Research ArticleShape Modification for 𝜆-Bézier Curves Based onConstrained Optimization of Position and Tangent Vector

Gang Hu,1 Xiaomin Ji,2 Xinqiang Qin,1 and Suxia Zhang1

1Department of Applied Mathematics, Xi’an University of Technology, Xi’an 710054, China2College of the Arts, Xi’an University of Technology, Xi’an 710048, China

Correspondence should be addressed to Gang Hu; peng gh@163.com

Received 7 October 2014; Revised 15 January 2015; Accepted 23 January 2015

Academic Editor: Chih-Cheng Hung

Copyright © 2015 Gang Hu et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Besides inheriting the properties of classical Bézier curves of degree n, the corresponding 𝜆-Bézier curves have a good performanceon adjusting their shapes by changing shape control parameter. Specially, in the case where the shape control parameter equals zero,the 𝜆-Bézier curves degenerate to the classical Bézier curves. In this paper, the shapemodification of 𝜆-Bézier curves by constrainedoptimization of position and tangent vector is investigated.The definition and properties of 𝜆-Bézier curves are given in detail, andthe shape modification is implemented by optimizing perturbations of control points. At the same time, the explicit formulas ofmodifying control points and shape parameter are obtained by Lagrange multiplier method. Using this algorithm, 𝜆-Bézier curvesare modified to satisfy the specified constraints of position and tangent vector, meanwhile the shape-preserving property is stillretained. In order to illustrate its ability on adjusting the shape of 𝜆-Bézier curves, some curve design applications are discussed,which show that the proposed method is effective and easy to implement.

1. Introduction

Bézier curves are widely used in Computer Aided GeometricDesign (CAGD) and computer graphics (CG) which havemany properties that are helpful for shape design. Developingmore convenient techniques for designing and modifyingBézier curves is an important problem in computer aideddesign (CAD), computer aided manufacturing (CAM), andNC technology fields; see [1]. However, shape design is time-consuming andusually cannot be accomplished in one stroke.After creating Bézier curves or surfaces, we often need tomodify them so that their shapes can satisfy our designrequirements.

Many efforts have beenmade to developmore convenientand effective methods for shape modification of parametriccurves and surfaces. Piegl [2] proposed two methods toalter the shape of NURBS curves, including control-point-basedmodification and weight-basedmodification. Sànchez-Reyes [3] developed a simple technique to modify NURBScurves based on a perspective functional transformation ofarbitrary origin. Juhász [4] provided a weight-based shapemodification method, by which one can prescribe not only

the new position of an arbitrary chosen point of a planeNURBS curves but the tangent direction as well. Hu et al.[5, 6] developed a method for shape modification of NURBScurves and surfaces with geometric constraint. However,developing more effective way for shape modification ofBézier curves is still an important problem. Inspired by theresults in [5, 6], Xu et al. [7] proposed a method to modifythe shape of Bézier curves by minimizing the changes of theshape by least square, where explicit formulas are deducedto calculate positions of new control points of the modi-fied curve. Wu and Xia [8] investigated the optimal shapemodification of Bézier curves by geometric constraint, whereshapemodification of Bézier curve with added end-point andtangent constraints is discussed. Wang et al. [9] presenteda method for shape modification of NURBS curves, whichis based on constrained optimization by means of alteringthe corresponding weights of their control points. Juhász andHoffmann [10] investigated the effect of the modification ofknot values on the shape of B-spline curves. Han and Ren [11]investigated geometric constrained optimization for shapemodification of Bézier curves and obtained precise formulafor it.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 735629, 12 pageshttp://dx.doi.org/10.1155/2015/735629

2 Mathematical Problems in Engineering

Bézier curves have now become a powerful tool forconstructing free-form curves in CAD/CAM. However, theBézier model is imperfect, and it has its own shortage. Thatis, after choosing the basis functions, the shape of a Béziercurve is well determined by its control points. In the recentyears, in order to overcome the shortage of Bézier curves,many scholars constructed new curves whose structuresare similar to the Bézier curves by introducing parametersinto basis functions; see [12–21]. These new curves sharemany basic properties with the Bézier curves and at thesame time hold flexible shape adjustable property. Yan andLiang [16] constructed a new kind of basis functions by arecursive approach and defined a kind of parametric curvescalled 𝜆-Bézier curves with shape parameter based on thesebasis functions. The new curves have most properties of thecorresponding classical Bézier curves. Moreover, the shapeparameter can adjust the shape of the new curves withoutchanging the control points. Focusing on the problem ofshape modification of 𝜆-Bézier curves, we study the shapemodification of 𝜆-Bézier curves by constrained optimizationof position and tangent vector and obtain the explicit formu-las of the modified control points and shape parameter.

The remainder of the paper is organized as follows. Thedefinition and properties of 𝜆-Bézier curves are given inSection 2. In Section 3, we present the shape modification of𝜆-Bézier curve by constrained optimization of single pointconstraint. Practical examples are given in Section 4, and wepresent some applications. At last, a short conclusion is givenin Section 5.

2. The Definition and Properties of𝜆-Bézier Curves

2.1. Extension Basis Function. The definition of extensionBernstein basis functions is given as follows [16].

Definition 1. Let 𝜆 ∈ [−1, 1]; for 𝑡 ∈ [0, 1], the followingpolynomial functions

𝑏0,2 (𝑡; 𝜆) = (1− 2𝜆𝑡 + 𝜆𝑡2) (1− 𝑡)2 ,

𝑏1,2 (𝑡; 𝜆) = 2𝑡 (1− 𝑡) (1+𝜆−𝜆𝑡 + 𝜆𝑡2) ,

𝑏2,2 (𝑡; 𝜆) = (1−𝜆+𝜆𝑡2) 𝑡

2,

𝑡 ∈ [0, 1]

(1)

are called the extension Bernstein basis functions of degree 2associated with the shape parameter 𝜆.

For any integer 𝑛 (𝑛 ≥ 3), the functions 𝑏𝑖,𝑛

(𝑡; 𝜆) (𝑖 =

0, 1, . . . , 𝑛), defined recursively by

𝑏𝑖,𝑛

(𝑡; 𝜆) = (1− 𝑡) 𝑏𝑖,𝑛−1 (𝑡; 𝜆) + 𝑡𝑏𝑖−1,𝑛−1 (𝑡; 𝜆) ,

𝑡 ∈ [0, 1] ,(2)

are called the extension Bernstein basis functions of degree 𝑛.In the case 𝑘 = −1 or 𝑘 > 𝑙, we set 𝑏

𝑘,𝑙

(𝑡; 𝜆) = 0.

Theorem 2. The extension Bernstein basis functions of degreen can be expressed explicitly as

𝑏𝑖,𝑛

(𝑡; 𝜆)

= (1+3𝐶𝑖−1

𝑛−2

+ 𝐶𝑖

𝑛−1

− 𝐶𝑖

𝑛

𝐶𝑖𝑛

𝜆−2𝐶𝑖

𝑛−1

𝐶𝑖𝑛

𝜆𝑡 + 𝜆𝑡2

)

⋅𝐶𝑖

𝑛

𝑡𝑖

(1 − 𝑡)𝑛−1

(𝑖 = 0, 1, . . . , 𝑛) ,

(3)

where 𝑛 ≥ 2, 𝐶𝑖𝑛

= 𝑛!/𝑖!(𝑛 − 𝑖)!.

The extension Bernstein basis functions (3) have thefollowing properties:

(a) Degeneracy. In the particular case where the shapecontrol parameter 𝜆 equals zero, the extension Bern-stein basis functions of degree 𝑛 are just the classicalones of the same degree.

(b) Nonnegativity. For any 𝜆 ∈ [−1, 1], 𝑏𝑖,𝑛

(𝑡; 𝜆) ≥ 0 (𝑖 =0, 1, . . . , 𝑛), where 𝑛 ≥ 2.

(c) Partition of Unity. One has ∑𝑛𝑖=0 𝑏𝑖,𝑛(𝑡; 𝜆) = 1.

(d) Symmetry. One has 𝑏𝑖,𝑛

(1 − 𝑡; 𝜆) = 𝑏𝑛−𝑖,𝑛

(𝑡; 𝜆) (𝑖 =

0, 1, . . . , 𝑛).

(e) Linear Independence. For any 𝜆 ∈ [−1, 1], theextension Bernstein basis functions 𝑏

𝑖,𝑛

(𝑡; 𝜆) (𝑖 =

0, 1, . . . , 𝑛) are linearly independent.

2.2. Construction of 𝜆-Bézier Curves

Definition 3. Given control points P𝑖

(𝑖 = 0, 1, . . . , 𝑛; 𝑛 ≥ 2)in 𝑅2 or 𝑅3, then

p (𝑡; 𝜆) =𝑛

∑

𝑖=0P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆) , 𝑡 ∈ [0, 1] , 𝜆 ∈ [−1, 1] (4)

is called a 𝜆-Bézier curve of degree 𝑛 with shape parameter𝜆, where basis functions 𝑏

𝑖,𝑛

(𝑡; 𝜆) (𝑖 = 0, 1, . . . , 𝑛; 𝑛 ≥ 2) aredefined by (3) (see Definition 3.2 in [16]).

From the properties of the extension Bernstein basisfunctions (3), the 𝜆-Bézier curves (4) have the followingproperties.

(a) End-Point Properties. For any 𝜆 ∈ [−1, 1], we have

p (0; 𝜆) = P0,

p (1; 𝜆) = P𝑛

.

(5)

And the derivative at end-points will satisfy

p (0; 𝜆) = (𝑛 + 2𝜆) (P1 −P0) ,

p (1; 𝜆) = (𝑛 + 2𝜆) (P𝑛

−P𝑛−1) .

(6)

Mathematical Problems in Engineering 3

Furthermore, the second derivative at end-points can berepresented as

p (0; 𝜆) = [2𝜆+ (4𝜆− 1) 𝑛 + 𝑛2]P0

+ [8𝜆+ (2− 8𝜆) 𝑛 − 2𝑛2]P1

+ [−10𝜆+ (4𝜆− 1) 𝑛 + 𝑛2]P2,

p (1; 𝜆) = [−10𝜆+ (4𝜆− 1) 𝑛 + 𝑛2]P𝑛−2

+ [8𝜆+ (2− 8𝜆) 𝑛 − 2𝑛2]P𝑛−1

+ [2𝜆+ (4𝜆− 1) 𝑛 + 𝑛2]P𝑛

.

(7)

(b) Symmetry. Considering P0,P1, . . . ,P𝑛 (𝑛 ≥ 2) andP𝑛

,P𝑛−1, . . . ,P0 (𝑛 ≥ 2), define the same 𝜆-Bézier curves; that

is,

p (𝑡; 𝜆;P0,P1, . . . ,P𝑛) = p (1− 𝑡; 𝜆;P𝑛,P𝑛−1, . . . ,P0) . (8)

(c) Convex Hull Property. The entire 𝜆-Bézier curves mustlie inside the convex hull of its control polygon spanned byP0,P1, . . . ,P𝑛 (𝑛 ≥ 2).

(d) Affine Invariance. The shape of 𝜆-Bézier curve is inde-pendent of the choice of coordinates; that is, (4) satisfies thefollowing two equations:

p (𝑡; 𝜆;P0 +Q,P1 +Q, . . . ,P𝑛 +Q)

= p (𝑡; 𝜆;P0,P1, . . . ,P𝑛) +Q

p (𝑡; 𝜆;MP0,MP1, . . . ,MP𝑛)

= Mp (𝑡; 𝜆;P0,P1, . . . ,P𝑛) ,

(9)

where Q is an arbitrary vector in 𝑅2 or 𝑅3 and M is anarbitrary 𝑙 × 𝑙 matrix, 𝑙 = 2 or 3.

(e) Variation Diminishing Property. Since the extension Bern-stein basis functions given in (3) form a group of (optimal)normalized totally positive basis functions, the correspond-ing 𝜆-Bézier curves possess variation diminishing property,which means that no plane intersects 𝜆-Bézier curve moreoften than it intersects the corresponding control polygon.

2.3. Performance Comparison of 𝜆-Bézier Curves, BézierCurves, and NURBS. A Bézier curve is defined as a para-metric one which forms the basis of the Bernstein function.However, once the control points and their correspondingBernstein polynomials are given, the shape of a Bézier curveis formed uniquely and there is no possibility to adjust itanymore. Modifying the shape of Bézier curves essentiallyrequires the adjustments of vertexes of the control polygon,which is very inconvenient. For these reasons, the problemof shape modification of curves is proposed. Although the

Table 1: Performance comparisons of 𝜆-Bézier curves, Béziercurves, and NURBS.

Property 𝜆-BéziercurvesBéziercurves

NURBScurves

Property of basis functionsNonnegativity ✓ ✓ ✓Partition of unity ✓ ✓ ✓Symmetry ✓ ✓ ✓Shape parameters ✓ M ∗Linear independence ✓ ✓ ✓Degeneracy ✓ M ✓

Property of the CurvesVariation diminishing property ✓ ✓ ✓Affine invariability ✓ ✓ ✓Convex hull property ✓ ✓ ✓Symmetry ✓ ✓ ✓End-point properties ✓ ✓ ✓Extra degree of freedom ✓ M ✓Computational complexity Low Low High

∗Theweights in NURBSmethods possess an effect for adjusting the shape ofthe curves.

weights in NURBS method can adjust the shapes of NURBScurve and the NURBS curve has good properties and canexpress the conic section, the NURBS curve also has dis-advantages, such as difficulty in choosing the value of theweight, the increased order of rational fraction caused bythe derivation, and the need for a numerical method ofintegration.

The shape parameters are applied to generate some curveswhose shape is adjustable as an extension of the existingmethod. The 𝜆-Bézier curves (4) have most properties ofthe corresponding classical Bézier curves. Moreover, theshape parameter can adjust the shape of the 𝜆-Bézier curveswithout changing the control points. With the increasing ofthe shape parameter, the 𝜆-Bézier curves approach to thecontrol polygon or control net, and the 𝜆-Bézier model canapproximate the control polygon or control net better thanthe classical Bézier model. In addition, the expressions of 𝜆-Bézier curves defined in this paper are more concise com-paredwith the Bézier curves andNURBS curves. Particularly,when the shape parameter 𝜆 equals zero, the 𝜆-Bézier curves(4) degenerate to the classicalBézier curves.

To sum up, with the extra degree of freedom providedby the shape parameter 𝜆 in 𝑏

𝑖,𝑛

(𝑡; 𝜆) ≥ 0 (𝑖 = 0, 1, . . . , 𝑛),the curves p(𝑡; 𝜆) can be freely adjusted and controlled bychanging the value of 𝜆 instead of changing the controlpointsP0,P1, . . . ,P𝑛. Performances of𝜆-Bézier curves, Béziercurves, and NURBS curves are compared in detail in Table 1.

Figure 1 shows graphs of 𝜆-Bézier curves with the samecontrol polygon but different shape parameters. Figure 1(a)shows the curves generated by the extension Bernstein basisfunctions with 𝑛 = 4 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 extension

4 Mathematical Problems in Engineering

0.2 0.4 0.6 0.80 1

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2−0.8

−0.6

−0.4

−0.2

(a) 𝜆-Bézier curves of degree 4

0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

0.3

0.4

0.5

0.6

0.7

0.8

(b) 𝜆-Bézier curves of degree 5

Figure 1: 𝜆-Bézier curves with the same control polygon but different shape parameters.

Bernstein basis functions with 𝑛 = 5 and p(𝑡; 1) (solidlines), p(𝑡; 0) (dashed lines), and p(𝑡; −1) (dot-dashed lines),respectively. From the figures, we can see that the 𝜆-Béziercurves approach to the control polygon when the shapeparameter is increased.

3. Shape Modification for 𝜆-Bézier Curves byConstrained Optimization

𝜆-Bézier curve of degree 𝑛 with control points P0,P1, . . . ,P𝑛

(𝑛 ≥ 2) is

p1 (𝑡; 𝜆1) =𝑛

∑

𝑖=0P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆1) , 𝑡 ∈ [0, 1] . (10)

Modified curves are

p2 (𝑡; 𝜆1) =𝑙−1∑

𝑖=0P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆1) +𝑚

∑

𝑖=𝑙

(P𝑖

+ 𝛿𝑖

) 𝑏𝑖,𝑛

(𝑡; 𝜆1)

+

𝑛

∑

𝑖=𝑚+1P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆1) ,

(11)

p3 (𝑡; 𝜆2) =𝑙−1∑

𝑖=0P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆2) +𝑚

∑

𝑖=𝑙

(P𝑖

+ 𝛿𝑖

) 𝑏𝑖,𝑛

(𝑡; 𝜆2)

+

𝑛

∑

𝑖=𝑚+1P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆2) .

(12)

Here, 𝛿𝑖

= (𝛿𝑥

𝑖

, 𝛿𝑦

𝑖

, 𝛿𝑧

𝑖

)𝑇

(𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚) are perturbationsof control points P

𝑖

(𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚).

3.1. Position Vector Constraint. Suppose the position vectorof 𝜆-Bézier curves (10) is p1(𝑡∗; 𝜆1) = S1 and then that ofmodified curve (11) satisfies p2(𝑡∗; 𝜆1) = S2; here

p2 (𝑡∗

; 𝜆1) = S1 +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) = S2. (13)

For 𝑛 ≥ 2 and 0 ≤ 𝑙 ≤ 𝑚 ≤ 𝑛, perturbation 𝛿𝑖

can becomputed by constrained optimization with ∑𝑚

𝑖=𝑙

‖𝛿𝑖

‖2 being

minimum.The Lagrange function is defined as follows:

𝐿 =

𝑚

∑

𝑖=𝑙

𝛿𝑖2+𝛼𝑇

[S2 − p2 (𝑡∗

; 𝜆1)] , (14)

where 𝛼𝑇 = (𝛼𝑥, 𝛼𝑦, 𝛼𝑧) is the Lagrange multiplier and ‖ ⋅ ‖ isEuclidean norm.

Theorem 4. When 𝑡∗ ∈ (0, 1) and 𝑛 ≥ 2, then (14) has aunique solution.

Proof. Let perturbation 𝛿𝑖

= (𝛿𝑥

𝑖

, 𝛿𝑦

𝑖

, 𝛿𝑧

𝑖

)𝑇 satisfy

𝜕𝐿

𝜕𝛿𝑥𝑖

=𝜕𝐿

𝜕𝛿𝑦

𝑖

=𝜕𝐿

𝜕𝛿𝑧

𝑖

= 0, 𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚

𝜕𝐿

𝜕𝛼𝑥=

𝜕𝐿

𝜕𝛼𝑦=

𝜕𝐿

𝜕𝛼𝑧= 0.

(15)

We can obtain

𝛿𝑖

=12

[𝑏𝑖,𝑛

(𝑡∗

; 𝜆1)𝛼] , 𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚 (16)

S2 = S1 +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) . (17)

Due to 𝑡∗ ∈ (0, 1), we know 𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) > 0 (𝑖 = 𝑙, 𝑙+1, . . . , 𝑚).Substituting (16) into (17), we can get the following equation:

[

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆1)]𝛼 = 2 (S2 − S1) . (18)

Namely,

𝛼 =2 (S2 − S1)

∑𝑚

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗; 𝜆1). (19)

Mathematical Problems in Engineering 5

Substituting (19) into (16) yields

𝛿𝑖

=𝑏𝑖,𝑛

(𝑡∗

; 𝜆1)

∑𝑚

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗; 𝜆1)(S2 − S1) , 𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚. (20)

The control points can be modified by (20) such thatp2(𝑡∗; 𝜆1) = S2 is satisfied.

3.2. Tangent Vector Constraint. Let tangent vector of 𝜆-Béziercurves (10) be p1(𝑡

∗

; 𝜆1) = T1 and that of modified curves(11) satisfies p2(𝑡

∗

; 𝜆1) = T2. Obviously, the shape of thecurves can be preservedwell by tangent vectorT2, whereT2 isdependent on control point p1(𝑡∗; 𝜆1) and the requirements.

For 𝑛 ≥ 3 and 1 ≤ 𝑙 < 𝑚 ≤ 𝑛 − 1, in order to fix theend-points of modified curve, let 𝛿

𝑖

= (𝛿𝑥

𝑖

, 𝛿𝑦

𝑖

, 𝛿𝑧

𝑖

)𝑇 be the

perturbation, and then 𝛿𝑧𝑖

= 0 if it is plane curve.According to (11), the following derivative is obtained:

p2 (𝑡∗

; 𝜆1) =𝑙−1∑

𝑖=0P𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆1) +𝑚

∑

𝑖=𝑙

(P𝑖

+ 𝛿𝑖

) 𝑏

𝑖,𝑛

(𝑡; 𝜆1)

+

𝑛

∑

𝑖=𝑚+1P𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆1)

=

𝑛

∑

𝑖=0P𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆1) +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆1)

= T1 +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆1) = T2.

(21)

Notice that 𝛿𝑖

is nonunique if 𝑚 > 𝑙, and it can becomputed by constrained optimization with ∑𝑚

𝑖=𝑙

‖𝛿𝑖

‖2 being

minimum.The Lagrange function is defined as follows:

𝐿 =

𝑚

∑

𝑖=𝑙

𝛿𝑖2+𝛽𝑇

[T2 − p

2 (𝑡∗

; 𝜆1)] , (22)

where 𝛽𝑇 = (𝛽𝑥, 𝛽𝑦, 𝛽𝑧) is the Lagrange multiplier and ‖ ⋅ ‖ isEuclidean norm.

Theorem 5. When 𝑡∗ ∈ (0, 1), (22) has a unique solution if𝑚 > 𝑙 or 𝑚 = 𝑙 and 𝑡∗ is not the maximum point.

Proof. Letting

𝜕𝐿

𝜕𝛿𝑥𝑖

=𝜕𝐿

𝜕𝛿𝑦

𝑖

=𝜕𝐿

𝜕𝛿𝑧

𝑖

= 0, 𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚

𝜕𝐿

𝜕𝛽𝑥=

𝜕𝐿

𝜕𝛽𝑦=

𝜕𝐿

𝜕𝛽𝑧= 0,

(23)

we can obtain

𝛿𝑖

=12

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)𝛽] , 𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚 (24)

T2 = T1 +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1) . (25)

When𝑚 > 𝑙, 𝑏𝑖,𝑛

(𝑡∗

; 𝜆) (𝑖 = 𝑙, 𝑙 +1, . . . , 𝑚) are not all zero.When 𝑚 > 𝑙 or 𝑚 = 𝑙 and 𝑡∗ is not the maximum point, wehave 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆) ̸= 0. Substituting (25) into (24), we can get thefollowing expression:

𝛽 =2 (T2 − T1)

∑𝑚

𝑖=𝑙

[𝑏𝑖,𝑛

(𝑡∗; 𝜆1)]2 . (26)

Combining (24) and (26), we have

𝛿𝑖

=𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)

∑𝑚

𝑖=𝑙

[𝑏𝑖,𝑛

(𝑡∗; 𝜆1)]2 (T2 −T1) ,

𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚.

(27)

The control points can bemodified according to (27) suchthat p2(𝑡

∗

; 𝜆1) = T2 is satisfied.

3.3. Position Vector and Tangent Vector Constraints. Letposition vector of 𝜆-Bézier curves (10) be p1(𝑡∗; 𝜆1) = S1 andlet tangent vector be p1(𝑡

∗

; 𝜆1) = T1; then, those of modifiedcurves (11) satisfy p2(𝑡∗; 𝜆1) = S2 and p2(𝑡

∗

; 𝜆1) = T2,respectively, and here

p2 (𝑡∗

; 𝜆1) = S1 +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) = S2,

p2 (𝑡∗

; 𝜆1) = T1 +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1) = T2.

(28)

When 𝑛 ≥ 3 and 1 ≤ 𝑙 < 𝑚 ≤ 𝑛 − 1, (11) has at leasttwo perturbations, which can be computed by constrainedoptimization with ∑𝑚

𝑖=𝑙

‖𝛿𝑖

‖2 being minimum. The Lagrange

function is defined as follows:

𝐿 =

𝑚

∑

𝑖=𝑙

𝛿𝑖2+𝛼𝑇

[S2 − p2 (𝑡∗

; 𝜆1)]

+𝛽𝑇

[T2 − p

2 (𝑡∗

; 𝜆1)] ,

(29)

where 𝛼𝑇 = (𝛼𝑥, 𝛼𝑦, 𝛼𝑧) and 𝛽𝑇 = (𝛽𝑥, 𝛽𝑦, 𝛽𝑧) are the La-grange multipliers and ‖ ⋅ ‖ is Euclidean norm.

Theorem 6. When 𝑡∗ ∈ (0, 1), (29) has a unique solution for𝑛 ≥ 3 and 1 ≤ 𝑙 < 𝑚 ≤ 𝑛 − 1.

Proof. Setting

𝜕𝐿

𝜕𝛿𝑥𝑖

=𝜕𝐿

𝜕𝛿𝑦

𝑖

=𝜕𝐿

𝜕𝛿𝑧

𝑖

= 0, 𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚

𝜕𝐿

𝜕𝛼𝑥=

𝜕𝐿

𝜕𝛼𝑦=

𝜕𝐿

𝜕𝛼𝑧= 0,

𝜕𝐿

𝜕𝛽𝑥=

𝜕𝐿

𝜕𝛽𝑦=

𝜕𝐿

𝜕𝛽𝑧= 0,

(30)

6 Mathematical Problems in Engineering

we can obtain

𝛿𝑖

=12

[𝑏𝑖,𝑛

(𝑡∗

; 𝜆1)𝛼+ 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)𝛽] ,

𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚(31)

S2 = S1 +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) (32)

T2 = T1 +𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1) . (33)

Substituting (31) into (32) and (33), we have the followingequations:

[

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆1)]𝛼+[𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)]𝛽

= 2 (S2 − S1) ,

[

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)]𝛼+𝑚

∑

𝑖=𝑙

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)]2𝛽

= 2 (T2 −T1) .

(34)

Denote

Δ = [

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆1)] [𝑚

∑

𝑖=𝑙

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)]2]

−[

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)]

2

.

(35)

We know Δ ≥ 0 by inequality of Cauchy-Schwarz. Further,we have

Δ = ∑

𝑙≤𝑖 0,(37)

which implies Δ > 0. The following equations are presentedby (34):

𝛼 =2Δ

[(

𝑚

∑

𝑖=𝑙

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)]2)(S2 − S1)

−(

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)) (T2 −T1)] ,

𝛽 =2Δ

[(

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆1)) (T2 −T1)

−(

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)) (S2 − S1)] .

(38)

Accordingly, when 𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚, substituting (38) into(31) leads to

𝛿𝑖

=S2 − S1

Δ[(

𝑚

∑

𝑖=𝑙

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)]2)𝑏𝑖,𝑛

(𝑡∗

; 𝜆1)

−(

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)]

+T2 − T1

Δ[(

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆1)) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)

−(

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆1) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆1)) 𝑏𝑖,𝑛 (𝑡∗

; 𝜆1)] .

(39)

The control points can be modified by (39) such thatp2(𝑡∗

; 𝜆1) = T2 and p2(𝑡∗

; 𝜆1) = T2.

3.4. Position Vector, Tangent Vector, and Shape ParameterConstraints. Let position vector of 𝜆-Bézier curves (10) bep1(𝑡∗; 𝜆1) = S1 and let tangent vector be p1(𝑡

∗

; 𝜆1) = T1;then position vector and tangent vector of modified curves(12) satisfy p3(𝑡∗; 𝜆2) = S2 and p3(𝑡

∗

; 𝜆2) = T2, respectively,and here

p3 (𝑡∗

; 𝜆2) = S1 +𝑙−1∑

𝑖=0P𝑖

[𝑏𝑖,𝑛

(𝑡; 𝜆2) − 𝑏𝑖,𝑛 (𝑡; 𝜆1)]

+

𝑚

∑

𝑖=𝑙

P𝑖

[𝑏𝑖,𝑛

(𝑡; 𝜆2) − 𝑏𝑖,𝑛 (𝑡; 𝜆1)]

+

𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆2)

+

𝑛

∑

𝑖=𝑚+1P𝑖

[𝑏𝑖,𝑛

(𝑡; 𝜆2) − 𝑏𝑖,𝑛 (𝑡; 𝜆1)]

= S2,

Mathematical Problems in Engineering 7

p3 (𝑡∗

; 𝜆2) = T1 +𝑙−1∑

𝑖=0P𝑖

[𝑏

𝑖,𝑛

(𝑡; 𝜆2) − 𝑏

𝑖,𝑛

(𝑡; 𝜆1)]

+

𝑚

∑

𝑖=𝑙

P𝑖

[𝑏

𝑖,𝑛

(𝑡; 𝜆2) − 𝑏

𝑖,𝑛

(𝑡; 𝜆1)]

+

𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆2)

+

𝑛

∑

𝑖=𝑚+1P𝑖

[𝑏

𝑖,𝑛

(𝑡; 𝜆2) − 𝑏

𝑖,𝑛

(𝑡; 𝜆1)]

= T2.(40)

When 𝑛 ≥ 3 and 1 ≤ 𝑙 < 𝑚 ≤ 𝑛 − 1, then (12) has at leasttwo perturbations which can be computed by constrainedoptimization with ∑𝑚

𝑖=𝑙

‖𝛿𝑖

‖2 being minimum. The Lagrange

function is defined as follows:

𝐿 =

𝑚

∑

𝑖=𝑙

𝛿𝑖2+𝛼𝑇

[S2 − p3 (𝑡∗

; 𝜆2)]

+𝛽𝑇

[T2 − p

3 (𝑡∗

; 𝜆2)] ,

(41)

where 𝛼𝑇 = (𝛼𝑥, 𝛼𝑦, 𝛼𝑧) and 𝛽𝑇 = (𝛽𝑥, 𝛽𝑦, 𝛽𝑧) are theLagrange multipliers and ‖ ⋅ ‖ is Euclidean norm.

Theorem 7. When 𝑡∗ ∈ (0, 1), (41) has a unique solution for𝑛 ≥ 3 and 1 ≤ 𝑙 < 𝑚 ≤ 𝑛 − 1.

Proof. Letting𝜕𝐿

𝜕𝛿𝑥𝑖

=𝜕𝐿

𝜕𝛿𝑦

𝑖

=𝜕𝐿

𝜕𝛿𝑧

𝑖

= 0, 𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚

𝜕𝐿

𝜕𝛼𝑥=

𝜕𝐿

𝜕𝛼𝑦=

𝜕𝐿

𝜕𝛼𝑧= 0,

𝜕𝐿

𝜕𝛽𝑥=

𝜕𝐿

𝜕𝛽𝑦=

𝜕𝐿

𝜕𝛽𝑧= 0,

(42)

we can obtain

𝛿𝑖

=12

[𝑏𝑖,𝑛

(𝑡∗

; 𝜆2)𝛼+ 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)𝛽] ,

𝑖 = 𝑙, 𝑙 + 1, . . . , 𝑚(43)

S2 = S1 +𝑙−1∑

𝑖=0P𝑖

[𝑏𝑖,𝑛

(𝑡; 𝜆2) − 𝑏𝑖,𝑛 (𝑡; 𝜆1)]

+

𝑚

∑

𝑖=𝑙

P𝑖

[𝑏𝑖,𝑛

(𝑡; 𝜆2) − 𝑏𝑖,𝑛 (𝑡; 𝜆1)]

+

𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆2)

+

𝑛

∑

𝑖=𝑚+1P𝑖

[𝑏𝑖,𝑛

(𝑡; 𝜆2) − 𝑏𝑖,𝑛 (𝑡; 𝜆1)]

(44)

T2 = T1 +𝑙−1∑

𝑖=0P𝑖

[𝑏

𝑖,𝑛

(𝑡; 𝜆2) − 𝑏

𝑖,𝑛

(𝑡; 𝜆1)]

+

𝑚

∑

𝑖=𝑙

P𝑖

[𝑏

𝑖,𝑛

(𝑡; 𝜆2) − 𝑏

𝑖,𝑛

(𝑡; 𝜆1)]

+

𝑚

∑

𝑖=𝑙

𝛿𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆2)

+

𝑛

∑

𝑖=𝑚+1P𝑖

[𝑏

𝑖,𝑛

(𝑡; 𝜆2) − 𝑏

𝑖,𝑛

(𝑡; 𝜆1)] .

(45)

Substituting (43) into (44) and (45), it follows that

[

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆2)]𝛼+[𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆2) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)]𝛽

= 2(S2 −𝑛

∑

𝑖=0P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆2)) ,

[

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆2) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)]𝛼+𝑚

∑

𝑖=𝑙

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)]2𝛽

= 2(T2 −𝑛

∑

𝑖=0P𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆2)) .

(46)

Let

Δ = [

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆2)] [𝑚

∑

𝑖=𝑙

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)]2]

−[

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆2) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)]

2

.

(47)

We know Δ ≥ 0 by inequality of Cauchy-Schwarz. Moreover,we have

Δ = ∑

𝑙≤𝑖 0,(49)

8 Mathematical Problems in Engineering

0.5

0 1 2 3 4 5 6 7 8 9

0

1

1.5

2

2.5

3

3.5

P0

P1

P2

P3

P4

P∗1

P∗2

S1

S2

(a) Modification of control points P1, P2

0.5

0 1 2 3 4 5 6 7 8 9

0

1

1.5

2

2.5

3

3.5

P0

P1

P2

P3

P4

P∗1

P∗2

P∗3

S1

S2

(b) Modification of control points P1, P2, and P3

0.5

0 1 2 3 4 5 6 7 8 9

0

1

1.5

2

2.5

3

3.5

P0

P1

P2

P3

P4

P∗1

P∗0

P∗2

P∗3

P∗4

S1

S2

(c) Modification of control points P0, P1, P2, P3, and P4

Figure 2: Shape modification for quartic 𝜆-Bézier curve based on position vector constraint of single target point.

which implies Δ > 0.The following equations are deduced by(46):

𝛼 =2Δ

[(

𝑚

∑

𝑖=𝑙

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)]2)(S2 −

𝑛

∑

𝑖=0P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆2))

−(

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆2) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2))

⋅(T2 −𝑛

∑

𝑖=0P𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆2))] ,

𝛽 =2Δ

[(

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆2))(T2 −𝑛

∑

𝑖=0P𝑖

𝑏

𝑖,𝑛

(𝑡; 𝜆2))

−(

𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆2) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2))

⋅(S2 −𝑛

∑

𝑖=0P𝑖

𝑏𝑖,𝑛

(𝑡; 𝜆2))] .

(50)

Accordingly, for 𝑖 = 𝑙, 𝑙 + 1, . . . 𝑚, substituting (50) into (43),we have

𝛿𝑖

=S2 − ∑

𝑛

𝑖=0 P𝑖𝑏𝑖,𝑛 (𝑡; 𝜆2)Δ

[(

𝑚

∑

𝑖=𝑙

[𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)]2)

⋅ 𝑏𝑖,𝑛

(𝑡∗

; 𝜆2) −(𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆2) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2))

⋅ 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2)]

+T2 − ∑

𝑛

𝑖=0 P𝑖𝑏

𝑖,𝑛

(𝑡; 𝜆2)

Δ[(

𝑚

∑

𝑖=𝑙

𝑏2𝑖,𝑛

(𝑡∗

; 𝜆2))

⋅ 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2) −(𝑚

∑

𝑖=𝑙

𝑏𝑖,𝑛

(𝑡∗

; 𝜆2) 𝑏

𝑖,𝑛

(𝑡∗

; 𝜆2))

⋅ 𝑏𝑖,𝑛

(𝑡∗

; 𝜆2)] .

(51)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

0.5

0

1

1.5

2

3

2.5

P0

P1

P2

P3

P4

P∗1

P∗2

S1

S2

(a) Modification of control points P1, P2

0 1 2 3 4 5 6 7 8 9

0.5

0

1

1.5

2

3

2.5

P0

P1

P2

P3

P4

P∗1

P∗2

P∗3

S1

S2

(b) Modification of control points P1, P2, and P3

Figure 3: Shape modification for quartic 𝜆-Bézier curve based on tangent vector constraint of single target point.

1.5

0.5

0 1 2 3 4 5 6 7 8 9

0

1

2

2.5

3

3.5

P0

P1

P2

P3

P4

P∗1

P∗2

S1

S2

(a) Modification of control points P1, P2

1.5

0.5

0 1 2 3 4 5 6 7 8 9

0

1

2

2.5

3

3.5

P0

P1

P2

P3

P4

P∗1

P∗2

P∗3

S1

S2

(b) Modification of control points P1, P2, and P3

Figure 4: Shape modification for quartic 𝜆-Bézier curve based on position vector and tangent vector constraint of single target point.

The control points can be modified by (51) such thatp3(𝑡∗; 𝜆2) = S2 and p3(𝑡

∗

; 𝜆2) = T2.

4. Practical Examples

In this section, we will give three examples to show the effectsof the proposed method.

Example 1. Given shape parameter 𝜆1

= −1 and controlpoints {P

0

= (0, 0),P1

= (2, 2.5),P2

= (4.5, 0.4),P3

=

(7, 2.8),P4

= (9, 0.5)}, we can construct an original quartic𝜆-Bézier curve. In Figures 2, 3, and 4, the shape modificationof 𝜆-Bézier curves by constrained optimization of positionvector and tangent vector with single target point S1 =p1(0.6; 𝜆1) are obtained, where original curves are shownas green solid lines and modified curves are shown as bluedashed lines. Original target points S

1

are black dot andmodified target points S2 = p2(0.6; 𝜆1) are blue square point.P∗𝑖

(𝑖 = 0, 1, 2, 3, 4) are control points of modified curves.

Figure 5 illustrates the shape modification for quartic 𝜆-Bézier curves by constrained optimization of position vector,tangent vector, and shape parameter of single target pointS1 = p1(0.6; 𝜆1). In Figure 5, the shape parameter 𝜆1 oforiginal curves equals −1, while the shape parameter 𝜆

2

of modified curves equals 1. The control points P𝑖

(𝑖 =

0, 1, 2, 3, 4) of original curves are the same as those of thecurves in Figure 4. P∗

𝑖

(𝑖 = 1, 2, 3) are control points ofmodified curves. Original curves are shown as green solidlines and modified curves are shown as blue dashed lines,where original target points S

1

are black dot and modifiedtarget points S2 = p3(0.6; 𝜆2) are blue square point.

Example 2. Given shape parameter 𝜆1

= 1 and controlpoints {P

0

= (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 an original 𝜆-Bézier curve ofdegree 7. Figure 6 illustrates the shape modification for it byconstrained optimization of position and tangent vector ofsingle target point S1 = p1(0.6; 𝜆1), where original curves are

10 Mathematical Problems in Engineering

1.5

0.5

0 1 2 3 4 5 6 7 8 9

0

1

2

2.5

3

3.5

P0

P1

P2

P3

P4

P∗1

P∗2 S1

S2

𝜆1 = −1

𝜆2 = 1

(a) Modification of control points P1, P2

1.5

0.5

0 1 2 3 4 5 6 7 8 9

0

1

2

2.5

3

3.5

P0

P1

P2

P3

P4

P∗1

P∗2

P∗3

S1

S2

𝜆1 = −1

𝜆2 = 1

(b) Modification of control points P1, P2, and P3

Figure 5: Shape modification for quartic 𝜆-Bézier curve based on position vector, tangent vector, and shape parameter constraint of singletarget point.

1.5

0.5

0 1 2 3 4 5 6 7 8 9

0

1

2

2.5

3

3.5

P0

P1P2

P3

P4

P5

P6

P7

P∗3

P∗4

S1

S2

−0.5

(a) Position vector constraint

1.5

0.5

6 9

0

1

2

2.5

3

3.5

0 1 2 3 4 5 7 8

P0

P1P2

P3

P4

P5

P6

P7P∗3

P∗4

S1

S2

−0.5

(b) Tangent vector constraint

1.5

0.5

0 1 2 3 4 5 6 7 8 9

0

1

2

2.5

3

3.5

P0

P1P2

P3

P4

P5

P6

P7

P∗3

P∗4

S1

S2

−0.5

(c) Position vector and tangent vector constraints

Figure 6: Shape modification for 𝜆-Bézier curves of degree seven with single target point.

Mathematical Problems in Engineering 11

00.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9

P0

P1

P2P3

P4

P5

P6

P7 P8

P9

P∗6

P∗7

S1

S2

(a) Position vector constraint

00.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9

P0

P1

P2 P3

P4

P5

P6

P7 P8

P9P∗6

P∗7

S1

S2

(b) Tangent vector constraint

00.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9

P0

P1

P2P3

P4

P5

P6

P7 P8

P9

P∗6

P∗7

S1

S2

(c) Position vector and tangent vector constraints

Figure 7: Shape modification for 𝜆-Bézier curves of degree nine with single target point.

shown as green solid lines and modified curves are shownas red dashed lines. Original target points S

1

are black dotand modified target points S2 = p2(0.6; 𝜆1) are red squarepoint. P∗

𝑖

(𝑖 = 3, 4) are modified control points, and the othercontrol points of original curves remain unchanged.

Example 3. Given shape parameter 𝜆1 = 1 and controlpoints {P0 = (0, 0.3),P1 = (1, 0.47),P2 = (2, 0.32), P3 =(3, 0.3),P4 = (4, 0.37),P5 = (5, 0.67),P6 = (6, 0.4),P7 =(7, 0.32),P8 = (8, 0.3),P9 = (9, 0.45)}, we can constructan original 𝜆-Bézier curve of degree 9. Figure 7 illustratesthe shape modification for it by constrained optimizationof position and tangent vector of single target point S1 =p1(0.6; 𝜆1), where original curves are shown as green solidlines and modified curves are shown as violet dashed lines.Original target points S1 are black dot and modified targetpoints S2 = p2(0.6; 𝜆1) are violet square point. P∗𝑖 (𝑖 = 6, 7)are modified control points and the other control points oforiginal curves remain unchanged.

5. Conclusions

In this paper, we give the definition of 𝜆-Bézier curvesand discuss their properties in detail. It is shown that

𝜆-Bézier curves of degree 𝑛 with shape parameter keep manyproperties of the corresponding traditional Bézier curves andare more convenient than traditional ones. We can alter theshape of 𝜆-Bézier curve by modifying the values of the shapeparameter without changing its control points. Further, weinvestigate the shape modification of 𝜆-Bézier curves forconstrained optimization of single point constraint (includ-ing modification of shape parameter and control points). Inorder to modify the shape of 𝜆-Bézier curves effectively, weobtain some explicit formulas for modifying control pointsand shape parameter. Three practical examples show thatthe method is applicable for computer aided design system.Future work will focus on studying the shape modificationfor 𝜆-Bézier surfaces.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors are very grateful to the referees for their helpfulsuggestions and comments which have improved the paper.

12 Mathematical Problems in Engineering

This work is supported by the National Natural ScienceFoundation of China (nos. 51305344 and 11426173).This workis also supported by the Research Fund of Department ofScience and Department of Education of Shaanxi, China (no.2013JK1029).

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