Abstract
Besides inheriting the properties of classical Bézier curves of degree , the corresponding -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 -Bézier curves in the -norm. By analysing the properties of -Bézier curves of degree , a method which can deal with approximating -Bézier curve of degree by -Bézier curve of degree is presented. Then, in unrestricted and , constraint conditions, the new control points of approximating -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.
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 curve of degree with Bézier points , find an approximate Bézier curve of lower degree , where , with the set of Bézier points , so that and satisfy boundary conditions at the end points, and the error between and 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 ()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 -approximation over the interval 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 -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 -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 of λ-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 of -Bézier Curves
2.1. Extension of Basis Function
The definition of extension Bernstein basis functions is given as follows [18].
Definition 1. Let ; for any , the polynomial functions are called the extension Bernstein basis functions of degree 2 associated with the shape parameter .
For any integer , the functions defined recursively byare called the extension Bernstein basis functions of degree . In the case or , we set .
Theorem 2. The extension Bernstein basis functions of degree can be expressed explicitly aswhere
2.2. Construction of -Bézier Curves
Definition 3. Given control points in or , thenis called a -Bézier curve of degree with shape parameter , where basis functions are defined by (3) (see Definition in [18]).
When the shape parameter is equal to zero, -Bézier curves degenerate to the classical Bézier curves. -Bézier curve inherits most properties of the classical Bézier curve, such as convex hull property, geometric invariance, symmetry, and the following terminal property: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 and (solid lines), (dashed lines), and (dot-dashed lines), respectively. Figure 1(b) shows the curves generated by the same basis functions as in Figure 1(a) with and (solid lines), (dashed lines), and (dot-dashed lines), respectively. From the figures, we can see that -Bézier curves approach the control polygon when the shape parameter is increasing.
(a) -Bézier curve of degree 3
(b) -Bézier curve of degree 4
3. Problem Description
Problem 4. Given control points (), -Bézier curve of degree is expressed as follows: where are basis functions of degree and is global shape parameter. Given control points , the corresponding -Bézier curves of degree () aresuch that the distance minimizes between and in certain distance function .
Here we are interested in obtaining explicit expression of approximating -Bézier curves , so we choose the following least square distance function:Then we can convert Problem 4 into subproblems, and every subproblem leads to a minimized component function:Let and ; then (8) is determined by the following formula:For subdistance function , it is sufficient to minimize . Therefore, we can just study the problem of minimum component function in the following.
Problem 5. Given a series of real numbers , from which we will determine -Bézier functions of degree ,where denotes a component of vector , then it is necessary to find real numbers with the corresponding -Bézier functions of degree such that minimizes by least square distance.
In order to determine the approximate function , primarily, we aim to obtain the coefficients .
4. Least Square Degree Reduction of -Bézier Curves
4.1. The Approximate Degree Reduction of -Bézier Curves under Unrestricted Condition
Theorem 6. If coefficients of approximating functions are solutions of Problem 5, the vector satisfies linear systems , where
Proof. By Problem 5, we obtainLet ; then the above equations can be simplified to the following ones:Furthermore, (15) can be represented in matrix form by calculation, which is described as follows:whereLet , and supposeThat is,We then get the following formula:thus . Since are linearly independent in interval , we have , which means vectors are linearly independent, and then solutions of linear systems (16) are uniquely determined.
4.2. The Approximate Degree Reduction of -Bézier Curves under Constraint Condition
When approximating degree reduction, we expect to satisfy continuity, that is, maintaining interpolation of terminal points, so two equations and are determined. The remaining control points are determined by the following theorem.
Theorem 7. If coefficients of approximating functions are solutions of Problem 5 and maintain continuity, the vector satisfies linear systems except for two equations and for terminal points, where
Proof. According to the condition of continuity, that is, and , it is easy to obtain two equations and . Then by applying Problem 5, we getLet . Equation (22) can be simplified to the following equations:Furthermore, these equations can be represented in matrix form as follows:whereLet , and supposeThat is,Because are linearly independent in interval , the vectors , as in Theorem 6, are linearly independent. Thus solutions of linear systems (24) are uniquely determined and maintain continuity.
4.3. The Approximate Degree Reduction of -Bézier Curves under Constraint Condition
When approximating degree reduction, if continuity is maintained (i.e., four equations , , , and are specified), the remaining control points are determined by the following theorem.
Theorem 8. If coefficients of approximate functions are solutions of Problem 5 and maintain C1 continuity, the vector satisfies linear systems except for four equations , , , and for terminal points, where
Proof. According to the condition of continuity, we getIt is easy to obtain the following four equations:Then by Problem 5, we obtainLet . Equation (31) can be simplified to the following form:Furthermore, this equation can be represented in matrix form as follows:whereLet , and supposeThat is,Because are linearly independent in interval , the vectors , as in Theorem 6, are linearly independent. Therefore, solutions of linear systems (33) exist uniquely and maintain continuity.
5. Numerical Examples
Example 1. Given the shape parameter and the coordinates of control points , , , , , , we can construct -Bézier curve of degree 5. Then this curve can be separately reduced to -Bézier curve of degree 4, respectively, under unrestricted and , 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 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 [] in (a), and those in (b) and (c) are [] and [], respectively.
(a) Under unrestricted condition
(b) Under constraint condition
(c) Under constraint condition
(a) Under unrestricted condition
(b) Under constraint condition
(c) Under constraint condition
Example 2. Given the shape parameter and the coordinates of control points , , , , , , , , 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 [] in (a), and those in (b) and (c) are [] and [], respectively.
(a) Under unrestricted condition
(b) Under constraint condition
(c) Under constraint condition
(a) Under unrestricted condition
(b) Under constraint condition
(c) Under constraint condition
Example 3. Given shape parameter , 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 and constraint conditions in Example 3 are and , 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 [].
6. 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.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 51305344 and no. 11426173). This work is also supported by the Research Fund of Department of Science and Department of Education of Shaanxi, China (no. 2013JK1029), and Research Fund of Shaanxi, China (2014K05-22).