Approximate Multidegree Reduction of λ-Bézier Curves

Besides inheriting the properties of classical Bézier curves of degree n, 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 L 2 -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 C , 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.


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 curve R  () of degree  with Bézier points {r  }  =0 , find an approximate Bézier curve R () of lower degree , where  < , with the set of Bézier points {r  }  =0 , so that R  and R satisfy boundary conditions at the end points, and the error between R  and R 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][6][7][8], algebraic means of basis function transformations [9][10][11][12][13][14], and B net and constrained optimization [15,16].Watkins and Worsey [9] presented an algorithm for generating ( − 1)st degree approximation to th 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  2 -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 √  −  2 -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  2norms, 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][19][20][21][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 -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 -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 -Bézier curves.
The remainder of the paper is organized as follows.The definition and properties of -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 -Bézier curve.Numerical examples are given in Section 5, and we present some applications.At last, a short conclusion is given in Section 6.

The Definition and Properties of
-Bézier Curves

Extension of Basis Function. The definition of extension
Bernstein basis functions is given as follows [18].
are called the extension Bernstein basis functions of degree 2 associated with the shape parameter .

Construction of
Because of introducing parameter , -Bézier curves have more powerful expressiveness than the classical Bézier curves.
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  = 3 and p * (; 1) (solid lines), p * (; 0) (dashed lines), and p * (; −1) (dot-dashed lines), respectively.Figure 1(b) shows the curves generated by the same basis functions as in Figure 1(a) with  = 4 and p * (; 1) (solid lines), p * (; 0) (dashed lines), and p * (; −1) (dot-dashed lines), respectively.From the figures, we can see that -Bézier curves approach the control polygon when the shape parameter is increasing.

The Approximate Degree Reduction of 𝜆-Bézier
Curves under  0 Constraint Condition.When approximating degree reduction, we expect to satisfy  0 continuity, that is, maintaining interpolation of terminal points, so two equations  0 =  * 0 and   =  * +1 are determined.The remaining  − 1 control points are determined by the following theorem.
We give the approximation error graphs of degree reduction of degree 5 in three conditions with different shape parameter, as shown in Figure 3. 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   3)}, we can construct a -Bézier curve of degree 7. Then this curve will be separately reduced to -Bézier curves of degree 5 under three conditions.Control points and errors for approximating -Bézier curve of degree 5 to a -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 of -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 Example 3. Given shape parameter  = −1, and two segments of -Bézier curves of degree 7 expressing patterned vase, then they will be separately reduced to two segments of -Bézier curve of degree 5 under unrestricted condition.Control points and error for approximating -Bézier curve of degree 5 to a -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  0 and  1 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 of -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 ].

Concluding Remarks
-Bézier curves of degree  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 for -Bézier curves is studied by least squared approximation.An algorithm for approximating degree reduction of -Bézier curves of degree  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 -Bézier surfaces in future work.

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

Figure
Figure4: 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.

Figure 5 :𝑐
Figure 5: Error graph of degree reduction of -Bézier of degree 7.

Figure 6 :
Figure 6: Degree reduction of -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.

5 Figure 7 :
Figure 7: Error graph of degree reduction of -Bézier of degree 7 which expresses patterned vase in unrestricted condition.

Table 1 :
Control points and approximation errors with different constraint conditions in Example 1 (from degree 5 to degree 4).

Table 2 :
Control points and approximation errors with different constraint conditions in Example 2 (from degree 7 to degree 5).