# Approximate Multidegree Reduction ofλ-Bézier Curves

Authors: Gang Hu; Huanxin Cao; Suxia Zhang
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Besides inheriting the properties of classical Bézier curves of degreen, the corresponding λ-Bézier curves have a good performance in adjusting their shapes by changing shape control parameter. In this paper, we derive an approximation algorithm for multidegree reduction of λ-Bézier curves in the L2-norm. By analysing the properties of λ-Bézier curves of degree n, a method which can deal with approximating λ-Bézier curve of degree n+1 by λ-Bézier curve of degree m(m≤n) is presented. Then, in unrestricted and C0, C1 constraint conditions, the new control points of approximating λ-Bézier curve can be obtained by solving linear equations, which can minimize the least square error between the approximating curves and the original ones. Finally, several numerical examples of degree reduction are given and the errors are computed in three conditions. The results indicate that the proposed method is effective and easy to implement.


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## Body

## 1. Introduction

Bézier curves are one of the main mathematical models in CAD/CAM system [1]. Degree reduction of Bézier curve is an important technique in geometric computation and geometric approximation [2] and has great significance for shape design. Firstly, it is embodied in data transfer and exchange between CAD systems or in CAD system, because the highest allowable degree of Bernstein polynomial for curve is generally different in various CAD systems or models. Next, degree reduction of curve is favorable for data compression. With the popularization of digitized and network product design, data communication between design systems becomes quite frequent [3], and geometric data in design system has come to mass [4]. Therefore, the operation of degree reduction attracts a good deal of attention.The issue of degree reduction of Bézier curves is concerned with the solution of the following problem: for a given Bézier curveRn(t) of degree n with Bézier points rii=0n, find an approximate Bézier curve R~m(t) of lower degree m, where m<n, with the set of Bézier points r~ii=0m, so that Rn and R~m satisfy boundary conditions at the end points, and the error between Rn and R~m is minimum. For degree reduction of Bézier curves, many scholars have done a lot of research that can be classified into three categories: geometry of approximate control point [5–8], algebraic means of basis function transformations [9–14], and B net and constrained optimization [15, 16]. Watkins and Worsey [9] presented an algorithm for generating (n-1)st degree approximation to nth degree Bézier curve. Eck [10] investigated a complete algorithm for performing the degree reduction within a prescribed error tolerance by help of subdivision. Chen and Wang [11] investigated the problem of optimal multidegree reduction of Bézier curves with constraints of endpoints continuity. Zheng and Wang [12] proved that the problem of finding a best L2-approximation over the interval [0,1] for constrained degree reduction is equivalent to that of finding a minimum perturbation vector in a certain weighted Euclidean norm. Using the transformation matrices, Lu and Wang [13] presented a method for the best multidegree reduction with respect to t-t2-weighted square norm for the unconstrained case. Tan and Fang [14] proposed three methods for degree reduction of interval generalized Ball curves of Wang-Said type. Degree reduction of Bézier curves has been conducted according to different norms, mostly L2-norms, for both unconstrained and constrained conditions. In general, unconstrained degree reduction gives lower error than the constrained one. However, Bézier curves are often a part of a piecewise curve, so constrained degree reduction is preferred.Although Bézier curves have now become a powerful tool for constructing free-form curves in CAD/CAM, they have their own disadvantages. Specifically, the shape of a Bézier curve is well-determined by its control points after choosing the basis functions [17]. In recent years, in order to improve the performance of Bézier curves, many scholars have constructed some new curves which are similar to the Bézier ones by introducing parameters into basis functions; see [18–22]. These new curves share many basic properties with the Bézier ones. Furthermore, they hold the property of flexible shape adjustability. Yan and Liang [18] constructed a new kind of basis function by a recursive approach; thus a kind of parametric curves with shape parameter is defined, which are called λ-Bézier curves. These new curves have most properties of the corresponding classical Bézier curves. Moreover, the shape parameter can adjust the shape of the new curves without changing the control points. Focusing on degree reduction of λ-Bézier curves, we study the corresponding problem by the least square method and obtain the new control points as well as the shape parameter of approximating λ-Bézier curves.The remainder of the paper is organized as follows. The definition and properties ofλ-Bézier curves are introduced in Section 2. In Section 3, we give the problem description of approximating degree reduction. In Section 4, we present the least square degree reduction of λ-Bézier curve. Numerical examples are given in Section 5, and we present some applications. At last, a short conclusion is given in Section 6.


## 2. The Definition and Properties ofλ-Bézier Curves

### 2.1. Extension of Basis Function

The definition of extension Bernstein basis functions is given as follows [18].Definition 1.
Letλ∈[-1,1]; for any t∈[0,1], the polynomial functions (1)b0,2t;λ=1-2λt+λt21-t2,b1,2t;λ=2t1-t1+λ-λt+λt2,b2,2t;λ=1-λ+λt2t2are called the extension Bernstein basis functions of degree 2 associated with the shape parameter λ.
For any integern(n≥3), the functions bi,n(t;λ)(i=0,1,…,n) defined recursively by(2)bi,nt;λ=1-tbi,n-1t;λ+tbi-1,n-1t;λ,t∈0,1,are called the extension Bernstein basis functions of degree n. In the case k=-1 or k>l, we set bk,l(t;λ)=0.Theorem 2.
The extension Bernstein basis functions of degreen can be expressed explicitly as(3)bi,nt;λ=1+3Cn-2i-1+Cn-1i-CniCniλ-2Cn-1iCniλt+λt2Cniti1-tn-ii=0,1,…,n,where n≥2,Cni=n!/i!(n-i)!.


### 2.2. Construction ofλ-Bézier Curves

Definition 3.
Given control pointsPi∗(i=0,1,…,n;n≥2) in R2 or R3, then(4)p∗t;λ=∑i=0nPi∗bi,nt;λ,t∈0,1,λ∈-1,1is called a λ-Bézier curve of degree n with shape parameter λ, where basis functions bi,n(t;λ)(i=0,1,…,n;n≥2) are defined by (3) (see Definition 3.2 in [18]).
When the shape parameterλ is equal to zero, λ-Bézier curves degenerate to the classical Bézier curves. λ-Bézier curve inherits most properties of the classical Bézier curve, such as convex hull property, geometric invariance, symmetry, and the following terminal property:(5)p∗0;λ=P0∗,p∗1;λ=Pn∗,dp∗t;λdtt=0=n+2λP1∗-P0∗,dp∗t;λdtt=1=n+2λPn∗-Pn-1∗.Because of introducing parameter λ, λ-Bézier curves have more powerful expressiveness than the classical Bézier curves.
Figure1 shows graphs of λ-Bézier curves with the same control polygon but different shape parameters. Figure 1(a) shows the curves generated by the extension Bernstein basis functions with n=3 and p∗(t;1) (solid lines), p∗(t;0) (dashed lines), and p∗(t;-1) (dot-dashed lines), respectively. Figure 1(b) shows the curves generated by the same basis functions as in Figure 1(a) with n=4 and p∗(t;1) (solid lines), p∗(t;0) (dashed lines), and p∗(t;-1) (dot-dashed lines), respectively. From the figures, we can see that λ-Bézier curves approach the control polygon when the shape parameter is increasing.Figure 1
λ-Bézier curves with the same control polygon but different shape parameters.
(a)
λ-Bézier curve of degree 3
(b)
λ-Bézier curve of degree 4


## 2.1. Extension of Basis Function

The definition of extension Bernstein basis functions is given as follows [18].Definition 1.
Letλ∈[-1,1]; for any t∈[0,1], the polynomial functions (1)b0,2t;λ=1-2λt+λt21-t2,b1,2t;λ=2t1-t1+λ-λt+λt2,b2,2t;λ=1-λ+λt2t2are called the extension Bernstein basis functions of degree 2 associated with the shape parameter λ.
For any integern(n≥3), the functions bi,n(t;λ)(i=0,1,…,n) defined recursively by(2)bi,nt;λ=1-tbi,n-1t;λ+tbi-1,n-1t;λ,t∈0,1,are called the extension Bernstein basis functions of degree n. In the case k=-1 or k>l, we set bk,l(t;λ)=0.Theorem 2.
The extension Bernstein basis functions of degreen can be expressed explicitly as(3)bi,nt;λ=1+3Cn-2i-1+Cn-1i-CniCniλ-2Cn-1iCniλt+λt2Cniti1-tn-ii=0,1,…,n,where n≥2,Cni=n!/i!(n-i)!.


## 2.2. Construction ofλ-Bézier Curves

Definition 3.
Given control pointsPi∗(i=0,1,…,n;n≥2) in R2 or R3, then(4)p∗t;λ=∑i=0nPi∗bi,nt;λ,t∈0,1,λ∈-1,1is called a λ-Bézier curve of degree n with shape parameter λ, where basis functions bi,n(t;λ)(i=0,1,…,n;n≥2) are defined by (3) (see Definition 3.2 in [18]).
When the shape parameterλ is equal to zero, λ-Bézier curves degenerate to the classical Bézier curves. λ-Bézier curve inherits most properties of the classical Bézier curve, such as convex hull property, geometric invariance, symmetry, and the following terminal property:(5)p∗0;λ=P0∗,p∗1;λ=Pn∗,dp∗t;λdtt=0=n+2λP1∗-P0∗,dp∗t;λdtt=1=n+2λPn∗-Pn-1∗.Because of introducing parameter λ, λ-Bézier curves have more powerful expressiveness than the classical Bézier curves.
Figure1 shows graphs of λ-Bézier curves with the same control polygon but different shape parameters. Figure 1(a) shows the curves generated by the extension Bernstein basis functions with n=3 and p∗(t;1) (solid lines), p∗(t;0) (dashed lines), and p∗(t;-1) (dot-dashed lines), respectively. Figure 1(b) shows the curves generated by the same basis functions as in Figure 1(a) with n=4 and p∗(t;1) (solid lines), p∗(t;0) (dashed lines), and p∗(t;-1) (dot-dashed lines), respectively. From the figures, we can see that λ-Bézier curves approach the control polygon when the shape parameter is increasing.Figure 1
λ-Bézier curves with the same control polygon but different shape parameters.
(a)
λ-Bézier curve of degree 3
(b)
λ-Bézier curve of degree 4


## 3. Problem Description

Problem 4.
Given control points{Pi∗}i=0n+1⊂Rs (s=2,3), λ-Bézier curve of degree n+1 is expressed as follows: (6)p∗t;λ=∑i=0n+1Pi∗bi,n+1t;λ,t∈0,1,where {bi,n+1(t;λ)}i=0n+1 are basis functions of degree n+1 and λ∈[-1,1] is global shape parameter. Given control points {Pi}i=0m⊂Rs, the corresponding λ-Bézier curves of degree m (m≤n) are(7)pt;λ=∑i=0mPibi,mt;λ,such that the distance minimizes between p∗(t;λ) and p(t;λ) in certain distance function d(p∗(t;λ),p(t;λ)).
Here we are interested in obtaining explicit expression of approximatingλ-Bézier curves p(t;λ), so we choose the following least square distance function:(8)d2p∗t;λ,pt;λ=∫01p∗t;λ-pt;λ22dt.
Then we can convert Problem4 into s subproblems, and every subproblem leads to a minimized component function:(9)d2pk∗t;λ,pkt;λ=∫01pk∗t;λ-pkt;λ2dtk=1,2,…,s.Let p∗t;λ=(p1∗t;λ,p2∗t;λ,…,ps∗(t;λ)) and p(t;λ)=(p1(t;λ),p2(t;λ),…,ps(t;λ)); then (8) is determined by the following formula:(10)dp∗t;λ,pt;λ=∑k=1sd2pk∗t;λ,pkt;λ1/2.For subdistance function d2(pk∗(t;λ),pk(t;λ)), it is sufficient to minimize d(p∗(t;λ),p(t;λ)). Therefore, we can just study the problem of minimum component function in the following.Problem 5.
Given a series of real numbers{Pi∗}i=0n+1, from which we will determine λ-Bézier functions of degree n+1,(11)f∗t;λ=∑i=0n+1Pi∗bi,n+1t;λ,where Pi∗ denotes a component of vector Pi∗, then it is necessary to find real numbers {Pj}j=0m with the corresponding λ-Bézier functions of degree m(12)ft;λ=∑j=0mPjbj,mt;λsuch that d2(f∗(t;λ),f(t;λ))=∫01[f∗(t;λ)-f(t;λ)]2dt minimizes by least square distance.
In order to determine the approximate functionf(t;λ), primarily, we aim to obtain the coefficients {Pj}j=0m.


## 4. Least Square Degree Reduction ofλ-Bézier Curves

### 4.1. The Approximate Degree Reduction ofλ-Bézier Curves under Unrestricted Condition

Theorem 6.
If coefficients{Pj}j=0m of approximating functions f(t;λ) are solutions of Problem 5, the vector P=(P0,P1,…,Pm)T satisfies linear systems AP=b, where(13)A=ai,jm+1,m+1,b=b1,b2,…,bm+1T,ai+1,j+1=∫01bi,mt;λbj,mt;λdt,bj+1=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt.i,j=0,1,…,m.Proof.
By Problem5, we obtain(14)S=d2f∗t;λ,ft;λ=∫01f∗t;λ-ft;λ2dt=∫01∑i=0n+1Pi∗bi,n+1t;λ-∑j=0mPjbj,mt;λ2dt.
Let∂S/∂Pj=0(j=0,1,…,m); then the above equations can be simplified to the following ones:(15)∑i=0mPi∫01bi,mt;λbj,mt;λdt=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdtj=0,1,…,m.
Furthermore, (15) can be represented in matrix form by calculation, which is described as follows:(16)AP=b,where(17)A=ai,jm+1,m+1,b=b1,b2,…,bm+1T,ai+1,j+1=∫01bi,mt;λbj,mt;λdt,bj+1=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt.i,j=0,1,…,m.Let ej+1=(a1,j+1,a2,j+1,…,am+1,j+1)T(j=0,1,…,m), and suppose(18)∑j=0mcj+1ej+1=c1a1,1a2,1⋮am+1,1+c2a1,2a2,2⋮am+1,2+⋯+cm+1a1,m+1a2,m+1⋮am+1,m+1=00⋮0.That is,(19)∑j=0mcj+1ai+1,j+1=∫01∑j=0mcj+1bj,mt;λbi,mt;λdt=0i=0,1,…,m.We then get the following formula:(20)∫01∑j=0mcj+1bj,mt;λ2dt=∫01∑j=0mcj+1bj,mt;λ∑i=0mci+1bi,mt;λdt=∑i=0mci+1∫01∑j=0mcj+1bj,mt;λbi,mt;λdt=0;thus ∑j=0mcj+1bj,m(t;λ)≡0. Since b0,mt;λ,b1,mt;λ,…,bm,m(t;λ) are linearly independent in interval t∈[0,1], we have cj+1≡0(j=0,1,…,m), which means vectors e1,e2,…,em+1 are linearly independent, and then solutions of linear systems (16) are uniquely determined.


### 4.2. The Approximate Degree Reduction ofλ-Bézier Curves under C0 Constraint Condition

When approximating degree reduction, we expect to satisfyC0 continuity, that is, maintaining interpolation of terminal points, so two equations P0=P0∗ and Pm=Pn+1∗ are determined. The remaining m-1 control points are determined by the following theorem.Theorem 7.
If coefficients{Pi}i=0m of approximating functions f(t;λ) are solutions of Problem 5 and maintain C0 continuity, the vector P=P1,P2,…,Pm-1Tsatisfies linear systems AP=b except for two equations P0=P0∗ and Pm=Pn+1∗ for terminal points, where(21)A=ai,jm-1,m-1,b=b1,b2,…,bm-1T,ai,j=∫01bi,mt;λbj,mt;λdt,bj=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt-∫01P0∗b0,mt;λ+Pn+1∗bm,mt;λbj,mt;λdt.i,j=1,2,…,m-1.Proof.
According to the condition ofC0 continuity, that is, f∗(0;λ)=f(0;λ) and f∗(1;λ)=f(1;λ), it is easy to obtain two equations P0=P0∗ and Pm=Pn+1∗. Then by applying Problem 5, we get(22)S=d2f∗,f=∫01f∗t;λ-ft;λ2dt=∫01∑i=0n+1Pi∗bi,n+1t;λ-∑j=0mPjbj,mt;λ2dt.
Let∂S/∂Pj=0(j=1,2,…,m-1). Equation (22) can be simplified to the following equations:(23)∑i=1m-1Pi∫01bi,mt;λbj,mt;λdt=∫01∑i=0n+1Pi∗bi,n+1t;λ-P0∗b0,mt;λ-Pn+1∗bm,mt;λbj,mt;λdt.
Furthermore, these equations can be represented in matrix form as follows:(24)AP=b,where(25)A=ai,jm-1,m-1,b=b1,b2,…,bm-1T,ai,j=∫01bi,mt;λbj,mt;λdt,bj=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt-∫01P0∗b0,mt;λ+Pn+1∗bm,mt;λbj,mt;λdt.i,j=1,2,…,m-1.Let ej=(a1,j,a2,j,…,am-1,j)T(j=1,2,…,m-1), and suppose(26)∑j=1m-1cjej=c1a1,1a2,1⋮am-1,1+c2a1,2a2,2⋮am-1,2+⋯+cm-1a1,m-1a2,m-1⋮am-1,m-1=00⋮0.That is,(27)∑j=1m-1cjai,j=∫01∑j=1m-1cjbj,mt;λbi,mt;λdt=0i=1,2,…,m-1.Because b1,mt;λ,b2,mt;λ,…,bm-1,m(t;λ) are linearly independent in interval t∈[0,1], the vectors e1,e2,…,em-1, as in Theorem 6, are linearly independent. Thus solutions of linear systems (24) are uniquely determined and maintain C0 continuity.


### 4.3. The Approximate Degree Reduction ofλ-Bézier Curves under C1 Constraint Condition

When approximating degree reduction, ifC1 continuity is maintained (i.e., four equations P0=P0∗, Pm=Pn+1∗, P1=P0∗+((n+1+2λ)/(m+2λ))(P1∗-P0∗), and Pm-1=Pn+1∗-((n+1+2λ)/(m+2λ))(Pn+1∗-Pn∗) are specified), the remaining m-3 control points are determined by the following theorem.Theorem 8.
If coefficients{Pi}i=0m of approximate functions f(t;λ) are solutions of Problem 5 and maintainC1 continuity, the vector P=P2,P3,…,Pm-2T satisfies linear systems AP=b except for four equations P0=P0∗, Pm=Pn+1∗, P1=P0∗+(n+1+2λ/(m+2λ))(P1∗-P0∗), and Pm-1=Pn+1∗-((n+1+2λ)/(m+2λ))(Pn+1∗-Pn∗) for terminal points, where(28)A=ai-1,j-1m-3,m-3,b=b1,b2,…,bm-3T,ai-1,j-1=∫01bi,mt;λbj,mt;λdt,bj-1=∫01∑i=0n+1Pi∗bi,n+1t;λ-P0b0,mt;λ-P1b1,mt;λbj,mt;λdt-∫01Pm-1bm-1,mt;λ+Pmbm,mt;λbj,mt;λdt.i,j=2,3,…,m-2.Proof.
According to the condition ofC1 continuity, we get(29)f∗0;λ=f0;λ,f∗1;λ=f1;λ,df∗t;λdtt=0=dft;λdtt=0,df∗t;λdtt=1=dft;λdtt=1.It is easy to obtain the following four equations:(30)P0=P0∗,Pm=Pn+1∗,P1=P0∗+n+1+2λm+2λP1∗-P0∗,Pm-1=Pn+1∗-n+1+2λm+2λPn+1∗-Pn∗.
Then by Problem5, we obtain(31)S=d2f∗,f=∫01f∗t;λ-ft;λ2dt=∫01∑i=0n+1Pi∗bi,n+1t;λ-∑j=0mPjbj,mt;λ2dt.
Let∂S/∂Pj=0(j=2,3,…,m-2). Equation (31) can be simplified to the following form:(32)∑i=2m-2Pi∫01bi,mt;λbj,mt;λdt=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt-∫01P0b0,mt;λ+P1b1,mt;λ+Pm-1bm-1,mt;λ+Pmbm,mt;λbj,mt;λdt.Furthermore, this equation can be represented in matrix form as follows:(33)AP=b,where(34)A=ai-1,j-1m-3,m-3,b=b1,b2,…,bm-3T,ai-1,j-1=∫01bi,mt;λbj,mt;λdt,bj-1=∫01∑i=0n+1Pi∗bi,n+1t;λ-P0b0,mt;λ-P1b1,mt;λbj,mt;λdt-∫01Pm-1bm-1,mt;λ+Pmbm,mt;λbj,mt;λdt.i,j=2,3…,m-2.Let ej=(a1,j,a2,j,…,am-3,j)T(j=1,2,…,m-3), and suppose(35)∑j=1m-3cjej=c1a1,1a2,1⋮am-3,1+c2a1,2a2,2⋮am-3,2+⋯+cm-3a1,m-3a2,m-3⋮am-3,m-3=00⋮0.That is,(36)∑j=1m-3cjai,j=∫01∑j=1m-3cjbj+1,mt;λbi+1,mt;λdt=0i=1,2,…,m-3.Because b2,m(t;λ),…,bm-2,m(t;λ) are linearly independent in interval t∈[0,1], the vectors e1,e2,…,em-3, as in Theorem 6, are linearly independent. Therefore, solutions of linear systems (33) exist uniquely and maintain C1 continuity.


## 4.1. The Approximate Degree Reduction ofλ-Bézier Curves under Unrestricted Condition

Theorem 6.
If coefficients{Pj}j=0m of approximating functions f(t;λ) are solutions of Problem 5, the vector P=(P0,P1,…,Pm)T satisfies linear systems AP=b, where(13)A=ai,jm+1,m+1,b=b1,b2,…,bm+1T,ai+1,j+1=∫01bi,mt;λbj,mt;λdt,bj+1=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt.i,j=0,1,…,m.Proof.
By Problem5, we obtain(14)S=d2f∗t;λ,ft;λ=∫01f∗t;λ-ft;λ2dt=∫01∑i=0n+1Pi∗bi,n+1t;λ-∑j=0mPjbj,mt;λ2dt.
Let∂S/∂Pj=0(j=0,1,…,m); then the above equations can be simplified to the following ones:(15)∑i=0mPi∫01bi,mt;λbj,mt;λdt=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdtj=0,1,…,m.
Furthermore, (15) can be represented in matrix form by calculation, which is described as follows:(16)AP=b,where(17)A=ai,jm+1,m+1,b=b1,b2,…,bm+1T,ai+1,j+1=∫01bi,mt;λbj,mt;λdt,bj+1=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt.i,j=0,1,…,m.Let ej+1=(a1,j+1,a2,j+1,…,am+1,j+1)T(j=0,1,…,m), and suppose(18)∑j=0mcj+1ej+1=c1a1,1a2,1⋮am+1,1+c2a1,2a2,2⋮am+1,2+⋯+cm+1a1,m+1a2,m+1⋮am+1,m+1=00⋮0.That is,(19)∑j=0mcj+1ai+1,j+1=∫01∑j=0mcj+1bj,mt;λbi,mt;λdt=0i=0,1,…,m.We then get the following formula:(20)∫01∑j=0mcj+1bj,mt;λ2dt=∫01∑j=0mcj+1bj,mt;λ∑i=0mci+1bi,mt;λdt=∑i=0mci+1∫01∑j=0mcj+1bj,mt;λbi,mt;λdt=0;thus ∑j=0mcj+1bj,m(t;λ)≡0. Since b0,mt;λ,b1,mt;λ,…,bm,m(t;λ) are linearly independent in interval t∈[0,1], we have cj+1≡0(j=0,1,…,m), which means vectors e1,e2,…,em+1 are linearly independent, and then solutions of linear systems (16) are uniquely determined.


## 4.2. The Approximate Degree Reduction ofλ-Bézier Curves under C0 Constraint Condition

When approximating degree reduction, we expect to satisfyC0 continuity, that is, maintaining interpolation of terminal points, so two equations P0=P0∗ and Pm=Pn+1∗ are determined. The remaining m-1 control points are determined by the following theorem.Theorem 7.
If coefficients{Pi}i=0m of approximating functions f(t;λ) are solutions of Problem 5 and maintain C0 continuity, the vector P=P1,P2,…,Pm-1Tsatisfies linear systems AP=b except for two equations P0=P0∗ and Pm=Pn+1∗ for terminal points, where(21)A=ai,jm-1,m-1,b=b1,b2,…,bm-1T,ai,j=∫01bi,mt;λbj,mt;λdt,bj=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt-∫01P0∗b0,mt;λ+Pn+1∗bm,mt;λbj,mt;λdt.i,j=1,2,…,m-1.Proof.
According to the condition ofC0 continuity, that is, f∗(0;λ)=f(0;λ) and f∗(1;λ)=f(1;λ), it is easy to obtain two equations P0=P0∗ and Pm=Pn+1∗. Then by applying Problem 5, we get(22)S=d2f∗,f=∫01f∗t;λ-ft;λ2dt=∫01∑i=0n+1Pi∗bi,n+1t;λ-∑j=0mPjbj,mt;λ2dt.
Let∂S/∂Pj=0(j=1,2,…,m-1). Equation (22) can be simplified to the following equations:(23)∑i=1m-1Pi∫01bi,mt;λbj,mt;λdt=∫01∑i=0n+1Pi∗bi,n+1t;λ-P0∗b0,mt;λ-Pn+1∗bm,mt;λbj,mt;λdt.
Furthermore, these equations can be represented in matrix form as follows:(24)AP=b,where(25)A=ai,jm-1,m-1,b=b1,b2,…,bm-1T,ai,j=∫01bi,mt;λbj,mt;λdt,bj=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt-∫01P0∗b0,mt;λ+Pn+1∗bm,mt;λbj,mt;λdt.i,j=1,2,…,m-1.Let ej=(a1,j,a2,j,…,am-1,j)T(j=1,2,…,m-1), and suppose(26)∑j=1m-1cjej=c1a1,1a2,1⋮am-1,1+c2a1,2a2,2⋮am-1,2+⋯+cm-1a1,m-1a2,m-1⋮am-1,m-1=00⋮0.That is,(27)∑j=1m-1cjai,j=∫01∑j=1m-1cjbj,mt;λbi,mt;λdt=0i=1,2,…,m-1.Because b1,mt;λ,b2,mt;λ,…,bm-1,m(t;λ) are linearly independent in interval t∈[0,1], the vectors e1,e2,…,em-1, as in Theorem 6, are linearly independent. Thus solutions of linear systems (24) are uniquely determined and maintain C0 continuity.


## 4.3. The Approximate Degree Reduction ofλ-Bézier Curves under C1 Constraint Condition

When approximating degree reduction, ifC1 continuity is maintained (i.e., four equations P0=P0∗, Pm=Pn+1∗, P1=P0∗+((n+1+2λ)/(m+2λ))(P1∗-P0∗), and Pm-1=Pn+1∗-((n+1+2λ)/(m+2λ))(Pn+1∗-Pn∗) are specified), the remaining m-3 control points are determined by the following theorem.Theorem 8.
If coefficients{Pi}i=0m of approximate functions f(t;λ) are solutions of Problem 5 and maintainC1 continuity, the vector P=P2,P3,…,Pm-2T satisfies linear systems AP=b except for four equations P0=P0∗, Pm=Pn+1∗, P1=P0∗+(n+1+2λ/(m+2λ))(P1∗-P0∗), and Pm-1=Pn+1∗-((n+1+2λ)/(m+2λ))(Pn+1∗-Pn∗) for terminal points, where(28)A=ai-1,j-1m-3,m-3,b=b1,b2,…,bm-3T,ai-1,j-1=∫01bi,mt;λbj,mt;λdt,bj-1=∫01∑i=0n+1Pi∗bi,n+1t;λ-P0b0,mt;λ-P1b1,mt;λbj,mt;λdt-∫01Pm-1bm-1,mt;λ+Pmbm,mt;λbj,mt;λdt.i,j=2,3,…,m-2.Proof.
According to the condition ofC1 continuity, we get(29)f∗0;λ=f0;λ,f∗1;λ=f1;λ,df∗t;λdtt=0=dft;λdtt=0,df∗t;λdtt=1=dft;λdtt=1.It is easy to obtain the following four equations:(30)P0=P0∗,Pm=Pn+1∗,P1=P0∗+n+1+2λm+2λP1∗-P0∗,Pm-1=Pn+1∗-n+1+2λm+2λPn+1∗-Pn∗.
Then by Problem5, we obtain(31)S=d2f∗,f=∫01f∗t;λ-ft;λ2dt=∫01∑i=0n+1Pi∗bi,n+1t;λ-∑j=0mPjbj,mt;λ2dt.
Let∂S/∂Pj=0(j=2,3,…,m-2). Equation (31) can be simplified to the following form:(32)∑i=2m-2Pi∫01bi,mt;λbj,mt;λdt=∫01∑i=0n+1Pi∗bi,n+1t;λbj,mt;λdt-∫01P0b0,mt;λ+P1b1,mt;λ+Pm-1bm-1,mt;λ+Pmbm,mt;λbj,mt;λdt.Furthermore, this equation can be represented in matrix form as follows:(33)AP=b,where(34)A=ai-1,j-1m-3,m-3,b=b1,b2,…,bm-3T,ai-1,j-1=∫01bi,mt;λbj,mt;λdt,bj-1=∫01∑i=0n+1Pi∗bi,n+1t;λ-P0b0,mt;λ-P1b1,mt;λbj,mt;λdt-∫01Pm-1bm-1,mt;λ+Pmbm,mt;λbj,mt;λdt.i,j=2,3…,m-2.Let ej=(a1,j,a2,j,…,am-3,j)T(j=1,2,…,m-3), and suppose(35)∑j=1m-3cjej=c1a1,1a2,1⋮am-3,1+c2a1,2a2,2⋮am-3,2+⋯+cm-3a1,m-3a2,m-3⋮am-3,m-3=00⋮0.That is,(36)∑j=1m-3cjai,j=∫01∑j=1m-3cjbj+1,mt;λbi+1,mt;λdt=0i=1,2,…,m-3.Because b2,m(t;λ),…,bm-2,m(t;λ) are linearly independent in interval t∈[0,1], the vectors e1,e2,…,em-3, as in Theorem 6, are linearly independent. Therefore, solutions of linear systems (33) exist uniquely and maintain C1 continuity.


## 5. Numerical Examples

Example 1.
Given the shape parameterλ=1 and the coordinates of control points {P0∗=(-5,0), P1∗=(-7,2), P2∗=(-3,5), P3∗=(2,6), P4∗=(5,3), P5∗=(3,0)}, we can construct λ-Bézier curve of degree 5. Then this curve can be separately reduced to λ-Bézier curve of degree 4, respectively, under unrestricted and C0, C1 constraint condition. Control points and errors for approximating λ-Bézier curve of degree 4 to a λ-Bézier curve of degree 5 are shown in Table 1. Degree reduction with various constraint conditions from degree 5 to degree 4 is shown in Figure 2.
We give the approximation error graphs of degree reduction of degree 5 in three conditions with different shape parameter, as shown in Figure3. From Figure 3, the approximation error value of degree reduction decreases at first and then increases when increasing the shape parameter. The range of error value is [0.27488×10-4,1.1886×10-4] in (a), and those in (b) and (c) are [0.60937×10-4,2.3013×10-4] and [0.85492×10-3,2.3364×10-3], respectively.Table 1
Control points and approximation errors with different constraint conditions in Example1 (from degree 5 to degree 4).
Constraint condition
Control points
Errors
Under unrestricted condition
P
0
=
(
-
5.0241
,
-
0.01868
), P1=(-7.2251,2.266), P2=(-0.24919,6.690), P3=(5.2589,3.672), P4=(3.0186,-0.03573)
d
2
(
p
5
∗
(
t
;
1
)
,
p
4
(
t
;
1
)
)
=
0.11866
×
1
0
-
3
UnderC0 constraint condition
P
0
=
(
-
5,0
), P1=(-7.2449,2.273), P2=(-0.24428,6.705), P3=(5.2705,3.639), P4=(3,0)
d
2
(
p
5
∗
(
t
;
1
)
,
p
4
(
t
;
1
)
)
=
0.23013
×
1
0
-
3
UnderC1 constraint condition
P
0
=
(
-
5,0
), P1=(-7.3333,2.3333), P2=(-0.22601,6.7616), P3=(5.3333,3.5000), P4=(3,0)
d
2
(
p
5
∗
(
t
;
1
)
,
p
4
(
t
;
1
)
)
=
0.21581
×
1
0
-
2Figure 2
Degree reduction with various constraint conditions (from degree 5 to degree 4). Blue solid: the given curve of degree 5; red dot-dashed line: the degree-reduced curve of degree 4.
(a)
Under unrestricted condition
(b)
UnderC0 constraint condition
(c)
UnderC1 constraint conditionFigure 3
Error graph of degree reduction ofλ-Bézier of degree 5.
(a)
Under unrestricted condition
(b)
UnderC0 constraint condition
(c)
UnderC1 constraint conditionExample 2.
Given the shape parameterλ=-1 and the coordinates of control points {P0∗=(0,0), P1∗=(1,2), P2∗=(2,2.3), P3∗=(3,0.4), P4∗=(5,0.8), P5∗=(6,2.1), P6∗=(8,2.7), P7∗=(9,0.3)}, we can construct a λ-Bézier curve of degree 7. Then this curve will be separately reduced to λ-Bézier curves of degree 5 under three conditions. Control points and errors for approximating λ-Bézier curve of degree 5 to a λ-Bézier curve of degree 7 are shown in Table 2. Degree reductions with various constraint conditions from degree 7 to degree 5 are shown in Figure 4.
We give the approximation error graphs of degree reduction ofλ-Bézier of degree 7 in three conditions with different shape parameter, as shown in Figure 5. From Figure 5, the approximation error value of degree reduction increases and slope decreases by increasing the shape parameter. The range of error value is [0.25675×10-4,1.812×10-4] in (a), and those in (b) and (c) are [0.45964×10-4,3.3081×10-4] and [0.30212×10-3,2.2919×10-3], respectively.Table 2
Control points and approximation errors with different constraint conditions in Example2 (from degree 7 to degree 5).
Constraint condition
Control points
Error
Under unrestricted condition
P
0
=
(
-
0.0073157
,
-
0.01488
), P1=(1.6003,3.489), P2=(2.7337,0.4590), P3=(4.8462,-0.5802), P4=(7.1691,4.336), P5=(9.0139,0.2929)
d
2
(
p
7
∗
(
t
;
1
)
,
p
5
(
t
;
1
)
)
=
0.25675
×
1
0
-
4
UnderC0 constraint condition
P
0
=
(
0,0
), P1=(1.6128,3.443), P2=(2.7475,0.5180), P3=(4.7940,-0.5994), P4=(7.2115,4.325), P5=(9,0.3)
d
2
(
p
7
∗
(
t
;
1
)
,
p
5
(
t
;
1
)
)
=
0.45964
×
1
0
-
4
UnderC1 constraint condition
P
0
=
(
0,0
), P1=(1.6667,3.3333), P2=(2.7655,0.7045), P3=(4.6134,-0.6613), P4=(7.3333,4.3), P5=(9,0.3)
d
2
(
p
7
∗
(
t
;
1
)
,
p
5
(
t
;
1
)
)
=
0.30212
×
1
0
-
3Figure 4
Degree reduction with various constraint conditions (from degree 7 to degree 5). Green solid: the given curve of degree 7; red dot-dashed line: the degree-reduced curve of degree 5.
(a)
Under unrestricted condition
(b)
UnderC0 constraint condition
(c)
UnderC1 constraint conditionFigure 5
Error graph of degree reduction ofλ-Bézier of degree 7.
(a)
Under unrestricted condition
(b)
UnderC0 constraint condition
(c)
UnderC1 constraint conditionExample 3.
Given shape parameterλ=-1, and two segments of λ-Bézier curves of degree 7 expressing patterned vase, then they will be separately reduced to two segments of λ-Bézier curve of degree 5 under unrestricted condition. Control points and error for approximating λ-Bézier curve of degree 5 to a λ-Bézier curve of degree 7 are shown in Table 3. Degree reductions of these two segments are shown in Figure 6. In addition, approximation errors with C0 and C1 constraint conditions in Example 3 are 0.81951×10-3 and 0.50732×10-2, respectively.
With the change of shape parameter, we present error graph of degree reduction ofλ-Bézier of degree 7 which expresses patterned vase in unrestricted condition, as is shown in Figure 7. From Figure 7, the error value increases with that of shape parameter. The range of error value is [0.45528×10-3,1.6712×10-3].Table 3
Control points and approximation error under unrestricted condition in Example3 (from degree 7 to degree 5).
Control points
Error
Before degree reduction
First segment
P
0,1
∗
=
(
4.5,8
), P1,1∗=(4.8,7.5), P2,1∗=(4.7,4), P3,1∗=(4.6,4), P4,1∗=(3,4), P5,1∗=(3,1), P6,1∗=(4,1), P7,1∗=(4.4,0.7)
d
(
p
7
∗
(
t
;
1
)
,
p
5
(
t
;
1
)
)
=
0.45528
×
1
0
-
3
Second segment
P
0,2
∗
=
(
5.5,8
), P1,2∗=(5.2,7.5), P2,2∗=(5.3,4), P3,2∗=(5.4,4), P4.2∗=(7,4), P5,2∗=(7,1), P6,2∗=(6,1), P7,2∗=(5.6,0.7)
After degree reduction
First segment
P
0,1
=
(
4.5165,8.067
), P1,1=(4.7885,6.478), P2,1=(5.4628,2.165), P3,1=(1.6892,4.636), P4,1=(3.7304,0.3241), P5,1=(4.4033,0.7784)
Second segment
P
0,2
=
(
5.4835,8.067
), P1,2=(5.2115,6.478), P2,2=(4.5372,2.165), P3,2=(8.3108,4.636), P4,2=(6.2696,0.3241), P5,2=(5.5967,0.7784)Figure 6
Degree reduction ofλ-Bézier curve of degree 7 which expresses patterned vase in unrestricted condition. Green solid and blue solid: the given curve of degree 7; red dot-dashed line: the degree-reduced curve of degree 5.Figure 7
Error graph of degree reduction ofλ-Bézier of degree 7 which expresses patterned vase in unrestricted condition.


## 6. Concluding Remarks

λ-Bézier curves of degree n have the same properties as Bézier curves. In addition, they have better performance when adjusting their shapes by changing the shape parameter, which includes shape adjustability and better approximation to control polygon as shown in Figure 1.Furthermore, the problem of degree reduction forλ-Bézier curves is studied by least squared approximation. An algorithm for approximating degree reduction of λ-Bézier curves of degree n is provided by adjusting control points under three conditions, which can minimize the least square error between the approximating curves and the original ones. Three practical examples show that the method is applicable for CAD/CAM modeling systems. We will focus on studying the degree reduction for λ-Bézier surfaces in future work.



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