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Approximate Multidegree Reduction of [lambda]-Bezier Curves.

1. Introduction

Bezier curves are one of the main mathematical models in CAD/CAM system [1]. Degree reduction of Bezier 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 Bezier curves is concerned with the solution of the following problem: for a given Bezier curve [R.sub.n](i) of degree n with Bezier points [{[r.sub.i]}.sup.n.sub.i=0], find an approximate Bezier curve [[??].sub.m](t) of lower degree m, where m < n, with the set of Bezier points [{[[??].sub.i]}.sup.m.sub.i=0], so that [R.sub.n] and [[??].sub.m] satisfy boundary conditions at the end points, and the error between [R.sub.n] and [[??].sub.m] is minimum. For degree reduction of Bezier 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 Bezier 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 Beezier curves with constraints of endpoints continuity. Zheng and Wang [12] proved that the problem of finding a best [L.sub.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 [square root of (t - [t.sup.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 Bezier curves has been conducted according to different norms, mostly [L.sub.2]-norms, for both unconstrained and constrained conditions. In general, unconstrained degree reduction gives lower error than the constrained one. However, Bezier curves are often a part of a piecewise curve, so constrained degree reduction is preferred.

Although Bezier 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 Bezier curve is well-determined by its control points after choosing the basis functions [17]. In recent years, in order to improve the performance of Bezier curves, many scholars have constructed some new curves which are similar to the Bezier ones by introducing parameters into basis functions; see [18-22]. These new curves share many basic properties with the Bezier 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 [lambda]-Bezier curves. These new curves have most properties of the corresponding classical Bezier curves. Moreover, the shape parameter can adjust the shape of the new curves without changing the control points. Focusing on degree reduction of [lambda]-Bezier 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 [lambda]-Bezier curves.

The remainder of the paper is organized as follows. The definition and properties of [lambda]-Bezier 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 [lambda]-Bezier 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 [lambda]-Bezier Curves

2.1. Extension of Basis Function. The definition of extension Bernstein basis functions is given as follows [18].

Definition 1. Let [lambda] [member of] [-1, 1]; for any t [member of] [0, 1], the polynomial functions

[mathematical expression not reproducible] (1)

are called the extension Bernstein basis functions of degree 2 associated with the shape parameter [lambda].

For any integer n (n [greater than or equal to] 3), the functions [b.sub.in](t; [lambda]) (i = 0, 1, ..., n) defined recursively by

[mathematical expression not reproducible], (2)

are called the extension Bernstein basis functions of degree n. In the case k = -1 or k > l, we set [b.sub.k,l](t; [lambda]) = 0.

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

[mathematical expression not reproducible], (3)

where n [greater than or equal to] 2, [C.sup.i.sub.n] = n!/i!(n - i)!.

2.2. Construction of X-Bezier Curves

Definition 3. Given control points [P.sup.*.sub.i] (i = 0, 1, ..., n; n [greater than or equal to] 2) in [R.sup.2] or [R.sup.3], then

[mathematical expression not reproducible] (4)

is called a [lambda]-Bezier curve of degree n with shape parameter [lambda], where basis functions [b.sub.i,n](t; [lambda]) (i = 0, 1, ..., n; n [greater than or equal to] 2) are defined by (3) (see Definition 3.2 in [18]).

When the shape parameter [lambda] is equal to zero, [lambda]-Bezier curves degenerate to the classical Bezier curves. [lambda]-Bezier curve inherits most properties of the classical Bezier curve, such as convex hull property, geometric invariance, symmetry, and the following terminal property:

[mathematical expression not reproducible]. (5)

Because of introducing parameter [lambda], [lambda]-Bezier curves have more powerful expressiveness than the classical Bezier curves.

Figure 1 shows graphs of [lambda]-Bezier 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.sup.*](t; 1) (solid lines), [p.sup.*](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.sup.*](i; 1) (solid lines), [p.sup.*](t; 0) (dashed lines), and [p.sup.*](t; -1) (dot-dashed lines), respectively. From the figures, we can see that [lambda]-Bezier curves approach the control polygon when the shape parameter is increasing.

3. Problem Description

Problem 4. Given control points [{[P.sup.*.subi]}.sup.n+1.sub.i=0] [subset] [R.sup.s] (s = 2, 3), [lambda]-Bezier curve of degree n + 1 is expressed as follows:

[mathematical expression not reproducible], (6)

where [{[b.sub.i,n+1](t; [lambda])}.sup.n+1.sub.i=0] are basis functions of degree n + 1 and [lambda] [member of] [-1, 1] is global shape parameter. Given control points [{[P.sub.i]}.sup.m.sub.i=0] [subset] [R.sup.s], the corresponding [lambda]-Bezier curves of degree m (m [less than or equal to] n) are

p (t; [lambda]) = [m.summation over (i=0)] [P.sub.i][b.sub.i,m] (t; [lambda]), (7)

such that the distance minimizes between [p.sup.*] (t; [lambda]) and p(t; [lambda]) in certain distance function d([p.sup.*] (t; [lambda]), p(t; [lambda])).

Here we are interested in obtaining explicit expression of approximating [lambda]-Bezier curves p(t; [lambda]), so we choose the following least square distance function:

[mathematical expression not reproducible]. (8)

Then we can convert Problem 4 into s subproblems, and every subproblem leads to a minimized component function:

[mathematical expression not reproducible]. (9)

Let [mathematical expression not reproducible]; then (8) is determined by the following formula:

[mathematical expression not reproducible]. (10)

For subdistance function [d.sup.2]([p.sup.*.sub.k](t; [lambda]), [p.sub.k](t; [lambda])), it is sufficient to minimize d([p.sup.*] (t; [lambda]), p(t; [lambda])). Therefore, we can just study the problem of minimum component function in the following.

Problem 5. Given a series of real numbers [{[P.sup.*.sub.i]}.sup.n+1.sub.i=0], from which we will determine [lambda]-Bezier functions of degree n + 1,

[f.sup.*] (t; [lambda]) = [n+1.summation over i=0] p.sup.*.sub.i] [b.sub.i,n+1] (t; [lambda]), (11)

where [P.sup.*.sub.i] denotes a component of vector [P.sup.*.sub.i], then it is necessary to find real numbers [{[P.sub.j]}.sup.m.sub.j=0] with the corresponding [lambda]-Bezier functions of degree m

f(t; [lambda]) = [m.summation over (j=0)] [P.sub.j][b.sub.j,m] (t; [lambda]) (12)

such that [mathematical expression not reproducible] minimizes by least square distance.

In order to determine the approximate function f(t; [lambda]), primarily, we aim to obtain the coefficients {[P.sub.j]}.sup.m.sub.j=0].

4. Least Square Degree Reduction of [lambda]-Bezier Curves

4.1. The Approximate Degree Reduction of X-Bezier Curves under Unrestricted Condition

Theorem 6. If coefficients [{[P.sub.j]}.sup.m.sub.j=0] of approximating functions f(t; [lambda]) are solutions of Problem 5, the vector P = [([P.sub.0], [P.sub.1], ..., [P.sub.m]).sup.T] satisfies linear systems AP = b, where

[mathematical expression not reproducible]. (13)

Proof. By Problem 5, we obtain

[mathematical expression not reproducible]. (14)

Let [partial derivative]S/[partial derivative][P.sub.j] = 0 (j = 0, 1, ..., m); then the above equations can be simplified to the following ones:

[mathematical expression not reproducible]. (15)

Furthermore, (15) can be represented in matrix form by calculation, which is described as follows:

AP = b, (16)

where

[mathematical expression not reproducible]. (17)

Let [e.sub.j+1] = [([a.sub.1,j+1], [a.sub.2,j+1], ..., [a.sub.m+1,j+1]).sup.T] (j = 0, 1, ..., m), and suppose

[mathematical expression not reproducible]. (18)

That is,

[mathematical expression not reproducible]. (19)

We then get the following formula:

[mathematical expression not reproducible]; (20)

thus [[summation].sup.m.sub.j=0] [c.sub.j+1] [b.sub.j,m] (t; [lambda]) [equivalent to] 0. Since {[b.sub.0,m](t; [lambda]), [b.sub.1,m] (t; [lambda]), ..., [b.sub.m,m] (t; [lambda])} are linearly independent in interval t [member of] [0, 1], we have [c.sub.j+1] [equivalent to] 0 (j = 0, 1, ..., m), which means vectors {[e.sub.1], [e.sub.2], ..., [e.sub.m+1]} are linearly independent, and then solutions of linear systems (16) are uniquely determined.

4.2. The Approximate Degree Reduction of [lambda]-Bezier Curves under [C.sup.0] Constraint Condition. When approximating degree reduction, we expect to satisfy [C.sup.0] continuity, that is, maintaining interpolation of terminal points, so two equations [P.sub.0] = [P.sup.*.sub.0] and [P.sub.m] = [P.sup.*.sub.n+1] are determined. The remaining m - 1 control points are determined by the following theorem.

Theorem 7. If coefficients [{[P.sub.i]}.sup.m.sub.i=0] of approximating functions f(t; [lambda]) are solutions of Problem 5 and maintain [C.sup.0] continuity, the vector P = [([P.sub.1], [P.sub.2], ..., [P.sub.m-1]).sup.T] satisfies linear systems AP = b except for two equations [P.sub.0] = [P.sup.*.sub.0] and [P.sub.m] = [P.sup.*.sub.n+1] for terminal points, where

[mathematical expression not reproducible]. (21)

Proof. According to the condition of [C.sup.0] continuity, that is, [f.sup.*](0; [lambda]) = f(0; [lambda]) and [f.sup.*](1; [lambda]) = f(1; [lambda]), it is easy to obtain two equations [P.sub.0] = [P.sup.*.sub.0] and [P.sub.m] = [P.sup.*.sub.n+1]. Then by applying Problem 5, we get

[mathematical expression not reproducible]. (22)

Let [partial derivative]S/[partial derivative][P.sub.j] = 0 (j = 1, 2, ..., m-1). Equation (22) can be simplified to the following equations:

[mathematical expression not reproducible]. (23)

Furthermore, these equations can be represented in matrix form as follows:

AP = b, (24)

where

[mathematical expression not reproducible]. (25)

Let [e.sub.j] = [([a.sub.1,j], [a.sub.2,j], ..., [a.sub.m-1,j]).sup.T] (j = 1, 2, ..., m-1), and suppose

[mathematical expression not reproducible]. (26)

That is,

[mathematical expression not reproducible]. (27)

Because {[b.sub.1,m] (t; [lambda]), [b.sub.2,m] (t; [lambda]), ..., [b.sub.m-1,m] (t; [lambda])} are linearly independent in interval t [member of] [0, 1], the vectors {[e.sub.1], [e.sub.2], ..., [e.sub.m-1]}, as in Theorem 6, are linearly independent. Thus solutions of linear systems (24) are uniquely determined and maintain [C.sup.0] continuity.

4.3. The Approximate Degree Reduction of [lambda]-Bezier Curves under [C.sup.1] Constraint Condition. When approximating degree reduction, if [C.sup.1] continuity is maintained (i.e., four equations [mathematical expression not reproducible] are specified), the remaining m-3 control points are determined by the following theorem.

Theorem 8. If coefficients {[P.sub.i]}.sup.m.sub.i=0] of approximate functions f(t; [lambda]) are solutions of Problem 5 and maintain [C.sup.1] continuity,

the vector P = [([P.sub.2], [P.sub.3], ..., [P.sub.m-2]).sup.T] satisfies linear systems AP = b except for four equations [mathematical expression not reproducible] for terminal points, where

[mathematical expression not reproducible]. (28)

Proof. According to the condition of [C.sup.1] continuity, we get

[mathematical expression not reproducible]. (29)

It is easy to obtain the following four equations:

[mathematical expression not reproducible]. (30)

Then by Problem 5, we obtain

[mathematical expression not reproducible]. (31)

Let [partial derivative]S/[partial derivative][P.sub.j] = 0 (j = 2, 3, ..., m-2). Equation (31) can be simplified to the following form:

[mathematical expression not reproducible]. (32)

Furthermore, this equation can be represented in matrix form as follows:

AP = b, (33)

where

[mathematical expression not reproducible]. (34)

Let [e.sub.j] = [([a.sub.1,j], [a.sub.2,j], ..., [a.sub.m-3,j]).sup.T] (j = 1, 2, ..., m-3), and suppose

[mathematical expression not reproducible]. (35)

That is,

[mathematical expression not reproducible]. (36)

Because {[b.sub.2,m] (t; [lambda]), ..., [b.sub.m-2,m] (t; [lambda])} are linearly independent in interval t [member of] [0, 1], the vectors {[e.sub.1], [e.sub.2], ..., [e.sub.m-3]}, as in Theorem 6, are linearly independent. Therefore, solutions of linear systems (33) exist uniquely and maintain [C.sup.1] continuity.

5. Numerical Examples

Example 1. Given the shape parameter [lambda] = 1 and the coordinates of control points {[P.sup.*.sub.0] = (-5, 0), [P.sup.*.sub.1] = (-7, 2), [P.sup.*.sub.2] = (-3, 5), [P.sup.*.sub.3] = (2, 6), [P.sub.*.sub.4] = (5, 3), [P.sub.*.sub.5] = (3, 0)}, we can construct [lambda]-Bezier curve of degree 5. Then this curve can be separately reduced to [lambda]-Bezier curve of degree 4, respectively, under unrestricted and [C.sup.0], [C.sup.1] constraint condition. Control points and errors for approximating [lambda]-Bezier curve of degree 4 to a [lambda]-Bezier 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 [0.27488 x [10.sup.-4], 1.1886 x [10.sup.-4]] in (a), and those in (b) and (c) are [0.60937 x [10.sup.-4], 2.3013 x [10.sup.-4]] and [0.85492 x [10.sup.-3], 2.3364 x [10.sup.-3]], respectively.

Example 2. Given the shape parameter [lambda] = -1 and the coordinates of control points {[P.sup.*.sub.0] = (0, 0), [P.sup.*.sub.1] = (1, 2), [P.sup.*.sub.2] = (2, 2.3), [P.sup.*.sub.3] = (3, 0.4), [P.sup.*.sub.4] = (5, 0.8), [P.sup.*.sub.5] = (6, 2.1), [P.sup.*.sub.6] = (8, 2.7), [P.sup.*.sub.7] = (9, 0.3)}, we can construct a [lambda]-Bezier curve of degree 7. Then this curve will be separately reduced to [lambda]-Bezier curves of degree 5 under three conditions. Control points and errors for approximating [lambda]-Bezier curve of degree 5 to a [lambda]-Bezier 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 [lambda]-Bezier 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 x [10.sup.-4], 1.812 x [10.sup.-4]] in (a), and those in (b) and (c) are [0.45964 x [10.sup.-4], 3.3081 x [10.sup.-4]] and [0.30212 x [10.sup.-3], 22919 x [10.sup.-3]], respectively.

Example 3. Given shape parameter [lambda] = -1, and two segments of [lambda]-Bezier curves of degree 7 expressing patterned vase, then they will be separately reduced to two segments of [lambda]-Bezier curve of degree 5 under unrestricted condition. Control points and error for approximating [lambda]-Bezier curve of degree 5 to a [lambda]-Bezier 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 [C.sup.0] and [C.sup.1] constraint conditions in Example 3 are 0.81951 x [10.sup.-3] and 0.50732 x [10.sup.-2], respectively.

With the change of shape parameter, we present error graph of degree reduction of [lambda]-Bezier 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 x [10.sup.-3], 1.6712 x [10.sup.-3]].

6. Concluding Remarks

[lambda]-Bezier curves of degree n have the same properties as Bezier 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 [lambda]-Bezier curves is studied by least squared approximation. An algorithm for approximating degree reduction of [lambda]-Bezier 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 [lambda]-Bezier surfaces in future work.

http://dx.doi.org/10.1155/2016/8140427

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).

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Gang Hu, Huanxin Cao, and Suxia Zhang

Department of Applied Mathematics, Xi'an University of Technology, Xian 710054, China

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

Received 28 December 2015; Accepted 12 April 2016

Academic Editor: George S. Dulikravich

Caption: Figure 1: [lambda]-Bezier curves with the same control polygon but different shape parameters.

Caption: Figure 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.

Caption: Figure 3: Error graph of degree reduction of [lambda]-Bezier of degree 5.

Caption: Figure 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.

Caption: Figure 5: Error graph of degree reduction of [lambda]-Bezier of degree 7.

Caption: Figure 6: Degree reduction of [lambda]-Bezier 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.

Caption: Figure 7: Error graph of degree reduction of [lambda]-Bezier 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).

Constraint        Control points              Errors
condition

Under             [P.sub.0] = (-5.0241,       [d.sup.2]([p.sup.*.sub.5]
unrestricted      -0.01868), [P.sub.1] =      (t; 1), [p.sub.4](t; 1))
condition         (-7.2251, 2.266),           = 011866 x [10.sup.-3]
                  [P.sub.2] = (-0.24919,
                  6.690), [P.sub.3] =
                  (5.2589, 3.672),
                  [P.sub.4] = (3.0186,
                  -0.03573)

Under             [P.sub.0] = (-5,0),         [d.sup.2]([p.sup.*.sub.5]
[C.sup.0]         [P.sub.1] = (-7.2449,       54 (t; 1), [p.sub.4](t;
constraint        2.273), [P.sub.2] =         1)) = 0.23013 x
condition         (-0.24428, 6.705),          [10.sup.-3]
                  [P.sub.3] = (5.2705,
                  3.639), [P.sub.4] = (3,
                  0)

Under             [P.sub.0] = (-5,0),         [d.sup.2]([p.sup.*.sub.5]
[C.sup.1]         [P.sub.1] = (-7.3333,       (t; 1), [p.sub.4](t; 1))
constraint        2.3333), [P.sub.2] =        = 0.21581 x [10.sup.-2]
condition         (-0.22601, 6.7616),
                  [P.sub.3] = (5.3333,
                  3.5000), [P.sub.4] = (3,
                  0)

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

Constraint        Control points              Error
condition

Under             [P.sub.0] =                 [d.sup.2]([p.sup.*.sub.7]
unrestricted      (-0.0073157, -0.01488),     (t; 1), [p.sub.5] (t;
condition         [P.sub.1] = (1.6003,        1)) = 0.25675 x
                  3.489), [P.sub.2] =         [10.sup.-4]
                  (2.7337, 0.4590),
                  [P.sub.3] = (4.8462,
                  -0.5802), [P.sub.4] =
                  (7.1691, 4.336),
                  [P.sub.5] = (9.0139,
                  0.2929)

Under             [P.sub.0] = (0, 0),         [d.sup.2]([p.sup.*.sub.7]
[C.sup.0]         [P.sub.1] = (1.6128,        (t; 1), [p.sub.5] (t;
constraint        3.443), [P.sub.2] =         1)) = 0.45964 x
condition         (2.7475, 0.5180),           [10.sup.-4]
                  [P.sub.3] = (4.7940,
                  -0.5994), [P.sub.4] =
                  (7.2115, 4.325),
                  [P.sub.5] = (9, 0.3)

Under             [P.sub.0] = (0, 0),         [d.sup.2]([p.sup.*.sub.7]
[C.sup.1]         [P.sub.1] = (1.6667,        (t; 1), [p.sub.5] (t;
constraint        3.3333), [P.sub.2] =        1)) = 0.30212 x
condition         (2.7655, 0.7045),           [10.sup.-3]
                  [P.sub.3] = (4.6134,
                  -0.6613), [P.sub.4] =
                  (7.3333, 4.3), [P.sub.5]
                  = (9, 0.3)

Table 3: Control points and approximation error under unrestricted
condition in Example 3 (from degree 7 to degree 5).

                      Control points          Error

Before      First     [P.sup.*.sub.0,1] =    d([P.sup.*.sub.7]
degree      segment   (4.5, 8),               (t; 1), [p.sub.5]
reduction             [P.sup.*.sub.1,1] =     (t; 1)) = 0.45528 x
                      (4.8, 7.5),             [10.sup.-3]
                      [P.sup.*.sub.2,1] =
                      (4.7, 4),
                      [P.sup.*.sub.3,1] =
                      (4.6, 4),
                      [P.sup.*.sub.4,1] =
                      (3, 4),
                      [P.sup.*.sub.5,1] =
                      (3, 1),
                      [P.sup.*.sub.6,1] =
                      (4, 1),
                      [P.sup.*.sub.7,1] =
                      (4.4, 0.7)

            Second    [P.sup.*.sub.0,2] =
            segment   (5.5, 8),
                      [P.sup.*.sub.1,2] =
                      (5.2, 7.5),
                      [P.sup.*.sub.2,2] =
                      (5.3, 4),
                      [P.sup.*.sub.3,2] =
                      (5.4, 4),
                      [P.sup.*.sub.4,2] =
                      (7, 4),
                      [P.sup.*.sub.5,2] =
                      (7, 1),
                      [P.sup.*.sub.6,2] =
                      (6, 1),
                      [P.sup.*.sub.7,2] =
                      (5.6, 0.7)

After       First     [P.sup.*.sub.0,1] =
degree      segment   (4.5165, 8.067),
reduction             [P.sup.*.sub.1,1] =
                      (4.7885, 6.478),
                      [P.sup.*.sub.2,1] =
                      (5.4628, 2.165),
                      [P.sup.*.sub.3,1] =
                      (1.6892, 4.636),
                      [P.sup.*.sub.4,1] =
                      (3.7304, 0.3241),
                      [P.sup.*.sub.5,1] =
                      (4.4033, 0.7784)

            Second    [P.sup.*.sub.0,2] =
            segment   (5.4835, 8.067),
                      [P.sup.*.sub.1,2] =
                      (5.2115, 6.478),
                      [P.sup.*.sub.2,2] =
                      (4.5372, 2.165),
                      [P.sup.*.sub.3,2] =
                      (8.3108, 4.636),
                      [P.sup.*.sub.4,2] =
                      (6.2696, 0.3241),
                      [P.sup.*.sub.5,2] =
                      (5.5967, 0.7784)
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Title Annotation:Research Article
Author:Hu, Gang; Cao, Huanxin; Zhang, Suxia
Publication:Mathematical Problems in Engineering
Date:Jan 1, 2016
Words:4825
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