# Quasi-Bézier Curves with Shape Parameters

Authors: Jun Chen
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The universal form of univariate Quasi-Bézier basis functions with multiple shape parameters and a series of corresponding Quasi-Bézier curves were constructed step-by-step in this paper, using the method of undetermined coefficients. The series of Quasi-Bézier curves had geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints and tangent edges at the endpoints, and shape adjustability while maintaining the control points. Various existing Quasi-Bézier curves became special cases in the series. The obvious geometric significance of shape parameters made the adjustment of the geometrical shape easier for the designer. The numerical examples indicated that the algorithm was valid and can easily be applied.


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## Body

## 1. Introduction

The Bézier curveγ1(t) listed as follows has a direct-viewing structure and can be computed using a simple process; it is also one of the most important tools in computer-aided geometric design (CAGD). Consider
(1)γ1(t)=∑i=0nPiBin(t),t∈[0,1].Here, Bernstein basis functions{Bin(t)}i=0n are defined as:
(2)Bin(t)=(ni)(1-t)n-iti,i=0,1,…,n.Given that the shape of the curve is characterized by the control polygon, the designer always adjusts the control point{Pi}i=0n when necessary. However, in the actual process, designing the geometrical shape is usually not completed at one time. The designer prefers to have more satisfactory geometrical shapes by maintaining control polygon, which allows him or her to make minute adjustments on the shape of the curve with fixed control points.The rational Bézier curveγ2(t) listed as follows is a natural choice to meet this requirement [1].
(3)γ2(t)=∑i=0nPiBin(t)∑i=0nPiωiBin(t),t∈[0,1].By assigning a weightωi for each control point Pi, the designer can adjust the shape of the curve by changing the value of the weights {ωi}i=0n [2, 3]. Although the rational Bézier curve has good properties and can express the conic section, it also has disadvantages, such as difficulty in choosing the value of the weight, the increased order of rational fraction caused by the derivation, and the need for a numerical method of integration.In addition, the algebraic trigonometric/hyperbolic curveγ3(t) with the definition domain α as the shape parameter is a feasible method [4–6]. Consider
(4)γ3(t)=∑i=0nPiuin(t),t∈[0,α].The simple form of the algebraic trigonometric/hyperbolic curveγ3(t) can express transcendental curves (e.g., spiral and cycloid) that cannot be expressed by the Bézier curve. Nevertheless, the basis functions {uin(t)}i=0n include trigonometric/hyperbolic functions, such as sint, cost, sinht, and cosht. So, the algebraic trigonometric/hyperbolic curve is incompatible with the existing NURBS system, thereby restricting its application in the actual project.In view of the fact that the expression of the parametric curve is determined by the control points and the basis functions, the properties of such functions identify the properties of the curve with its fixed control points. Therefore, several kinds of polynomial basis functions with shape parameters [7–12] and the corresponding curve have been constructed as follows.By letting{bin1,n2(t;λ1,λ2,…,λm)}i=0n1 be n1+1 polynomial functions of degree n2 (called order n1 and degree n2) and {Pi}i=0n1 be n1+1 points in spaces, the parametric curve with multiple shape parameters {λi}i=0m is constructed as follows:
(5)P(t;λ1,λ2,…,λm)=∑i=0n1Pibin1,n2(t;λ1,λ2,…,λm).For the sake of concision, the notations{bin1,n2(t;λ1,λ2,…,λm)}i=0n1 and P(t;λ1,λ2,…,λm) will be replaced by {bin1,n2(t)}i=0n1 and P(t). And  {bin1,n2(t)}i=0n1 and P(t) will be called Quasi-Bernstein basis and Quasi-Bézier curve, respectively.With the extra degree of freedom provided by the shape parameters{λi}i=1m in {bin1,n2(t)}i=0n1, the curve P(t) can be freely adjusted and controlled by changing the value of {λi}i=1m instead of changing the control points {Pi}i=0n1. The existing works are compared in detail in Table 1.Table 1
Properties of the basis functions and the curves with shape parameters.
Property
[7]
[8]
[9]
[10]
[11]
[12]
This paper
Basis functions with multiple shape parameters
Nonnegativity
✓
✓
✓
✓
✓
✓
✓
Partition of unity
✓
✓
✓
✓
✓
✓
✓
Symmetry
✓
*
*
*
×
✓
✓
Multiple shape parameters
×
✓
✓
✓
✓
×
✓
Linear independence
✓
×
×
✓
✓
✓
✓
Degeneracy
✓
✓
✓
×
×
✓
✓
Curve with multiple shape parameters
Geometric and affine invariability
✓
✓
✓
✓
✓
✓
✓
Convex hull property
✓
✓
✓
✓
✓
✓
✓
Symmetry
✓
*
*
*
×
✓
✓
Interpolation at the endpoints
✓
✓
✓
✓
✓
✓
✓
Tangent at the end edge
✓
✓
✓
✓
×
✓
✓
*The property of symmetry in [8–10] is based on the shape parameters.The construction of the basis functions with shape parameters is the key step in [7–12]. Although many kinds of basis functions with shape parameters have been obtained in the existing research, two problems need to be solved.(1)
In all existing research, the basis functions with shape parameters are initially given, and whether or not these functions and the corresponding curves have inherited the characteristics of the Bernstein basis functions and the Bézier curve, respectively, is examined. However, the method of obtaining the complex expressions of the basis functions remains unclear. Are these basis functions obtained through intuition or through an aimless attempt?(2)
There are numerous known basis functions with shape parameters in varying forms. Is there a type of Quasi-Bernstein basis function, which makes existing basis functions with shape parameters be its special case?To answer the previous two questions, this paper uses the method of undetermined coefficients, which clarifies the construction process of the Quasi-Bernstein basis functions. A series of Quasi-Bernstein basis functions are finally obtained, rendering the existing basis function with shape parameters as their special case.


## 2. Quasi-Bézier Curve

### 2.1.  Notation

First, the following vectors are introduced:(6)bn1,n2=(b0n1,n2(t),b1n1,n2(t),…,bn1n1,n2(t)),Pn1=(P0,P1,…,Pn1)T.Equation (5) can be rewritten as
(7)P(t)=bn1,n2Pn1.Given that{bin1,n2(t)}i=0n1 are polynomials with degree n2, they can be seen as the linear combination of the Bernstein basis functions {Bin2(t)}i=0n2 with degree n2 given by
(8)bn1,n2=Bn2Mn2,n1,Bn2=(B0n2(t),B1n2(t),…,Bn2n2(t)),Mn2,n1=(mij)i,j=0i=n2,j=n1.Thus, as long as the elements in the matrixMn2,n1 are determined, the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 with order n1 and degree n2 are completely constructed. Except for several elements that can be determined in Mn2,n1, the rest are shape parameters of the Quasi-Bernstein basis functions and the Quasi-Bézier curve. Here, the matrix Mn2,n1 is called the shape parameter matrix.


### 2.2. Construction of the Shape Parameter MatrixMn2,n1

The(n2+1)(n1+1) elements of mij in Mn2,n1 must be determined so that {bin1,n2(t)}i=0n1 and P(t) become the Quasi-Bernstein basis functions and the Quasi-Bézier curve, respectively.


#### 2.2.1. Determination ofmij according to the Characteristics of the Quasi-Bernstein Basis Functions

The Quasi-Bernstein basis functions{bin1,n2(t)}i=0n1 with order n1 and degree n2 must satisfy the characteristics of nonnegativity, normalization, symmetry, linear independence, and degeneracy.Proposition 1 (nonnegativity).
A sufficient condition forbin1,n2(t)≥0(i=0,1,…,n2) is
(9)mij≥0(i=0,1,…,n2,j=0,1,…,n1).Proof.
Here,bjn1,n2(t)=∑i=0n2mijBin2(t) is known to have been extracted from (8). Based on the non-negativity of the Bernstein basis functions {Bin2(t)}i=0n2, a sufficient condition for the non-negativity of the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 is the non-negativity of the elements mij in Mn2,n1. Hence, mij must satisfy (9).Note 1.
Clearly, there is no row with all elements being 0 inMn2,n1. In other words,
(10)∑i=0n2mij≠0(j=0,1…,n1).Proposition 2 (normalization).
The necessary and sufficient condition for∑j=0n1bjn1,n2(t)=1 is given by
(11)∑j=0n1mij=1(i=0,1,…,n2).Proof.
It is known that(12)∑j=0n1bjn1,n2(t)-1=∑j=0n1(∑i=0n2mijBin2(t))-1=∑i=0n2(∑j=0n1mij)Bin2(t)-1=∑i=0n2(∑j=0n1mij)Bin2(t)-∑i=0n2Bin2(t)=∑i=0n2(∑j=0n1mij-1)Bin2(t).
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for ∑j=0n1bjn1,n2(t)-1=0 is ∑j=0n1mij=1(i=0,1,…,n2).Note 2.
By combining (9) and (11), mij satisfies 0≤mij≤1(i=0,1,…,n2,j=0,1,…,n1).Proposition 3 (symmetry).
The necessary and sufficient condition forbjn1,n2(t)=bn1-jn1,n2(1-t)(j=0,1,…,n1) is given by
(13)mij=mn2-i,n1-j(i=0,1,…,n2,j=0,1,…,n1).Proof.
According to the symmetry of the Bernstein basis functions{Bin2(t)}i=0n2 of Bin2(t)=Bn2-in2(1-t), the following can be derived:
(14)bn1-jn1,n2(1-t)-bjn1,n2(t)=∑i=0n2Bin2(1-t)mi,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bn2-in2(t)mi,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bin2(t)mn2-i,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bin2(t)(mn2-i,n1-j-mij),j=0,1,…,n1.
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for bjn1,n2(t)-bn1-jn1,n2(1-t)=0 is (13).Proposition 4 (linear independence).
The necessary and sufficient condition for the linear independence of{bin1,n2(t)}i=0n1 is given by
(15)r(Mn2,n1)=n1+1.Proof.
It is known that(16)∑j=0n1kjbjn1,n2(t)=∑j=0n1kj(∑i=0n2mijBin2(t))=∑i=0n2(∑j=0n1kjmij)Bin2(t).
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for ∑j=0n1kjbjn1,n2(t)=0 is given by
(17)∑j=0n1kjmij=0(i=0,1,…,n2).
M
j
=
(
m
0
,
j
,
m
1
,
j
,
…
,
m
n
2
,
j
)
T is defined as the jth column vector of Mn2,n1. Equation (17) is equivalent to ∑j=0n1kjMj=0. Thus, the necessary and sufficient condition for the linear independence of {bin1,n2(t)}i=0n1 is also the linear independence of the column vectors {Mj}i=0n1 of the matrix Mn2,n1. Consequently, the necessary and sufficient condition for the linear independence of {bin1,n2(t)}i=0n1 is that the rank of the shape parameter matrix Mn2,n1 satisfies r(Mn2,n1)=n1+1.Note 3.
When (15) is true, n2≥n1.Proposition 5 (degeneracy).
If the elements{mij}i,j=0i=n2,j=n1 in the matrix Mn2,n1 are represented by (18), the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 with order n1 and degree n2 are degenerated into the Bernstein basis functions {Bin1(t)}i=0n1 with order n1. 											(18)mij={(n2-n1i-j)(n1j)(n2i),max(0,i-(n2-n1))≤j≤min(n1,i),0≤i≤n2,0≤j≤n1;0,else.Proof.
When the elements{mij}i,j=0i=n2,j=n1 in the matrix Mn2,n1 are represented by (18), the following is obtained:
(19)Bn1=Bn2Mn2,n1.
Comparing (19) with (8), Proposition 5 is proven.Note 4.
Ifn1=n2,Mn2,n1 is an identity matrix here.


#### 2.2.2. Determination ofmij according to the Characteristics of the Quasi-Bézier Curve

Proposition 6 (interpolation at the endpoints).
The necessary and sufficient condition forP(0)=P0 and P(1)=Pn1 is given by
(20)m00=1,m0j=0(j=1,2,…,n1),(21)mn2,n1=1,mn2,j=0(j=0,1,…,n1-1).Proof.
Clearly, the necessary and sufficient condition forP(0)=P0 is the Quasi-Bernstein primary functions {bin1,n2(t)}i=0n1 satisfying the following:
(22)bin1,n2(0)={1,i=0,0,i≠0.
It is known that(23)(b0n1,n2(0),b1n1,n2(0),…,bn1n1,n2(0))=(B0n2(0),B1n2(0),…,Bn2n2(0))·(mij)i,j=0i=n2,j=n1=(1,0,…,0)·(mij)i,j=0i=n2,j=n1=(m00,m01,…,m0,n1-1,m0n1).
Thus, the necessary and sufficient condition for(b0n1,n2(0),b1n1,n2(0),…,bn1n1,n2(0))=(1,0,…,0) is (20).
Similarly, the necessary and sufficient condition forP(1)=Pn1 is the Quasi-Bernstein primary functions {bin1,n2(t)}i=0n1 satisfying the following:
(24)bin1,n2(1)={1,i=n1,0,i≠n1.
It is known that(25)(b0n1,n2(1),b1n1,n2(1),…,bn1n1,n2(1))=(B0n2(1),B1n2(1),…,Bn2n2(1))·(mij)i,j=0i=n2,j=n1=(0,…,0,1)·(mij)i,j=0i=n2,j=n1=(mn2,0,mn2,1,…,mn2,n1-1,mn2,n1).
Thus, the necessary and sufficient condition for(b0n1,n2(1),b1n1,n2(1),…,bn1n1,n2(1))=(0,…,0,1) is (21).
Hence, the necessary and sufficient condition forP(1)=Pn1 is (21).Note 5.
When{bin1,n2(t)}i=0n1 have the property of symmetry, (21) is equivalent to (20) according to (13).Proposition 7 (tangent edges at the endpoints).
The necessary and sufficient condition forP'(0)∥P1P0,P'(1)∥PnPn-1 is given by
(26)m10+m11=1,m10≠1,m1j=0(j=2,3,…,n1),(27)mn2-1,n1-1+mn2-1,n1=1,mn2-1,n1≠1,mn2-1,j=0(j=0,1,…,n1-2).Proof.
It is known that(28)P′(0)=P′(t)|t=0=(b0n1,n2(t),b1n1,n2(t),…,bn1n1,n2(t))′|t=0×(P0,P1,…,Pn1)T=(B0n2(t),B1n2(t),…,Bn2n2(t))′|t=0·(mij)i,j=0i=n2,j=n1·(P0,P1,…,Pn1)T=n2(-1,1,0,…,0)·(mij)i,j=0i=n2,j=n1·(P0,P1,…,Pn1)T=n2(m10-m00,m11-m01,…,m1n1-m0n1)×(P0,P1,…,Pn1)T.
Clearly, the necessary and sufficient condition forP'(0)∥P1P0 is (m10-m00)/(m11-m01)=-1,m1j-m0j=0(j=2,3,…,n1) which verifies (26).
Similarly, the necessary and sufficient condition forP'(1)∥PnPn-1 is (27).Note 6.
When{bin1,n2(t)}i=0n1 have the property of symmetry, (27) is equivalent to (26) according to (13).


#### 2.2.3. Form of Shape Parameter MatrixMn2,n1

All shape parameter matrixes that satisfy (9), (11), (13), (15), (20), (21), (26), and (27) have the following form:
(29)Mn2,n1=(10⋯00m101-m10⋯00m20m21⋯m2,n1-1m2n1⋮⋮⋱⋮⋮m2n1m2,n1-1⋯m21m2000⋯1-m10m1000⋯01)(n2+1)×(n1+1).Here,mij are variable shape parameters that satisfy
(30)∑j=0n1mij=1(i=2,3,…,[(n2+1)2]),0≤m10<1,0≤mij≤1(i=2,3,…,[(n2+1)2],j=0,1,…,n1).


### 2.3. The Characteristics of the Quasi-Bézier Curve

In summary, the Quasi-Bézier curveP(t) based on the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 has the characteristics listed as follows:(a)
shape adjustability: the shape of the Quasi-Bézier curve can still be adjusted by maintaining the control points.(b)
geometric invariability: the Quasi-Bézier curve only relies on the control points, whereas it has nothing to do with the position and direction of the coordinate system; in other words, the curve shape remains invariable after translation and revolving in the coordinate system;(c)
affine invariability: barycentric combinations are invariant under affine maps; therefore, (9) and (11) give the algebraic verification of this property;(d)
symmetry: whether the control points are labeledP0P1⋯Pn1 or Pn1Pn1-1⋯P0, the curves that correspond to the two different orderings look the same; they differ only in the direction in which they are traversed, and this is written as
(31)∑i=0n1Pibin1,n2(t)=∑i=0n1Pn1-ibn1-in1,n2(1-t),which follows the inspection of (13);(e)
convex hull property: this property exists since the Quasi-Bernstein basis functions{bin1,n2(t)}i=0n1 have the properties of non-negativity and normalization; the Quasi-Bézier curve is the convex linear combination of control points, and as such, it is located in the convex hull of the control points;(f)
interpolation at the endpoints and tangent edges at the endpoint: the Quasi-Bézier curveP(t) interpolates the first and the last control points P(0)=P0 and P(1)=Pn1; the first and last edges of the control polygon are the tangent lines at the endpoints, where P'(0)∥P1P0  and  P'(1)∥PnPn-1.


### 2.4.  Geometric Significance of the Shape Parameters

According to (29), when mi,j0(i=0,1,…,n2,j0=0,1,…,n1) increases, bj0n1,n2(t) and bn1-j0n1,n2(t) increase as well; specifically, P(t) comes close to the control points Pj0 and Pn1-j0. The geometric significance of the shape parameters is shown in Section 3.


## 2.1.  Notation

First, the following vectors are introduced:(6)bn1,n2=(b0n1,n2(t),b1n1,n2(t),…,bn1n1,n2(t)),Pn1=(P0,P1,…,Pn1)T.Equation (5) can be rewritten as
(7)P(t)=bn1,n2Pn1.Given that{bin1,n2(t)}i=0n1 are polynomials with degree n2, they can be seen as the linear combination of the Bernstein basis functions {Bin2(t)}i=0n2 with degree n2 given by
(8)bn1,n2=Bn2Mn2,n1,Bn2=(B0n2(t),B1n2(t),…,Bn2n2(t)),Mn2,n1=(mij)i,j=0i=n2,j=n1.Thus, as long as the elements in the matrixMn2,n1 are determined, the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 with order n1 and degree n2 are completely constructed. Except for several elements that can be determined in Mn2,n1, the rest are shape parameters of the Quasi-Bernstein basis functions and the Quasi-Bézier curve. Here, the matrix Mn2,n1 is called the shape parameter matrix.


## 2.2. Construction of the Shape Parameter MatrixMn2,n1

The(n2+1)(n1+1) elements of mij in Mn2,n1 must be determined so that {bin1,n2(t)}i=0n1 and P(t) become the Quasi-Bernstein basis functions and the Quasi-Bézier curve, respectively.


### 2.2.1. Determination ofmij according to the Characteristics of the Quasi-Bernstein Basis Functions

The Quasi-Bernstein basis functions{bin1,n2(t)}i=0n1 with order n1 and degree n2 must satisfy the characteristics of nonnegativity, normalization, symmetry, linear independence, and degeneracy.Proposition 1 (nonnegativity).
A sufficient condition forbin1,n2(t)≥0(i=0,1,…,n2) is
(9)mij≥0(i=0,1,…,n2,j=0,1,…,n1).Proof.
Here,bjn1,n2(t)=∑i=0n2mijBin2(t) is known to have been extracted from (8). Based on the non-negativity of the Bernstein basis functions {Bin2(t)}i=0n2, a sufficient condition for the non-negativity of the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 is the non-negativity of the elements mij in Mn2,n1. Hence, mij must satisfy (9).Note 1.
Clearly, there is no row with all elements being 0 inMn2,n1. In other words,
(10)∑i=0n2mij≠0(j=0,1…,n1).Proposition 2 (normalization).
The necessary and sufficient condition for∑j=0n1bjn1,n2(t)=1 is given by
(11)∑j=0n1mij=1(i=0,1,…,n2).Proof.
It is known that(12)∑j=0n1bjn1,n2(t)-1=∑j=0n1(∑i=0n2mijBin2(t))-1=∑i=0n2(∑j=0n1mij)Bin2(t)-1=∑i=0n2(∑j=0n1mij)Bin2(t)-∑i=0n2Bin2(t)=∑i=0n2(∑j=0n1mij-1)Bin2(t).
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for ∑j=0n1bjn1,n2(t)-1=0 is ∑j=0n1mij=1(i=0,1,…,n2).Note 2.
By combining (9) and (11), mij satisfies 0≤mij≤1(i=0,1,…,n2,j=0,1,…,n1).Proposition 3 (symmetry).
The necessary and sufficient condition forbjn1,n2(t)=bn1-jn1,n2(1-t)(j=0,1,…,n1) is given by
(13)mij=mn2-i,n1-j(i=0,1,…,n2,j=0,1,…,n1).Proof.
According to the symmetry of the Bernstein basis functions{Bin2(t)}i=0n2 of Bin2(t)=Bn2-in2(1-t), the following can be derived:
(14)bn1-jn1,n2(1-t)-bjn1,n2(t)=∑i=0n2Bin2(1-t)mi,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bn2-in2(t)mi,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bin2(t)mn2-i,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bin2(t)(mn2-i,n1-j-mij),j=0,1,…,n1.
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for bjn1,n2(t)-bn1-jn1,n2(1-t)=0 is (13).Proposition 4 (linear independence).
The necessary and sufficient condition for the linear independence of{bin1,n2(t)}i=0n1 is given by
(15)r(Mn2,n1)=n1+1.Proof.
It is known that(16)∑j=0n1kjbjn1,n2(t)=∑j=0n1kj(∑i=0n2mijBin2(t))=∑i=0n2(∑j=0n1kjmij)Bin2(t).
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for ∑j=0n1kjbjn1,n2(t)=0 is given by
(17)∑j=0n1kjmij=0(i=0,1,…,n2).
M
j
=
(
m
0
,
j
,
m
1
,
j
,
…
,
m
n
2
,
j
)
T is defined as the jth column vector of Mn2,n1. Equation (17) is equivalent to ∑j=0n1kjMj=0. Thus, the necessary and sufficient condition for the linear independence of {bin1,n2(t)}i=0n1 is also the linear independence of the column vectors {Mj}i=0n1 of the matrix Mn2,n1. Consequently, the necessary and sufficient condition for the linear independence of {bin1,n2(t)}i=0n1 is that the rank of the shape parameter matrix Mn2,n1 satisfies r(Mn2,n1)=n1+1.Note 3.
When (15) is true, n2≥n1.Proposition 5 (degeneracy).
If the elements{mij}i,j=0i=n2,j=n1 in the matrix Mn2,n1 are represented by (18), the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 with order n1 and degree n2 are degenerated into the Bernstein basis functions {Bin1(t)}i=0n1 with order n1. 											(18)mij={(n2-n1i-j)(n1j)(n2i),max(0,i-(n2-n1))≤j≤min(n1,i),0≤i≤n2,0≤j≤n1;0,else.Proof.
When the elements{mij}i,j=0i=n2,j=n1 in the matrix Mn2,n1 are represented by (18), the following is obtained:
(19)Bn1=Bn2Mn2,n1.
Comparing (19) with (8), Proposition 5 is proven.Note 4.
Ifn1=n2,Mn2,n1 is an identity matrix here.


### 2.2.2. Determination ofmij according to the Characteristics of the Quasi-Bézier Curve

Proposition 6 (interpolation at the endpoints).
The necessary and sufficient condition forP(0)=P0 and P(1)=Pn1 is given by
(20)m00=1,m0j=0(j=1,2,…,n1),(21)mn2,n1=1,mn2,j=0(j=0,1,…,n1-1).Proof.
Clearly, the necessary and sufficient condition forP(0)=P0 is the Quasi-Bernstein primary functions {bin1,n2(t)}i=0n1 satisfying the following:
(22)bin1,n2(0)={1,i=0,0,i≠0.
It is known that(23)(b0n1,n2(0),b1n1,n2(0),…,bn1n1,n2(0))=(B0n2(0),B1n2(0),…,Bn2n2(0))·(mij)i,j=0i=n2,j=n1=(1,0,…,0)·(mij)i,j=0i=n2,j=n1=(m00,m01,…,m0,n1-1,m0n1).
Thus, the necessary and sufficient condition for(b0n1,n2(0),b1n1,n2(0),…,bn1n1,n2(0))=(1,0,…,0) is (20).
Similarly, the necessary and sufficient condition forP(1)=Pn1 is the Quasi-Bernstein primary functions {bin1,n2(t)}i=0n1 satisfying the following:
(24)bin1,n2(1)={1,i=n1,0,i≠n1.
It is known that(25)(b0n1,n2(1),b1n1,n2(1),…,bn1n1,n2(1))=(B0n2(1),B1n2(1),…,Bn2n2(1))·(mij)i,j=0i=n2,j=n1=(0,…,0,1)·(mij)i,j=0i=n2,j=n1=(mn2,0,mn2,1,…,mn2,n1-1,mn2,n1).
Thus, the necessary and sufficient condition for(b0n1,n2(1),b1n1,n2(1),…,bn1n1,n2(1))=(0,…,0,1) is (21).
Hence, the necessary and sufficient condition forP(1)=Pn1 is (21).Note 5.
When{bin1,n2(t)}i=0n1 have the property of symmetry, (21) is equivalent to (20) according to (13).Proposition 7 (tangent edges at the endpoints).
The necessary and sufficient condition forP'(0)∥P1P0,P'(1)∥PnPn-1 is given by
(26)m10+m11=1,m10≠1,m1j=0(j=2,3,…,n1),(27)mn2-1,n1-1+mn2-1,n1=1,mn2-1,n1≠1,mn2-1,j=0(j=0,1,…,n1-2).Proof.
It is known that(28)P′(0)=P′(t)|t=0=(b0n1,n2(t),b1n1,n2(t),…,bn1n1,n2(t))′|t=0×(P0,P1,…,Pn1)T=(B0n2(t),B1n2(t),…,Bn2n2(t))′|t=0·(mij)i,j=0i=n2,j=n1·(P0,P1,…,Pn1)T=n2(-1,1,0,…,0)·(mij)i,j=0i=n2,j=n1·(P0,P1,…,Pn1)T=n2(m10-m00,m11-m01,…,m1n1-m0n1)×(P0,P1,…,Pn1)T.
Clearly, the necessary and sufficient condition forP'(0)∥P1P0 is (m10-m00)/(m11-m01)=-1,m1j-m0j=0(j=2,3,…,n1) which verifies (26).
Similarly, the necessary and sufficient condition forP'(1)∥PnPn-1 is (27).Note 6.
When{bin1,n2(t)}i=0n1 have the property of symmetry, (27) is equivalent to (26) according to (13).


### 2.2.3. Form of Shape Parameter MatrixMn2,n1

All shape parameter matrixes that satisfy (9), (11), (13), (15), (20), (21), (26), and (27) have the following form:
(29)Mn2,n1=(10⋯00m101-m10⋯00m20m21⋯m2,n1-1m2n1⋮⋮⋱⋮⋮m2n1m2,n1-1⋯m21m2000⋯1-m10m1000⋯01)(n2+1)×(n1+1).Here,mij are variable shape parameters that satisfy
(30)∑j=0n1mij=1(i=2,3,…,[(n2+1)2]),0≤m10<1,0≤mij≤1(i=2,3,…,[(n2+1)2],j=0,1,…,n1).


## 2.2.1. Determination ofmij according to the Characteristics of the Quasi-Bernstein Basis Functions

The Quasi-Bernstein basis functions{bin1,n2(t)}i=0n1 with order n1 and degree n2 must satisfy the characteristics of nonnegativity, normalization, symmetry, linear independence, and degeneracy.Proposition 1 (nonnegativity).
A sufficient condition forbin1,n2(t)≥0(i=0,1,…,n2) is
(9)mij≥0(i=0,1,…,n2,j=0,1,…,n1).Proof.
Here,bjn1,n2(t)=∑i=0n2mijBin2(t) is known to have been extracted from (8). Based on the non-negativity of the Bernstein basis functions {Bin2(t)}i=0n2, a sufficient condition for the non-negativity of the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 is the non-negativity of the elements mij in Mn2,n1. Hence, mij must satisfy (9).Note 1.
Clearly, there is no row with all elements being 0 inMn2,n1. In other words,
(10)∑i=0n2mij≠0(j=0,1…,n1).Proposition 2 (normalization).
The necessary and sufficient condition for∑j=0n1bjn1,n2(t)=1 is given by
(11)∑j=0n1mij=1(i=0,1,…,n2).Proof.
It is known that(12)∑j=0n1bjn1,n2(t)-1=∑j=0n1(∑i=0n2mijBin2(t))-1=∑i=0n2(∑j=0n1mij)Bin2(t)-1=∑i=0n2(∑j=0n1mij)Bin2(t)-∑i=0n2Bin2(t)=∑i=0n2(∑j=0n1mij-1)Bin2(t).
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for ∑j=0n1bjn1,n2(t)-1=0 is ∑j=0n1mij=1(i=0,1,…,n2).Note 2.
By combining (9) and (11), mij satisfies 0≤mij≤1(i=0,1,…,n2,j=0,1,…,n1).Proposition 3 (symmetry).
The necessary and sufficient condition forbjn1,n2(t)=bn1-jn1,n2(1-t)(j=0,1,…,n1) is given by
(13)mij=mn2-i,n1-j(i=0,1,…,n2,j=0,1,…,n1).Proof.
According to the symmetry of the Bernstein basis functions{Bin2(t)}i=0n2 of Bin2(t)=Bn2-in2(1-t), the following can be derived:
(14)bn1-jn1,n2(1-t)-bjn1,n2(t)=∑i=0n2Bin2(1-t)mi,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bn2-in2(t)mi,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bin2(t)mn2-i,n1-j-∑i=0n2Bin2(t)mij=∑i=0n2Bin2(t)(mn2-i,n1-j-mij),j=0,1,…,n1.
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for bjn1,n2(t)-bn1-jn1,n2(1-t)=0 is (13).Proposition 4 (linear independence).
The necessary and sufficient condition for the linear independence of{bin1,n2(t)}i=0n1 is given by
(15)r(Mn2,n1)=n1+1.Proof.
It is known that(16)∑j=0n1kjbjn1,n2(t)=∑j=0n1kj(∑i=0n2mijBin2(t))=∑i=0n2(∑j=0n1kjmij)Bin2(t).
According to the linear independence of the Bernstein basis functions{Bin2(t)}i=0n2, the necessary and sufficient condition for ∑j=0n1kjbjn1,n2(t)=0 is given by
(17)∑j=0n1kjmij=0(i=0,1,…,n2).
M
j
=
(
m
0
,
j
,
m
1
,
j
,
…
,
m
n
2
,
j
)
T is defined as the jth column vector of Mn2,n1. Equation (17) is equivalent to ∑j=0n1kjMj=0. Thus, the necessary and sufficient condition for the linear independence of {bin1,n2(t)}i=0n1 is also the linear independence of the column vectors {Mj}i=0n1 of the matrix Mn2,n1. Consequently, the necessary and sufficient condition for the linear independence of {bin1,n2(t)}i=0n1 is that the rank of the shape parameter matrix Mn2,n1 satisfies r(Mn2,n1)=n1+1.Note 3.
When (15) is true, n2≥n1.Proposition 5 (degeneracy).
If the elements{mij}i,j=0i=n2,j=n1 in the matrix Mn2,n1 are represented by (18), the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 with order n1 and degree n2 are degenerated into the Bernstein basis functions {Bin1(t)}i=0n1 with order n1. 											(18)mij={(n2-n1i-j)(n1j)(n2i),max(0,i-(n2-n1))≤j≤min(n1,i),0≤i≤n2,0≤j≤n1;0,else.Proof.
When the elements{mij}i,j=0i=n2,j=n1 in the matrix Mn2,n1 are represented by (18), the following is obtained:
(19)Bn1=Bn2Mn2,n1.
Comparing (19) with (8), Proposition 5 is proven.Note 4.
Ifn1=n2,Mn2,n1 is an identity matrix here.


## 2.2.2. Determination ofmij according to the Characteristics of the Quasi-Bézier Curve

Proposition 6 (interpolation at the endpoints).
The necessary and sufficient condition forP(0)=P0 and P(1)=Pn1 is given by
(20)m00=1,m0j=0(j=1,2,…,n1),(21)mn2,n1=1,mn2,j=0(j=0,1,…,n1-1).Proof.
Clearly, the necessary and sufficient condition forP(0)=P0 is the Quasi-Bernstein primary functions {bin1,n2(t)}i=0n1 satisfying the following:
(22)bin1,n2(0)={1,i=0,0,i≠0.
It is known that(23)(b0n1,n2(0),b1n1,n2(0),…,bn1n1,n2(0))=(B0n2(0),B1n2(0),…,Bn2n2(0))·(mij)i,j=0i=n2,j=n1=(1,0,…,0)·(mij)i,j=0i=n2,j=n1=(m00,m01,…,m0,n1-1,m0n1).
Thus, the necessary and sufficient condition for(b0n1,n2(0),b1n1,n2(0),…,bn1n1,n2(0))=(1,0,…,0) is (20).
Similarly, the necessary and sufficient condition forP(1)=Pn1 is the Quasi-Bernstein primary functions {bin1,n2(t)}i=0n1 satisfying the following:
(24)bin1,n2(1)={1,i=n1,0,i≠n1.
It is known that(25)(b0n1,n2(1),b1n1,n2(1),…,bn1n1,n2(1))=(B0n2(1),B1n2(1),…,Bn2n2(1))·(mij)i,j=0i=n2,j=n1=(0,…,0,1)·(mij)i,j=0i=n2,j=n1=(mn2,0,mn2,1,…,mn2,n1-1,mn2,n1).
Thus, the necessary and sufficient condition for(b0n1,n2(1),b1n1,n2(1),…,bn1n1,n2(1))=(0,…,0,1) is (21).
Hence, the necessary and sufficient condition forP(1)=Pn1 is (21).Note 5.
When{bin1,n2(t)}i=0n1 have the property of symmetry, (21) is equivalent to (20) according to (13).Proposition 7 (tangent edges at the endpoints).
The necessary and sufficient condition forP'(0)∥P1P0,P'(1)∥PnPn-1 is given by
(26)m10+m11=1,m10≠1,m1j=0(j=2,3,…,n1),(27)mn2-1,n1-1+mn2-1,n1=1,mn2-1,n1≠1,mn2-1,j=0(j=0,1,…,n1-2).Proof.
It is known that(28)P′(0)=P′(t)|t=0=(b0n1,n2(t),b1n1,n2(t),…,bn1n1,n2(t))′|t=0×(P0,P1,…,Pn1)T=(B0n2(t),B1n2(t),…,Bn2n2(t))′|t=0·(mij)i,j=0i=n2,j=n1·(P0,P1,…,Pn1)T=n2(-1,1,0,…,0)·(mij)i,j=0i=n2,j=n1·(P0,P1,…,Pn1)T=n2(m10-m00,m11-m01,…,m1n1-m0n1)×(P0,P1,…,Pn1)T.
Clearly, the necessary and sufficient condition forP'(0)∥P1P0 is (m10-m00)/(m11-m01)=-1,m1j-m0j=0(j=2,3,…,n1) which verifies (26).
Similarly, the necessary and sufficient condition forP'(1)∥PnPn-1 is (27).Note 6.
When{bin1,n2(t)}i=0n1 have the property of symmetry, (27) is equivalent to (26) according to (13).


## 2.2.3. Form of Shape Parameter MatrixMn2,n1

All shape parameter matrixes that satisfy (9), (11), (13), (15), (20), (21), (26), and (27) have the following form:
(29)Mn2,n1=(10⋯00m101-m10⋯00m20m21⋯m2,n1-1m2n1⋮⋮⋱⋮⋮m2n1m2,n1-1⋯m21m2000⋯1-m10m1000⋯01)(n2+1)×(n1+1).Here,mij are variable shape parameters that satisfy
(30)∑j=0n1mij=1(i=2,3,…,[(n2+1)2]),0≤m10<1,0≤mij≤1(i=2,3,…,[(n2+1)2],j=0,1,…,n1).


## 2.3. The Characteristics of the Quasi-Bézier Curve

In summary, the Quasi-Bézier curveP(t) based on the Quasi-Bernstein basis functions {bin1,n2(t)}i=0n1 has the characteristics listed as follows:(a)
shape adjustability: the shape of the Quasi-Bézier curve can still be adjusted by maintaining the control points.(b)
geometric invariability: the Quasi-Bézier curve only relies on the control points, whereas it has nothing to do with the position and direction of the coordinate system; in other words, the curve shape remains invariable after translation and revolving in the coordinate system;(c)
affine invariability: barycentric combinations are invariant under affine maps; therefore, (9) and (11) give the algebraic verification of this property;(d)
symmetry: whether the control points are labeledP0P1⋯Pn1 or Pn1Pn1-1⋯P0, the curves that correspond to the two different orderings look the same; they differ only in the direction in which they are traversed, and this is written as
(31)∑i=0n1Pibin1,n2(t)=∑i=0n1Pn1-ibn1-in1,n2(1-t),which follows the inspection of (13);(e)
convex hull property: this property exists since the Quasi-Bernstein basis functions{bin1,n2(t)}i=0n1 have the properties of non-negativity and normalization; the Quasi-Bézier curve is the convex linear combination of control points, and as such, it is located in the convex hull of the control points;(f)
interpolation at the endpoints and tangent edges at the endpoint: the Quasi-Bézier curveP(t) interpolates the first and the last control points P(0)=P0 and P(1)=Pn1; the first and last edges of the control polygon are the tangent lines at the endpoints, where P'(0)∥P1P0  and  P'(1)∥PnPn-1.


## 2.4.  Geometric Significance of the Shape Parameters

According to (29), when mi,j0(i=0,1,…,n2,j0=0,1,…,n1) increases, bj0n1,n2(t) and bn1-j0n1,n2(t) increase as well; specifically, P(t) comes close to the control points Pj0 and Pn1-j0. The geometric significance of the shape parameters is shown in Section 3.


## 3. Numerical Examples

Example 1.
The shape parameter matrixM3,2 is constructed from (29). The corresponding Quasi-Bernstein basis functions and the Quasi-Bézier curves with different shape parameter m10 are given as follows:
(32)M3,2=(100m101-m10001-m10m10001),0≤m10<1.
The geometric significance of the shape parametersm10 is shown in Figure 1. As the value of the shape parameter m10 increases, the elements in the second column of M3,2 decrease. According to (8), the second Quasi-Bernstein basis function b12,3(t) decreases. So, the corresponding Quasi-Bézier curve moves away from the control point P1 (see Figure 1(b)).Quasi-Bernstein basis functions and Quasi-Bézier curves whenn1=2  and  n2=3.
(a)
(b)Example 2.
The shape parameter matrixM4,2 is constructed from (29). The corresponding Quasi-Bernstein basis functions and the Quasi-Bézier curves with different shape parameters m10 and m20 are given as follows:
(33)M4,2=(100m101-m100m201-2m20m2001-m10m10001),0≤m10<1,0≤m20≤12.
The geometric significance of the shape parametersm10 and m20 is shown in Figure 2. When we increase the value of m10 and keep m20 unchanged, the elements in the second column of M4,2 decrease. According to (8), the second Quasi-Bernstein basis function b12,4(t) decreases. Compare the blue curve with the red one, and we will find that the Quasi-Bézier curve moves away from the control point P1 (see Figure 2(b)).
If we increase the value ofm20 and keep m10 unchanged, similar result is also obtained. Compare the red curve with the green one, and we will find that the Quasi-Bézier curve moves away from the control point P1 (see Figure 2(b)).Quasi-Bernstein basis functions and Quasi-Bézier curves whenn1=2  and n2=4.
(a)
(b)Example 3.
The shape parameter matrixM3,3 is constructed from (29). The corresponding Quasi-Bernstein basis functions and the Quasi-Bézier curves with different shape parameter m10 are given as follows:
(34)M3,3=(1000m101-m1000001-m10m100001),0≤m10<1.
The geometric significance of the shape parametersm10 is shown in Figure 3. As the value of the shape parameter m10 increases, the elements in the second and the third column of M3,3 decrease. According to (8), the second Quasi-Bernstein basis function b13,3(t) and the third Quasi-Bernstein basis function b23,3(t) decrease. So, the corresponding Quasi-Bézier curve moves away from the control points P1 and P2 (see Figure 3(b)).Quasi-Bernstein basis functions and Quasi-Bézier curves whenn1=3  and n2=3.
(a)
(b)Example 4.
The shape parameter matrixM4,3 is constructed from (29). The corresponding Quasi-Bernstein basis functions and the Quasi-Bézier curves with different shape parameter m10 and m20 are given as follows:
(35)M4,3=(1000m101-m1000m2012-m2012-m20m20001-m10m100001),0≤m10<1,0≤m20≤12.
The geometric significance of the shape parametersm10 and m20 is shown in Figure 4. When we increase the value of m10 and keep m20 unchanged, the elements in the second and the third column of M4,3 decrease. According to (8), the second Quasi-Bernstein basis function b13,4(t) and the third Quasi-Bernstein basis function b23,4(t) decrease. Compare the blue curve with the red one, and we will find that the Quasi-Bézier curve moves away from the control points P1 and P2 (see Figure 4(b)).
If we increase the value ofm20 and keep m10 unchanged, similar result is also obtained. Compare the red curve with the green one, and we will find that the Quasi-Bézier curve moves away from the control points P1 and P2 (see Figure 4(b)).Quasi-Bernstein basis functions and Quasi-Bézier curves whenn1=3 and n2=4.
(a)
(b)Note 7.
Several Quasi-Bernstein basis functions for low degree and low order are presented aforementioned. The corresponding basis functions for higher degree and higher order are defined recursively as follows [7, 12]:
(36)bin1+1,n2+1(t)=(1-t)bin1,n2(t)+tbi-1n1,n2(t),i=0,1,…,n1.
Here, we setb-1n1,n2(t)=bn1+1n1,n2(t)=0.Example 5.
Figure5 presents three kinds of flowers with six petals, defined by six symmetric control polygons. Similar flowers are obtained from the same control polygons with different shape parameters.Figure 5
Three kinds of flowers with six petals.Example 6.
Figure6 presents three kinds of outlines of the vase, all of which are similar to the control polygons. So, the designer can make minute adjustments with the same control polygons by changing the value of the shape parameters.Figure 6
Three kinds of outlines of the vase.


## 4. Discussion

### 4.1. Special Cases

Several existing basis functions containing just one shape parameter in [7, 12] are considered as the special cases in this paper. Meanwhile, for the polynomial basis functions with multiple shape parameters in [8–11], the symmetry was not discussed by authors. In fact, when these shape parameters satisfy certain relations, the corresponding basis functions and curves become symmetrical. Then, the curves have geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints, and tangent edges at the endpoints, and the corresponding shape parameter matrices are the special cases of (29).We take [9] as example. When the shape parameters satisfy certain relations in [9], the shape parameter matrix is (37)Mn+1,n=(10⋯001-λ1n+11-1-λ1n+1⋯0002-λ2n+1⋯00⋮⋮⋱⋮⋮00⋯2-λ2n+1000⋯1-1-λ1n+11-λ1n+100⋯01)(n+2)×(n+1).It is the special case of (29).


### 4.2. Degree and Order of the Curve

In the previous work, the difference between the degree and order of the curve is fixed (i.e.,n2-n1=0 in [10, 11], n2-n1=1 in [7–9], and n2-n1=2 in [12]), and the scope of the curve is also fixed with the same control points.However, comparing Figures1(a) and 1(b), it is found that the greater the difference between degree and order, the larger the scope of the Quasi-Bézier curve acquired. In order to obtain the Quasi-Bézier curves with broader scope with the same control points, the designer can increase the difference between the degree and the order n2-n1.


## 4.1. Special Cases

Several existing basis functions containing just one shape parameter in [7, 12] are considered as the special cases in this paper. Meanwhile, for the polynomial basis functions with multiple shape parameters in [8–11], the symmetry was not discussed by authors. In fact, when these shape parameters satisfy certain relations, the corresponding basis functions and curves become symmetrical. Then, the curves have geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints, and tangent edges at the endpoints, and the corresponding shape parameter matrices are the special cases of (29).We take [9] as example. When the shape parameters satisfy certain relations in [9], the shape parameter matrix is (37)Mn+1,n=(10⋯001-λ1n+11-1-λ1n+1⋯0002-λ2n+1⋯00⋮⋮⋱⋮⋮00⋯2-λ2n+1000⋯1-1-λ1n+11-λ1n+100⋯01)(n+2)×(n+1).It is the special case of (29).


## 4.2. Degree and Order of the Curve

In the previous work, the difference between the degree and order of the curve is fixed (i.e.,n2-n1=0 in [10, 11], n2-n1=1 in [7–9], and n2-n1=2 in [12]), and the scope of the curve is also fixed with the same control points.However, comparing Figures1(a) and 1(b), it is found that the greater the difference between degree and order, the larger the scope of the Quasi-Bézier curve acquired. In order to obtain the Quasi-Bézier curves with broader scope with the same control points, the designer can increase the difference between the degree and the order n2-n1.


## 5. Conclusion and Further Work

In this paper, a series of univariate Quasi-Bernstein basis functions are constructed, thereby creating a series of Quasi-Bézier curves. The shape of the series of curves can be adjusted even with the control points fixed. The Quasi-Bézier curves also possess geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints, and tangent edges at the endpoints.Quasi-Bernstein basis functions with shape parameters have been directly studied in the previous research. However, in this paper, each function has been gradually inferred and constructed using a clear method of undetermined coefficients, where each shape parameter is proposed according to the properties of the Quasi-Bernstein basis functions and the Quasi-Bézier curve. Under the premise of satisfying symmetry, the former basis functions are all considered as the special cases in this paper.In the existing CAD/CAM systems, the triangular Bézier surface and the spline curve are widely used. The shape parameters also have been brought into the triangular surface in [12–14] and the spline curve [15–17]. The method in this paper also can be extended to construct the basis functions of the triangular surface and the spline curve with shape parameters directly, and more details can be seen in our other papers submitted.



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*Source: 171392-2013-03-30.xml*