Quasi-Bézier Curves with Shape Parameters

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.

1. Introduction

The Bézier curve 𝛾₁(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 𝛾₁(𝑡)=^𝑛_𝑖=0𝐏_𝑖𝐵^𝑛_𝑖(𝑡),𝑡∈[0,1]. (1)

Here, Bernstein basis functions {𝐵^𝑛_𝑖(𝑡)}^𝑛_𝑖=0 are defined as: 𝐵^𝑛_𝑖(𝑡)=𝑛𝑖(1−𝑡)^𝑛−𝑖𝑡^𝑖,𝑖=0,1,…,𝑛. (2)

Given that the shape of the curve is characterized by the control polygon, the designer always adjusts the control point {𝐏_𝑖}^𝑛_𝑖=0 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 curve 𝛾₂(𝑡) listed as follows is a natural choice to meet this requirement [1]. 𝛾₂(𝑡)=∑^𝑛_𝑖=0𝐏_𝑖𝐵^𝑛_𝑖(𝑡)∑^𝑛_𝑖=0𝐏_𝑖𝜔_𝑖𝐵^𝑛_𝑖(𝑡),𝑡∈[0,1]. (3)

By assigning a weight 𝜔_𝑖 for each control point 𝐏_𝑖, the designer can adjust the shape of the curve by changing the value of the weights {𝜔_𝑖}^𝑛_𝑖=0 [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 curve 𝛾₃(𝑡) with the definition domain 𝛼 as the shape parameter is a feasible method [4–6]. Consider 𝛾₃(𝑡)=^𝑛_𝑖=0𝐏_𝑖𝑢^𝑛_𝑖(𝑡),𝑡∈[0,𝛼]. (4)

The simple form of the algebraic trigonometric/hyperbolic curve 𝛾₃(𝑡) can express transcendental curves (e.g., spiral and cycloid) that cannot be expressed by the Bézier curve. Nevertheless, the basis functions {𝑢^𝑛_𝑖(𝑡)}^𝑛_𝑖=0 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 {𝑏^𝑛1,𝑛2_𝑖(𝑡;𝜆₁,𝜆₂,…,𝜆_𝑚)}^𝑛1_𝑖=0 be 𝑛₁+1 polynomial functions of degree 𝑛₂ (called order 𝑛₁ and degree 𝑛₂) and {𝐏_𝑖}^𝑛1_𝑖=0 be 𝑛₁+1 points in spaces, the parametric curve with multiple shape parameters {𝜆_𝑖}^𝑚_𝑖=0 is constructed as follows: 𝐏𝑡;𝜆₁,𝜆₂,…,𝜆_𝑚=^𝑛1_𝑖=0𝐏_𝑖𝑏^𝑛1,𝑛2_𝑖𝑡;𝜆₁,𝜆₂,…,𝜆_𝑚. (5)

For the sake of concision, the notations {𝑏^𝑛1,𝑛2_𝑖(𝑡;𝜆₁,𝜆₂,…,𝜆_𝑚)}^𝑛1_𝑖=0 and 𝐏(𝑡;𝜆₁,𝜆₂,…,𝜆_𝑚) will be replaced by {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 and 𝐏(𝑡). And {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 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 {𝜆_𝑖}^𝑚_𝑖=1 in {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0, the curve 𝐏(𝑡) can be freely adjusted and controlled by changing the value of {𝜆_𝑖}^𝑚_𝑖=1 instead of changing the control points {𝐏_𝑖}^𝑛1_𝑖=0. The existing works are compared in detail in Table 1.

Table 1 

Properties of the basis functions and the curves with 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: 𝐛^𝑛1,𝑛2=𝑏^𝑛1,𝑛2₀(𝑡),𝑏^𝑛1,𝑛2₁(𝑡),…,𝑏^𝑛1,𝑛2_𝑛1(𝑡,𝐏_𝑛1)=𝐏₀,𝐏₁,…,𝐏_𝑛1^𝑇. (6)

Equation (5) can be rewritten as 𝐏(𝑡)=𝐛^𝑛1,𝑛2𝐏^𝑛1. (7)

Given that {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 are polynomials with degree 𝑛₂, they can be seen as the linear combination of the Bernstein basis functions {𝐵^𝑛2_𝑖(𝑡)}^𝑛2_𝑖=0 with degree 𝑛₂ given by 𝐛^𝑛1,𝑛2=𝐁^𝑛2𝐌^𝑛2,𝑛1,𝐁^𝑛2=𝐵^𝑛2₀(𝑡),𝐵^𝑛2₁(𝑡),…,𝐵^𝑛2_𝑛2(𝑡),𝐌^𝑛2,𝑛1=𝑚_𝑖𝑗^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0. (8)

Thus, as long as the elements in the matrix 𝐌^𝑛2,𝑛1 are determined, the Quasi-Bernstein basis functions {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 with order 𝑛₁ and degree 𝑛₂ are completely constructed. Except for several elements that can be determined in 𝐌^𝑛2,𝑛1, the rest are shape parameters of the Quasi-Bernstein basis functions and the Quasi-Bézier curve. Here, the matrix 𝐌^𝑛2,𝑛1 is called the shape parameter matrix.

2.2. Construction of the Shape Parameter Matrix M^𝑛2,𝑛1

The (𝑛₂+1)(𝑛₁+1) elements of 𝑚_𝑖𝑗 in 𝐌^𝑛2,𝑛1 must be determined so that {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 and 𝐏(𝑡) become the Quasi-Bernstein basis functions and the Quasi-Bézier curve, respectively.

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

The Quasi-Bernstein basis functions {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 with order 𝑛₁ and degree 𝑛₂ must satisfy the characteristics of nonnegativity, normalization, symmetry, linear independence, and degeneracy.

Proposition 1 (nonnegativity). A sufficient condition for 𝑏^𝑛1,𝑛2_𝑖(𝑡)≥0(𝑖=0,1,…,𝑛₂) is 𝑚_𝑖𝑗≥0𝑖=0,1,…,𝑛₂,𝑗=0,1,…,𝑛₁. (9)

Proof. Here, 𝑏^𝑛1,𝑛2_𝑗(𝑡)=∑^𝑛2_𝑖=0𝑚_𝑖𝑗𝐵^𝑛2_𝑖(𝑡) is known to have been extracted from (8). Based on the non-negativity of the Bernstein basis functions {𝐵^𝑛2_𝑖(𝑡)}^𝑛2_𝑖=0, a sufficient condition for the non-negativity of the Quasi-Bernstein basis functions {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 is the non-negativity of the elements 𝑚_𝑖𝑗 in 𝐌^𝑛2,𝑛1. Hence, 𝑚_𝑖𝑗 must satisfy (9).

Note 1. Clearly, there is no row with all elements being 0 in 𝐌^𝑛2,𝑛1. In other words, ^𝑛2_𝑖=0𝑚_𝑖𝑗≠0𝑗=0,1…,𝑛₁. (10)

Proposition 2 (normalization). The necessary and sufficient condition for ∑^𝑛1_𝑗=0𝑏^𝑛1,𝑛2_𝑗(𝑡)=1 is given by ^𝑛1_𝑗=0𝑚_𝑖𝑗=1𝑖=0,1,…,𝑛₂. (11)

Proof. It is known that ^𝑛1_𝑗=0𝑏^𝑛1,𝑛2_𝑗(𝑡)=−1_{𝑛1𝑗=0𝑛2𝑖=0}𝑚_𝑖𝑗𝐵^𝑛2_𝑖(𝑡)=−1_{𝑛2𝑖=0𝑛1𝑗=0}𝑚_𝑖𝑗𝐵^𝑛2_𝑖(𝑡)=−1_{𝑛2𝑖=0𝑛1𝑗=0}𝑚_𝑖𝑗𝐵^𝑛2_𝑖(𝑡)−^𝑛2_𝑖=0𝐵^𝑛2_𝑖(𝑡)=^𝑛2_𝑖=0^𝑛1_𝑗=0𝑚_𝑖𝑗−1𝐵^𝑛2_𝑖(𝑡). (12)
According to the linear independence of the Bernstein basis functions {𝐵^𝑛2_𝑖(𝑡)}^𝑛2_𝑖=0, the necessary and sufficient condition for ∑^𝑛1_𝑗=0𝑏^𝑛1,𝑛2_𝑗(𝑡)−1=0 is ∑^𝑛1_𝑗=0𝑚_𝑖𝑗=1(𝑖=0,1,…,𝑛₂).

Note 2. By combining (9) and (11), 𝑚_𝑖𝑗 satisfies 0≤𝑚_𝑖𝑗≤1(𝑖=0,1,…,𝑛₂,𝑗=0,1,…,𝑛₁).

Proposition 3 (symmetry). The necessary and sufficient condition for 𝑏^𝑛1,𝑛2_𝑗(𝑡)=𝑏^𝑛1,𝑛2_𝑛1−𝑗(1−𝑡)(𝑗=0,1,…,𝑛₁) is given by 𝑚_𝑖𝑗=𝑚_{𝑛2−𝑖,𝑛1−𝑗}𝑖=0,1,…,𝑛₂,𝑗=0,1,…,𝑛₁. (13)

Proof. According to the symmetry of the Bernstein basis functions {𝐵^𝑛2_𝑖(𝑡)}^𝑛2_𝑖=0 of 𝐵^𝑛2_𝑖(𝑡)=𝐵^𝑛2_𝑛2−𝑖(1−𝑡), the following can be derived: 𝑏^𝑛1,𝑛2_𝑛1−𝑗(1−𝑡)−𝑏^𝑛1,𝑛2_𝑗(𝑡)=^𝑛2_𝑖=0𝐵^𝑛2_𝑖(1−𝑡)𝑚_{𝑖,𝑛1−𝑗}−^𝑛2_𝑖=0𝐵^𝑛2_𝑖(𝑡)𝑚_𝑖𝑗=^𝑛2_𝑖=0𝐵^𝑛2_𝑛2−𝑖(𝑡)𝑚_{𝑖,𝑛1−𝑗}−^𝑛2_𝑖=0𝐵^𝑛2_𝑖(𝑡)𝑚_𝑖𝑗=^𝑛2_𝑖=0𝐵^𝑛2_𝑖(𝑡)𝑚_{𝑛2−𝑖,𝑛1−𝑗}−^𝑛2_𝑖=0𝐵^𝑛2_𝑖(𝑡)𝑚_𝑖𝑗=^𝑛2_𝑖=0𝐵^𝑛2_𝑖(𝑡)𝑚_{𝑛2−𝑖,𝑛1−𝑗}−𝑚_𝑖𝑗,𝑗=0,1,…,𝑛₁. (14)
According to the linear independence of the Bernstein basis functions {𝐵^𝑛2_𝑖(𝑡)}^𝑛2_𝑖=0, the necessary and sufficient condition for 𝑏^𝑛1,𝑛2_𝑗(𝑡)−𝑏^𝑛1,𝑛2_𝑛1−𝑗(1−𝑡)=0 is (13).

Proposition 4 (linear independence). The necessary and sufficient condition for the linear independence of {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 is given by 𝑟𝐌^𝑛2,𝑛1=𝑛₁+1. (15)

Proof. It is known that ^𝑛1_𝑗=0𝑘_𝑗𝑏^𝑛1,𝑛2_𝑗(𝑡)=^𝑛1_𝑗=0𝑘_𝑗^𝑛2_𝑖=0𝑚_𝑖𝑗𝐵^𝑛2_𝑖(𝑡=_{𝑛2𝑖=0𝑛1𝑗=0})𝑘_𝑗𝑚_𝑖𝑗𝐵^𝑛2_𝑖(𝑡). (16)
According to the linear independence of the Bernstein basis functions {𝐵^𝑛2_𝑖(𝑡)}^𝑛2_𝑖=0, the necessary and sufficient condition for ∑^𝑛1_𝑗=0𝑘_𝑗𝑏^𝑛1,𝑛2_𝑗(𝑡)=0 is given by ^𝑛1_𝑗=0𝑘_𝑗𝑚_𝑖𝑗=0𝑖=0,1,…,𝑛₂. (17)
𝐌^𝑗=(𝑚_0,𝑗,𝑚_1,𝑗,…,𝑚_𝑛2,𝑗)^𝑇 is defined as the 𝑗th column vector of 𝐌^𝑛2,𝑛1. Equation (17) is equivalent to ∑^𝑛1_𝑗=0𝑘_𝑗𝐌^𝑗=0. Thus, the necessary and sufficient condition for the linear independence of {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 is also the linear independence of the column vectors {𝐌^𝑗}^𝑛1_𝑖=0 of the matrix 𝐌^𝑛2,𝑛1. Consequently, the necessary and sufficient condition for the linear independence of {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 is that the rank of the shape parameter matrix 𝐌^𝑛2,𝑛1 satisfies 𝑟(𝐌^𝑛2,𝑛1)=𝑛₁+1.

Note 3. When (15) is true, 𝑛₂≥𝑛₁.

Proposition 5 (degeneracy). If the elements {𝑚_𝑖𝑗}^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0 in the matrix 𝐌^𝑛2,𝑛1 are represented by (18), the Quasi-Bernstein basis functions {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 with order 𝑛₁ and degree 𝑛₂ are degenerated into the Bernstein basis functions {𝐵^𝑛1_𝑖(𝑡)}^𝑛1_𝑖=0 with order 𝑛₁. 𝑚_𝑖𝑗=^{𝑛2−𝑛1𝑖−𝑗𝑛1𝑗𝑛2𝑖},max0,𝑖−𝑛₂−𝑛₁≤𝑗≤min𝑛₁,𝑖,0≤𝑖≤𝑛₂,0≤𝑗≤𝑛₁;0,else. (18)

Proof. When the elements {𝑚_𝑖𝑗}^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0 in the matrix 𝐌^𝑛2,𝑛1 are represented by (18), the following is obtained: 𝐁^𝑛1=𝐁^𝑛2𝐌^𝑛2,𝑛1. (19)
Comparing (19) with (8), Proposition 5 is proven.

Note 4. If 𝑛₁=𝑛₂,𝐌^𝑛2,𝑛1 is an identity matrix here.

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

Proposition 6 (interpolation at the endpoints). The necessary and sufficient condition for 𝐏(0)=𝐏₀ and 𝐏(1)=𝐏_𝑛1 is given by 𝑚₀₀=1,𝑚_0𝑗=0𝑗=1,2,…,𝑛₁,𝑚_𝑛2,𝑛1=1,𝑚_𝑛2,𝑗=0𝑗=0,1,…,𝑛₁−1. (20)(21)

Proof. Clearly, the necessary and sufficient condition for 𝐏(0)=𝐏₀ is the Quasi-Bernstein primary functions {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 satisfying the following: 𝑏^𝑛1,𝑛2_𝑖(0)=1,𝑖=0,0,𝑖≠0. (22)
It is known that 𝑏^𝑛1,𝑛2₀(0),𝑏^𝑛1,𝑛2₁(0),…,𝑏^𝑛1,𝑛2_𝑛1(0=𝐵_𝑛20)(0),𝐵^𝑛2₁(0),…,𝐵^𝑛2_𝑛2(0)⋅𝑚_𝑖𝑗^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0=(1,0,…,0)⋅𝑚_𝑖𝑗^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0=𝑚₀₀,𝑚₀₁,…,𝑚_0,𝑛1−1,𝑚_0𝑛1. (23)
Thus, the necessary and sufficient condition for (𝑏^𝑛1,𝑛2₀(0),𝑏^𝑛1,𝑛2₁(0),…,𝑏^𝑛1,𝑛2_𝑛1(0))=(1,0,…,0) is (20).
Similarly, the necessary and sufficient condition for 𝐏(1)=𝐏_𝑛1 is the Quasi-Bernstein primary functions {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 satisfying the following: 𝑏^𝑛1,𝑛2_𝑖(1)=1,𝑖=𝑛₁,0,𝑖≠𝑛₁. (24)
It is known that 𝑏^𝑛1,𝑛2₀(1),𝑏^𝑛1,𝑛2₁(1),…,𝑏^𝑛1,𝑛2_𝑛1(1=𝐵_𝑛20)(1),𝐵^𝑛2₁(1),…,𝐵^𝑛2_𝑛2(1)⋅𝑚_𝑖𝑗^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0=(0,…,0,1)⋅𝑚_𝑖𝑗^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0=𝑚_𝑛2,0,𝑚_𝑛2,1,…,𝑚_{𝑛2,𝑛1−1},𝑚_𝑛2,𝑛1. (25)
Thus, the necessary and sufficient condition for (𝑏^𝑛1,𝑛2₀(1),𝑏^𝑛1,𝑛2₁(1),…,𝑏^𝑛1,𝑛2_𝑛1(1))=(0,…,0,1) is (21).
Hence, the necessary and sufficient condition for 𝐏(1)=𝐏_𝑛1 is (21).

Note 5. When {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 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 for 𝐏^′(0)‖𝐏₁𝐏₀,𝐏^′(1)‖𝐏_𝑛𝐏_𝑛−1 is given by 𝑚₁₀+𝑚₁₁=1,𝑚₁₀𝑚≠1,_1𝑗=0𝑗=2,3,…,𝑛₁,𝑚_{𝑛2−1,𝑛1−1}+𝑚_{𝑛2−1,𝑛1}=1,𝑚_{𝑛2−1,𝑛1}𝑚≠1,_{𝑛2−1,𝑗}=0𝑗=0,1,…,𝑛₁−2. (26)(27)

Proof. It is known that 𝐏^′(0)=𝐏^′(𝑡)_𝑡=0=𝑏^𝑛1,𝑛2₀(𝑡),𝑏^𝑛1,𝑛2₁(𝑡),…,𝑏^𝑛1,𝑛2_𝑛1(𝑡)^′_𝑡=0×𝐏₀,𝐏₁,…,𝐏_𝑛1^𝑇=𝐵^𝑛2₀(𝑡),𝐵^𝑛2₁(𝑡),…,𝐵^𝑛2_𝑛2(𝑡)^′_𝑡=0⋅𝑚_𝑖𝑗^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0⋅𝐏₀,𝐏₁,…,𝐏_𝑛1^𝑇=𝑛₂(−1,1,0,…,0)⋅𝑚_𝑖𝑗^{𝑖=𝑛2,𝑗=𝑛1}_𝑖,𝑗=0⋅𝐏₀,𝐏₁,…,𝐏_𝑛1^𝑇=𝑛₂𝑚₁₀−𝑚₀₀,𝑚₁₁−𝑚₀₁,…,𝑚_1𝑛1−𝑚_0𝑛1×𝐏₀,𝐏₁,…,𝐏_𝑛1^𝑇. (28)
Clearly, the necessary and sufficient condition for 𝐏^′(0)‖𝐏₁𝐏₀ is (𝑚₁₀−𝑚₀₀)/(𝑚₁₁−𝑚₀₁)=−1,𝑚_1𝑗−𝑚_0𝑗=0(𝑗=2,3,…,𝑛₁) which verifies (26).
Similarly, the necessary and sufficient condition for 𝐏^′(1)‖𝐏_𝑛𝐏_𝑛−1 is (27).

Note 6. When {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 have the property of symmetry, (27) is equivalent to (26) according to (13).

2.2.3. Form of Shape Parameter Matrix 𝐌^𝑛2,𝑛1

All shape parameter matrixes that satisfy (9), (11), (13), (15), (20), (21), (26), and (27) have the following form: 𝐌^𝑛2,𝑛1=𝑚10⋯00₁₀1−𝑚₁₀𝑚⋯00₂₀𝑚₂₁⋯𝑚_2,𝑛1−1𝑚_2𝑛1𝑚⋮⋮⋱⋮⋮_2𝑛1𝑚_2,𝑛1−1⋯𝑚₂₁𝑚₂₀00⋯1−𝑚₁₀𝑚₁₀00⋯01_{(𝑛2+1)×(𝑛1+1)}. (29)

Here, 𝑚_𝑖𝑗 are variable shape parameters that satisfy ^𝑛1_𝑗=0𝑚_𝑖𝑗𝑛=1𝑖=2,3,…,²+12,0≤𝑚₁₀<1,0≤𝑚_𝑖𝑗𝑛≤1𝑖=2,3,…,²+12,𝑗=0,1,…,𝑛₁. (30)

2.3. The Characteristics of the Quasi-Bézier Curve

In summary, the Quasi-Bézier curve 𝐏(𝑡) based on the Quasi-Bernstein basis functions {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 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 labeled 𝐏₀𝐏₁⋯𝐏_𝑛1 or 𝐏_𝑛1𝐏_𝑛1−1⋯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 ^𝑛1_𝑖=0𝐏_𝑖𝑏^𝑛1,𝑛2_𝑖(𝑡)=^𝑛1_𝑖=0𝐏_𝑛1−𝑖𝑏^𝑛1,𝑛2_𝑛1−𝑖(1−𝑡), (31) which follows the inspection of (13);(e) convex hull property: this property exists since the Quasi-Bernstein basis functions {𝑏^𝑛1,𝑛2_𝑖(𝑡)}^𝑛1_𝑖=0 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 curve 𝐏(𝑡) interpolates the first and the last control points 𝐏(0)=𝐏₀ and 𝐏(1)=𝐏_𝑛1; the first and last edges of the control polygon are the tangent lines at the endpoints, where 𝐏^′(0)‖𝐏₁𝐏₀  and  𝐏^′(1)‖𝐏_𝑛𝐏_𝑛−1.

2.4. Geometric Significance of the Shape Parameters

According to (29), when 𝑚_𝑖,𝑗0(𝑖=0,1,…,𝑛₂,𝑗₀=0,1,…,𝑛₁) increases, 𝑏^𝑛1,𝑛2_𝑗0(𝑡) and 𝑏^𝑛1,𝑛2_{𝑛1−𝑗0}(𝑡) increase as well; specifically, 𝐏(𝑡) comes close to the control points 𝐏_𝑗0 and 𝐏_{𝑛1−𝑗0}. The geometric significance of the shape parameters is shown in Section 3.

3. Numerical Examples

Example 1. The shape parameter matrix 𝐌^3,2 is constructed from (29). The corresponding Quasi-Bernstein basis functions and the Quasi-Bézier curves with different shape parameter 𝑚₁₀ are given as follows: 𝐌^3,2=𝑚100₁₀1−𝑚₁₀001−𝑚₁₀𝑚₁₀001,0≤𝑚₁₀<1. (32)
The geometric significance of the shape parameters 𝑚₁₀ is shown in Figure 1. As the value of the shape parameter 𝑚₁₀ increases, the elements in the second column of 𝐌^3,2 decrease. According to (8), the second Quasi-Bernstein basis function 𝑏₁^2,3(𝑡) decreases. So, the corresponding Quasi-Bézier curve moves away from the control point 𝐏₁ (see Figure 1(b)).

(a)

(b)

Figure 1 

Quasi-Bernstein basis functions and Quasi-Bézier curves when 𝑛1=2  and  𝑛2=3.

Example 2. The shape parameter matrix 𝐌^4,2 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: 𝐌^4,2=𝑚100₁₀1−𝑚₁₀0𝑚₂₀1−2𝑚₂₀𝑚₂₀01−𝑚₁₀𝑚₁₀,0010≤𝑚₁₀<1,0≤𝑚₂₀≤12. (33)
The geometric significance of the shape parameters 𝑚₁₀ 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 𝐌^4,2 decrease. According to (8), the second Quasi-Bernstein basis function 𝑏₁^2,4(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 of 𝑚₂₀ 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)).

(a)

(b)

Figure 2 

Quasi-Bernstein basis functions and Quasi-Bézier curves when 𝑛1=2  and 𝑛2=4.

Example 3. The shape parameter matrix 𝐌^3,3 is constructed from (29). The corresponding Quasi-Bernstein basis functions and the Quasi-Bézier curves with different shape parameter 𝑚₁₀ are given as follows: 𝐌^3,3=𝑚1000₁₀1−𝑚₁₀00001−𝑚₁₀𝑚₁₀0001,0≤𝑚₁₀<1. (34)
The geometric significance of the shape parameters 𝑚₁₀ is shown in Figure 3. As the value of the shape parameter 𝑚₁₀ increases, the elements in the second and the third column of 𝐌^3,3 decrease. According to (8), the second Quasi-Bernstein basis function 𝑏₁^3,3(𝑡) and the third Quasi-Bernstein basis function 𝑏₂^3,3(𝑡) decrease. So, the corresponding Quasi-Bézier curve moves away from the control points 𝐏₁ and 𝐏₂ (see Figure 3(b)).

(a)

(b)

Figure 3 

Quasi-Bernstein basis functions and Quasi-Bézier curves when 𝑛1=3  and 𝑛2=3.

Example 4. The shape parameter matrix 𝐌^4,3 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: 𝐌^4,3=𝑚1000₁₀1−𝑚₁₀𝑚00₂₀12−𝑚₂₀12−𝑚₂₀𝑚₂₀001−𝑚₁₀𝑚₁₀,00010≤𝑚₁₀<1,0≤𝑚₂₀≤12. (35)
The geometric significance of the shape parameters 𝑚₁₀ 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 𝐌^4,3 decrease. According to (8), the second Quasi-Bernstein basis function 𝑏₁^3,4(𝑡) and the third Quasi-Bernstein basis function 𝑏₂^3,4(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 of 𝑚₂₀ 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)).

(a)

(b)

Figure 4 

Quasi-Bernstein basis functions and Quasi-Bézier curves when 𝑛1=3 and 𝑛2=4.

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]: 𝑏^{𝑛1+1,𝑛2}_𝑖⁺¹(𝑡)=(1−𝑡)𝑏^𝑛1,𝑛2_𝑖(𝑡)+𝑡𝑏^𝑛1,𝑛2_𝑖−1(𝑡),𝑖=0,1,…,𝑛₁. (36)
Here, we set 𝑏^𝑛1,𝑛2₋₁(𝑡)=𝑏^𝑛1,𝑛2_𝑛1+1(𝑡)=0.

Example 5. Figure 5 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. Figure 6 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 𝐌^𝑛+1,𝑛=10⋯001−𝜆₁𝑛+11−1−𝜆₁0𝑛+1⋯002−𝜆₂𝑛+1⋯00⋮⋮⋱⋮⋮00⋯2−𝜆₂0𝑛+100⋯1−1−𝜆₁𝑛+11−𝜆₁𝑛+100⋯01_{(𝑛+2)×(𝑛+1)}. (37)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 Figures 1(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.

Acknowledgments

The author is very grateful to the anonymous referees for the inspiring comments and the valuable suggestions which improved the paper considerably. This work has been supported by the National Natural Science Foundation of China (no. Y5090377) and the Natural Science Foundation of Ningbo (nos. 2011A610174 and 2012A610029).