quantum Bézier curve ℛ ( v ; p , q ) . However, p and q will not act as shape parameter of ℛ ( v ; p , q ) if the weights w l are taken as shape parameters (in (5.1), shape parameters are w ¯ l ). For any configuration ( w ¯ 0 ⋆ , w ¯ 1 ⋆ , … , w ¯ μ ⋆ ) , we can obtain weights w l for arbitrarily p and q by the relation w l = w ¯ l ⋆ δ l , l = 0 , 1 , 2 , … , μ .

5.3 Remark 3

There will be nonuniformity in the distribution of weights ( w ¯ 0 , w ¯ 1 , … , w ¯ n ) if p and q are not close to 1. For p > q > 1 : μ l p , q ≥ μ l q ≥ μ l ≥ 1 . Since w ¯ 0 = w 0 p n ( n − 1 ) 2 , w ¯ n = w n q n ( n − 1 ) 2 and from relation (5.2), each w ¯ l increases quickly as p and q increase. This means the nonuniformity will be very high if degree μ is high or p and q are large numbers.

Author contributions: AK supervised and made the formal analysis. MI wrote the original draft. KK made verification. MM made editing and final writing. All the authors read and approved the final manuscript.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-03-18

Revised: 2022-04-30

Accepted: 2022-05-02

Published Online: 2022-08-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0016

Keywords for this article

post-quantum integers; weighted Lupaş post-quantum Bézier curve; rational Bézier curve; normalized totally positive basis; shape parameter; conic sections

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