Shape Modification for 𝜆 -Bézier Curves Based on Constrained Optimization of Position and Tangent Vector

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Introduction
Bézier curves are widely used in Computer Aided Geometric Design (CAGD) and computer graphics (CG) which have many properties that are helpful for shape design.Developing more convenient techniques for designing and modifying Bézier curves is an important problem in computer aided design (CAD), computer aided manufacturing (CAM), and NC technology fields; see [1].However, shape design is timeconsuming and usually cannot be accomplished in one stroke.After creating Bézier curves or surfaces, we often need to modify them so that their shapes can satisfy our design requirements.
Many efforts have been made to develop more convenient and effective methods for shape modification of parametric curves and surfaces.Piegl [2] proposed two methods to alter the shape of NURBS curves, including control-pointbased modification and weight-based modification.Sànchez-Reyes [3] developed a simple technique to modify NURBS curves based on a perspective functional transformation of arbitrary origin.Juhász [4] provided a weight-based shape modification method, by which one can prescribe not only the new position of an arbitrary chosen point of a plane NURBS curves but the tangent direction as well.Hu et al. [5,6] developed a method for shape modification of NURBS curves and surfaces with geometric constraint.However, developing more effective way for shape modification of Bézier curves is still an important problem.Inspired by the results in [5,6], Xu et al. [7] proposed a method to modify the shape of Bézier curves by minimizing the changes of the shape by least square, where explicit formulas are deduced to calculate positions of new control points of the modified curve.Wu and Xia [8] investigated the optimal shape modification of Bézier curves by geometric constraint, where shape modification of Bézier curve with added end-point and tangent constraints is discussed.Wang et al. [9] presented a method for shape modification of NURBS curves, which is based on constrained optimization by means of altering the corresponding weights of their control points.Juhász and Hoffmann [10] investigated the effect of the modification of knot values on the shape of B-spline curves.Han and Ren [11] investigated geometric constrained optimization for shape modification of Bézier curves and obtained precise formula for it.

Mathematical Problems in Engineering
Bézier curves have now become a powerful tool for constructing free-form curves in CAD/CAM.However, the Bézier model is imperfect, and it has its own shortage.That is, after choosing the basis functions, the shape of a Bézier curve is well determined by its control points.In the recent years, in order to overcome the shortage of Bézier curves, many scholars constructed new curves whose structures are similar to the Bézier curves by introducing parameters into basis functions; see [12][13][14][15][16][17][18][19][20][21].These new curves share many basic properties with the Bézier curves and at the same time hold flexible shape adjustable property.Yan and Liang [16] constructed a new kind of basis functions by a recursive approach and defined a kind of parametric curves called -Bézier curves with shape parameter based on these basis functions.The 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 the problem of shape modification of -Bézier curves, we study the shape modification of -Bézier curves by constrained optimization of position and tangent vector and obtain the explicit formulas of the modified control points and shape parameter.
The remainder of the paper is organized as follows.The definition and properties of -Bézier curves are given in Section 2. In Section 3, we present the shape modification of -Bézier curve by constrained optimization of single point constraint.Practical examples are given in Section 4, and we present some applications.At last, a short conclusion is given in Section 5.

The Definition and Properties of
-Bézier Curves

Extension Basis Function. The definition of extension
Bernstein basis functions is given as follows [16].
are called the extension Bernstein basis functions of degree 2 associated with the shape parameter .
The extension Bernstein basis functions (3) have the following properties: (a) Degeneracy.In the particular case where the shape control parameter  equals zero, the extension Bernstein basis functions of degree  are just the classical ones of the same degree.
(e) Variation Diminishing Property.Since the extension Bernstein basis functions given in (3) form a group of (optimal) normalized totally positive basis functions, the corresponding -Bézier curves possess variation diminishing property, which means that no plane intersects -Bézier curve more often than it intersects the corresponding control polygon.

Performance Comparison of 𝜆-Bézier Curves, Bézier
Curves, and NURBS.A Bézier curve is defined as a parametric one which forms the basis of the Bernstein function.However, once the control points and their corresponding Bernstein polynomials are given, the shape of a Bézier curve is formed uniquely and there is no possibility to adjust it anymore.Modifying the shape of Bézier curves essentially requires the adjustments of vertexes of the control polygon, which is very inconvenient.For these reasons, the problem of shape modification of curves is proposed.Although the High * The weights in NURBS methods possess an effect for adjusting the shape of the curves.
weights in NURBS method can adjust the shapes of NURBS curve and the NURBS curve has good properties and can express the conic section, the NURBS curve 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.The shape parameters are applied to generate some curves whose shape is adjustable as an extension of the existing method.The -Bézier curves (4) have most properties of the corresponding classical Bézier curves.Moreover, the shape parameter can adjust the shape of the -Bézier curves without changing the control points.With the increasing of the shape parameter, the -Bézier curves approach to the control polygon or control net, and the -Bézier model can approximate the control polygon or control net better than the classical Bézier model.In addition, the expressions of -Bézier curves defined in this paper are more concise compared with the Bézier curves and NURBS curves.Particularly, when the shape parameter  equals zero, the -Bézier curves (4) degenerate to the classicalBézier curves.
To sum up, with the extra degree of freedom provided by the shape parameter  in  , (; ) ≥ 0 ( = 0, 1, . . ., ), the curves p(; ) can be freely adjusted and controlled by changing the value of  instead of changing the control points P 0 , P 1 , . . ., P  .Performances of -Bézier curves, Bézier curves, and NURBS curves are compared in detail in Table 1.
Figure 1 shows graphs of -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  = 4 and p(; 1) (solid lines), p(; 0) (dashed lines), and p(; −1) (dot-dashed lines), respectively.Figure 1 Bernstein basis functions with  = 5 and p(; 1) (solid lines), p(; 0) (dashed lines), and p(; −1) (dot-dashed lines), respectively.From the figures, we can see that the -Bézier curves approach to the control polygon when the shape parameter is increased.
For  ≥ 3 and 1 ≤  <  ≤  − 1, in order to fix the end-points of modified curve, let   = (   ,    ,    )  be the perturbation, and then    = 0 if it is plane curve.According to (11), the following derivative is obtained: Notice that   is nonunique if  > , and it can be computed by constrained optimization with ∑  = ‖  ‖ 2 being minimum.The Lagrange function is defined as follows: where   = (  ,   ,   ) is the Lagrange multiplier and ‖ ⋅ ‖ is Euclidean norm.

Practical Examples
In this section, we will give three examples to show the effects of the proposed method.
Figure 5 illustrates the shape modification for quartic -Bézier curves by constrained optimization of position vector, tangent vector, and shape parameter of single target point S 1 = p 1 (0.6;  1 ).In Figure 5, the shape parameter  1 of original curves equals −1, while the shape parameter  2 of modified curves equals 1.The control points P  ( = 0, 1, 2, 3, 4) of original curves are the same as those of the curves in Figure 4. P *  ( = 1, 2, 3) are control points of modified curves.Original curves are shown as green solid lines and modified curves are shown as blue dashed lines, where original target points S 1 are black dot and modified target points S 2 = p 3 (0.6;  2 ) are blue square point.

Conclusions
In this paper, we give the definition of -Bézier curves and discuss their properties in detail.It is shown that -Bézier curves of degree  with shape parameter keep many properties of the corresponding traditional Bézier curves and are more convenient than traditional ones.We can alter the shape of -Bézier curve by modifying the values of the shape parameter without changing its control points.Further, we investigate the shape modification of -Bézier curves for constrained optimization of single point constraint (including modification of shape parameter and control points).In order to modify the shape of -Bézier curves effectively, we obtain some explicit formulas for modifying control points and shape parameter.Three practical examples show that the method is applicable for computer aided design system.Future work will focus on studying the shape modification for -Bézier surfaces.

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

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

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