research article the representation of circular arc by using ...research article the representation...

Research ArticleThe Representation of Circular Arc byUsing Rational Cubic Timmer Curve

Muhammad Abbas12 Norhidayah Ramli2 Ahmad Abd Majid2 and Jamaludin Md Ali2

1 Department of Mathematics University of Sargodha 40100 Sargodha Pakistan2 School of Mathematical Sciences Universiti Sains Malaysia 11800 Penang Malaysia

Correspondence should be addressed to Muhammad Abbas mabbasuosedupk

Received 24 June 2013 Accepted 12 December 2013 Published 16 January 2014

Academic Editor Metin O Kaya

Copyright copy 2014 Muhammad Abbas et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In CADCAM systems rational polynomials in particular the Bezier or NURBS forms are useful to approximate the circular arcsIn this paper a new representation method by means of rational cubic Timmer (RCT) curves is proposed to effectively representa circular arc The turning angle of a rational cubic Bezier and rational cubic Ball circular arcs without negative weight is still notmore than 41205873 and 120587 respectively The turning angle of proposed approach is more than Bezier and Ball circular arcs with easiercalculation and determination of control pointsThe proposedmethod also provides the easier modification in the shape of circulararc showing in several numerical examples

1 Introduction

Thestudy of curves plays a significant role inComputerAidedGeometric Design (CAGD) and Computer Graphics (CG)in particular parametric forms because it is easy to modelcurves interactively [1] CAGD copes with the representationof free form curves In parametric form it is importantwhich basis functions are used to represent the circular arcsCircular arcs are extensively used in the fields of CAGD andCADCAM systems since circular arcs can be represented byparametric (rational) polynomials instead of polynomials inexplicit form Faux and Pratt [2] represented only an ellipticsegment whose turning angle is less than 120587 by a rationalquadratic Bezier curve

Generally a rational cubic Bezier is used to extend theturning angle for conics The maximum turning angle of arational cubic circular arc is not more than 41205873 [3] Onlythe negative weight conditions can extend its expressingrange to 2120587 (not equal to 2120587) and such conditions are notcooperative in CAD systems because they lose the convexhull property [4] A rational quartic Bezier curve can expressany circular arc whose central angle is less than 2120587 andit requires at least rational Bezier curve of degree five to

represent a full circle without resorting to negative weights[5] Fang [6] presented a special representation for conicsections by a rational quartic Bezier curve which has the sameweight for all control points but the middle one G-J Wangand G-Z Wang [7] presented the necessary and sufficientconditions for the rational cubic Bezier representation ofconics by applying coordinate transformation and parametertransformation Hu and Wang [8] derived the necessaryand sufficient conditions on control points and weightsfor the rational quartic Bezier representation of conics byusing two specials kinds degree reducible and improperlyparameterization Usually rational cubic and rational quarticBezier curves are used for representation of conic sections orcircular arcs representationwhich can also be found in severalpapers [9ndash12] Hu and Wang [13] constructed necessary andsufficient conditions for conic representation in rational lowdegree Bezier form and the transformation formula fromBernstein basis to Said-Ball basis

In this work we present a rational cubic representationof a circular arc using Timmer basis functions It is ableto approximate a circular arc up to 2120587 (but not including2120587) without resorting to negative weights in contrast ofrational cubic Bezier and rational cubic Ball circular arcs

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 408492 6 pageshttpdxdoiorg1011552014408492

2 Mathematical Problems in Engineering

representation methods Necessary and sufficient conditionsare derived for a representation of RCT circular arc

The remainder of this paper is as follows The cubicalternative basis functions with geometric properties aregiven in Section 2 The RCT curve with some geometriccurve properties is presented in Section 3 and the weightformulation for Timmer circular arc is also part of thissection The RCT curve as conic segment is represented inSection 4 The Timmer circular arc representation methodis discussed in Section 5 and some numerical examples ofthe proposed method are presented in Section 6 Section 7 isdevoted to conclusions

2 Beacutezier-Like Cubic Basis Functions

In this section the Bezier-like cubic basis functions119861119894(119906) 119894 =

0 1 2 3 are defined in [14 15] as

1198610 (119906) = (1 minus 119906)

2(1 + (2 minus 119898) 119906)

1198611 (119906) = 119898(1 minus 119906)

2119906

1198612 (119906) = 119898 (1 minus 119906) 119906

2

1198613 (119906) = 119906

2(1 + (2 minus 119898) (1 minus 119906)) 119906 isin [0 1]

(1)

where 119898 is a shape parameter For 119898 = 2 the cubic basisfunction are the Ball basis functions [14] 119898 = 3 the basisfunctions are Bernstein Bezier basis functions [1] and if119898 =

4 then (1) represents a Timmer cubic basis functions [15 16]see Figure 1

Theorem 1 (see [15]) The Bezier-like cubic basis functions119861119894(119906) 119894 = 0 1 2 3 satisfy the following properties

(a) Positivity For 0 le 119898 le 3 the basis function (1) arenonnegative on the interval 119906 isin [0 1] but this property is nomore satisfied when119898 = 4

(b) Partition of Unity The sum of the Bezier-like cubic basisfunctions is one on the interval 119906 isin [0 1] That is

3

sum

119894=0

119861119894 (119906) = 1 (2)

(c) Monotonicity For the given value of the parameter 1198981198610(119906) is monotonically decreasing and 119861

3(119906) is monotonically

increasing

(d) Symmetry

119861119894 (119906119898) = 1198613minus119894 (1 minus 119906119898) 119894 = 0 1 2 3 (3)

3 A Rational Cubic Beacutezier-Like Curve

The rational cubic Bezier-like curve with single shape param-eter119898 can be defined as follows

Definition 2 Given the control points 119875119894(119894 = 0 1 2 3) inR2

the rational cubic Bezier-like curve is

119903 (119906) =sum3

119894=0119908119894119875119894119861119894 (119906)

sum3

119894=0119908119894119861119894 (119906)

119906 isin [0 1] (4)

In standard representation the weights 1199080= 1199083= 1 and

the middle weights are positive and in this paper the middleweights are considered to be equal to 119908

1= 1199082= 119908 Figure 2

illustrates cubic curves with 119898 = 2 3 and 4 with 119908 = 1 torepresent the ordinary cubic curves

There are some geometric properties of rational cubicBezier-like curve as follows

Theorem 3 The rational cubic Bezier-like curve satisfies thefollowing properties

(a) End Point Properties Consider

119903 (0) = 1198750 119903 (1) = 1198753

1199031015840(0) = 119898 (119875

1minus 1198750) 119908 119903

1015840(1) = 119898 (119875

3minus 1198752) 119908

11990310158401015840(0) = 2 ( (3 minus 119898) 1198753 + 1198981199081198752 + 119898119908 (minus2 + 119898 minus 119898119908)1198751

+ (minus3 + 1198982(minus1 + 119908)119908 + 119898 (1 + 119908)) 1198750)


+ (minus3 + 1198982(minus1 + 119908)119908 + 119898 (1 + 119908)) 1198753)

(5)

(b) Symmetry The control points 119875119894and 119875

3minus119894define the same

rational cubic Bezier-like curve in different parameterizationthat is 119903(119906119898 119875

119894) = 119903(1 minus 119906119898 119875

3minus119894)

(c) Geometric Invariance The partition of unity property ofBezier-like cubic basis functions assures the invariance of theshape of the rational cubic Bezier-like curve under translationand rotation of its control points

(d) Convex Hull Property In general rational cubic curve doesnot satisfy the convex hull property for119898 isin R But if119898 isin [0 3]it is always confined to the convex hull of its defining controlpoints For 119898 = 4 and 119908 gt 0 edge 119875

11198752of the control polygon

is tangent to the curve at the midpoint of 11987511198752and the curve

(4) always lies inside the control polygon see Figure 3

Proposition 4 The rational cubic Bezier-like curve (4) repre-sents a conic segment if and only if the weight of curve (4) is

119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (6)

Proof Let 119904(119906) = sum3

119894=0119861119894119875119894119908119894be the rational cubic poly-

nomial in (4) Since the middle weights are equal thedenominator is quadratic then for 119904(119906) to be a conic by

Mathematical Problems in Engineering 3

10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)

B2(u)

(a)

10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)

B2(u)

(b)

10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)B2(u)

(c)

Figure 1 (a) Ball cubic basis functions for119898 = 2 (b) Bernstein cubic Bezier basis functions for119898 = 3 and (c) Timmer cubic basis functionfor119898 = 4

20

15

10

05

minus2 minus1 1 2

P0

P1 P2

P3

Figure 2 Cubic Said-Ball curve for119898 = 2 (red) cubic Bezier curvefor119898 = 3 (green) and Timmer PC curve for119898 = 4 (magenta) witharbitrary control points 119875

119894

taking the condition that the numerator of (4) to be quadraticwe get

(2 minus 119898) (1198750 minus 1198753) + 119898 (1198751minus 1198752) 119908 = 0

(1198752minus 1198751) 119908 =

119898 minus 2

119898(1198753minus 1198750)

119908 = (119898 minus 2

119898)(1198753minus 1198750)

(1198752minus 1198751)

(7)

15

10

05

minus2 minus1 1 2

y-axis

x-axis

Figure 3 Rational cubic Timmer curve for 119898 = 4 with differentlocal values of weight119908 = 100 (blue dashed)119908 = 55 (blue)119908 = 35

(red dashed) 119908 = 2 (red) 119908 = 15 (brown dashed) 119908 = 1(brown)119908 = 05 (black) and 119908 = 022 (black dashed)

Showing that 1198752minus 1198751 1198753minus 1198750

119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (8)

For119898 = 4

119908 =1

2

(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (9)


S P2P1

P3P0

Figure 4 Different rational cubic Timmer curves with differentvalues of 119908

2

S P2P1

P3P0

Figure 5 Family of rational cubic Timmer with midpoint 119878 and redcurve is a conic

4 A Rational Cubic Timmer (RCT) Curve asa Conic Segment

In this section we illustrate simple determination of middleweights for a curve to interpolate the intermediate point 119878Sarfraz [17] has shown the construction using rational Beziercubic It is easy to determine the weights using RCT insteadof Bezier because 119878 is the intermediate point of RCT curve (4)at point 119905 = 12 such that

119903 (05) =11990811198751+ 11990821198752

1199081+ 1199082

(10)

Since 119878 is known intermediate point of Timmer thusfrom (9) we obtain

119878 =11990811198751+ 11990821198752

1199081+ 1199082

(11)

From (11) we get

1199081

1199082

= 119896 (12)

where

119896 =(1198752minus 119878) sdot (119875

2minus 119878)

(119878 minus 1198751) sdot (1198752minus 119878)

(13)

As long as the ratio does not change the curve passesthrough 119878 Figure 4 illustrates the different RCT curves

1 2minus2 minus1

15

10

05

S

C

rr120572

120579

OP0

P1 P2

P3

Figure 6 Rational cubic Timmer circular arc for119898 = 4

From (12) if 119878 is the midpoint then the ratio is one If 119908is equal to (9) then the RCT is a conic Figure 5 shows thatthe family of RCT with 119878 is the midpoint of 119875

11198752

Sometimes only the end points end tangents and theintermediate point are given The middle control points areobtained by taking the intersection points when the linethrough 119878 is parallel to the line defined by two end pointsintersecting with end tangents

5 A Representation of Circular Arc

In this section the representation of circular arc using RCTcurve (4) is discussed Let 119903(119906) = (119909 119910) be a RCT curvecentered at119862with 120579 as a turning angle 120579 isin (0 2120587)with radius119903 = csc120572 where 120572 = 120587 minus 1205792 is given Let 119879

0and 119879

1be the

end point tangents to this circular arc Since 1198752minus1198751 1198753minus1198750

referring to Figure 6 the control points of 119903(119906) are

1198750 (minus1 0)

1198751(minus119903cot120572

2 tan(120587 minus 120572

2))

1198752(119903cot120572

2 tan(120587 minus 120572

2))

1198753 (1 0)

(14)

with equal internal weights

119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (15)

The parametric coordinates of the curve (4) are

119909 (119906) =minus1 + 2119906

1 + 2 (minus1 + 119906) 119906 + (minus2 + 119898) (minus1 + 119906) 119906Cos120572

119910 (119906) = minus(minus2 + 119898) (minus1 + 119906) 119906 Sin120572


(16)

Theorem 5 A RCT curve 119903(119906) = (119909 119910) with control points119875119894 119894 = 0 1 2 3 and weight 119908 represents the circular arc with

central angle 120579 isin (0 2120587) if and only if 1199092(119906)+(119910(119906)minuscot120572)2 =csc2120572


Proof Let 119903(119906) be a circular arc with control points 119875119894 119894 =

0 1 2 3 The coordinates of any point (119909(119906) 119910(119906)) on thecurve (4) with equal internals weights 119908 are given by

119909 (119906) =(1 minus 2119906)

2 (1 minus 119906) 119906 (1 + Cos120572) minus 1

119910 (119906) = minus2119906 (1 minus 119906) Sin120572

2119906 (1 minus 119906) (1 + Cos120572) minus 1

(17)

Then

1199092(119906) + (119910 (119906) minus cot120572)2

= (1 minus 2119906

2119906(1 minus 119906)(1 + cos120572) minus 1)

2

+ (minus2119906 (1 minus 119906) sin120572

2119906 (1 minus 119906) (1 + cos120572) minus 1

minus cot120572)2

= ((1 minus 2119906)2sin2120572

+ (2119906 (1 minus 119906) sin2120572 + cos120572

times (2119906 (1 minus 119906) (1 + cos120572) minus 1))2

)

times ( (2119906 (1 minus 119906)

times (1 + cos120572) minus 1)2sin2120572)minus1

= (2119906 (1 minus 119906) (1 + cos120572) minus 1)2

times ( (2119906 (1 minus 119906)

times (1 + cos 120572) minus 1)2sin2120572)minus1

= csc2120572

= 1199032

(18)

Proposition 6 Given 119898 = 4 and 120579 isin (0 2120587) as the centralangle of the circular arc 119903(119906) with control points 119875

119894 then the

weight is 119908 = sin2(1205722)This is the outcome of (9)

Proposition 7 For 119908 = 0 the RCT curve (4) becomes astraight line which is parallel to both 119909-axis and 119875

01198753

6 Numerical Examples

We apply our representation method with examples that areshown in Figures 7 and 8 using different values of turningangle 120579 and values of weight 119908

120579 = 5912058736

120579 = 141205879

120579 = 131205879

120579 = 41205873

120579 = 101205879

120579 = 120587

120579 = 51205873

P0 P3

0 1 2minus1minus2

Figure 7 The representation of circular arcs with different turningangles and weights 119908 = 05 (blue) 119908 = 0413 (blue dashed) 119908 =

025 (red) 119908 = 0178 (red dashed) 119908 = 0116 (green dashed) 119908 =

0078 (green) and 119908 = 0066 (black)

70

60

50

40

30

20

10

minus30 minus20 minus10 10 20 30

Figure 8 RCT curve as a circular arc with weight 119908 = 000017 andturning angle 120579 = 11912058760

7 Conclusion

In this paper the representation of circular arcs using RCTcurves was presented The largest turning angle of a rationalcubic Bezier and rational cubic Ball circular arcs is stillnot more than 41205873 and 120587 respectively Therefore we haveincreased the range for representation of circular arc up to 2120587(but not including 2120587) without resorting to negative weightsusing RCT curve Conic segments have also represented


with RCT curves thus studying the necessary and sufficientconditions for RCT representation of circular arc is worthy

Conflict of Interests

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

Acknowledgments

The authors are indebted to the anonymous reviewers fortheir helpful valuable suggestions and comments for theimprovement of the paper The authors also would like toextend the gratitude to the School of Mathematical SciencesUniversiti Sains Malaysia Penang Malaysia for the financialsupport from FRGS Grant no 203PMATHS6711324

References

[1] G Farin Curves and Surfaces for CAGD A Practical GuideMorgan Kaufmann Boston Mass USA 5th edition 2002

[2] I D Faux and M J Pratt Computational Geometry for Designand Manufacture Ellis Horwood Chichester UK 1979

[3] G-J Wang ldquoRational cubic circular arcs and their applicationin CADrdquoComputers in Industry vol 16 no 3 pp 283ndash288 1991

[4] C Blanc and C Schlick ldquoAccurate parametrization of conics byNURBSrdquo IEEE Computer Graphics and Applications vol 16 no6 pp 64ndash71 1996

[5] J J Chou ldquoHigher order Bezier circlesrdquoComputer-AidedDesignvol 27 no 4 pp 303ndash309 1995

[6] L Fang ldquoA rational quartic Bezier representation for conicsrdquoComputer Aided Geometric Design vol 19 no 5 pp 297ndash3122002

[7] G-J Wang and G-Z Wang ldquoThe rational cubic Bezier repre-sentation of conicsrdquo Computer Aided Geometric Design vol 9no 6 pp 447ndash455 1992

[8] Q-Q Hu and G-J Wang ldquoNecessary and sufficient conditionsfor rational quartic representation of conic sectionsrdquo Journalof Computational and Applied Mathematics vol 203 no 1 pp190ndash208 2007

[9] Q-Q Hu ldquoApproximating conic sections by constrained Beziercurves of arbitrary degreerdquo Journal of Computational andApplied Mathematics vol 236 no 11 pp 2813ndash2821 2012

[10] Y J Ahn and H O Kim ldquoApproximation of circular arcs byBezier curvesrdquo Journal of Computational and Applied Mathe-matics vol 81 no 1 pp 145ndash163 1997

[11] S-H Kim and Y J Ahn ldquoAn approximation of circular arcs byquartic Bezier curvesrdquo Computer-Aided Design vol 39 no 6pp 490ndash493 2007

[12] Y J Ahn ldquoApproximation of conic sections by curvaturecontinuous quartic Bezier curvesrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 1986ndash1993 2010

[13] Q-Q Hu and G-J Wang ldquoRational cubicquartic Said-Ballconicsrdquo Applied Mathematics vol 26 no 2 pp 198ndash212 2011

[14] H B Said ldquoA generalized Ball curve and its recursive algo-rithmrdquoACMTransactions onGraphics vol 8 no 4 pp 360ndash3711989

[15] JM Ali ldquoAn alternative derivation of Said basis functionrdquo SainsMalaysiana vol 23 no 3 pp 42ndash56 1994

[16] H G Timmer ldquoAlternative representation for parametric cubiccurves and surfacesrdquo Computer-Aided Design vol 12 no 1 pp25ndash28 1980

[17] M Sarfraz ldquoOptimal curve fitting to digital datardquo Journal ofWSCG vol 11 no 3 8 pages 2003

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representation methods Necessary and sufficient conditionsare derived for a representation of RCT circular arc

The remainder of this paper is as follows The cubicalternative basis functions with geometric properties aregiven in Section 2 The RCT curve with some geometriccurve properties is presented in Section 3 and the weightformulation for Timmer circular arc is also part of thissection The RCT curve as conic segment is represented inSection 4 The Timmer circular arc representation methodis discussed in Section 5 and some numerical examples ofthe proposed method are presented in Section 6 Section 7 isdevoted to conclusions

2 Beacutezier-Like Cubic Basis Functions

In this section the Bezier-like cubic basis functions119861119894(119906) 119894 =

0 1 2 3 are defined in [14 15] as

1198610 (119906) = (1 minus 119906)

2(1 + (2 minus 119898) 119906)

1198611 (119906) = 119898(1 minus 119906)

2119906

1198612 (119906) = 119898 (1 minus 119906) 119906

2

1198613 (119906) = 119906

2(1 + (2 minus 119898) (1 minus 119906)) 119906 isin [0 1]

(1)

where 119898 is a shape parameter For 119898 = 2 the cubic basisfunction are the Ball basis functions [14] 119898 = 3 the basisfunctions are Bernstein Bezier basis functions [1] and if119898 =

4 then (1) represents a Timmer cubic basis functions [15 16]see Figure 1

Theorem 1 (see [15]) The Bezier-like cubic basis functions119861119894(119906) 119894 = 0 1 2 3 satisfy the following properties

(a) Positivity For 0 le 119898 le 3 the basis function (1) arenonnegative on the interval 119906 isin [0 1] but this property is nomore satisfied when119898 = 4

(b) Partition of Unity The sum of the Bezier-like cubic basisfunctions is one on the interval 119906 isin [0 1] That is

3

sum

119894=0

119861119894 (119906) = 1 (2)

(c) Monotonicity For the given value of the parameter 1198981198610(119906) is monotonically decreasing and 119861

3(119906) is monotonically

increasing

(d) Symmetry

119861119894 (119906119898) = 1198613minus119894 (1 minus 119906119898) 119894 = 0 1 2 3 (3)

3 A Rational Cubic Beacutezier-Like Curve

The rational cubic Bezier-like curve with single shape param-eter119898 can be defined as follows

Definition 2 Given the control points 119875119894(119894 = 0 1 2 3) inR2

the rational cubic Bezier-like curve is

119903 (119906) =sum3

119894=0119908119894119875119894119861119894 (119906)

sum3

119894=0119908119894119861119894 (119906)

119906 isin [0 1] (4)

In standard representation the weights 1199080= 1199083= 1 and

the middle weights are positive and in this paper the middleweights are considered to be equal to 119908

1= 1199082= 119908 Figure 2

illustrates cubic curves with 119898 = 2 3 and 4 with 119908 = 1 torepresent the ordinary cubic curves

There are some geometric properties of rational cubicBezier-like curve as follows

Theorem 3 The rational cubic Bezier-like curve satisfies thefollowing properties

(a) End Point Properties Consider

119903 (0) = 1198750 119903 (1) = 1198753

1199031015840(0) = 119898 (119875

1minus 1198750) 119908 119903

1015840(1) = 119898 (119875

3minus 1198752) 119908


+ (minus3 + 1198982(minus1 + 119908)119908 + 119898 (1 + 119908)) 1198750)


+ (minus3 + 1198982(minus1 + 119908)119908 + 119898 (1 + 119908)) 1198753)

(5)

(b) Symmetry The control points 119875119894and 119875

3minus119894define the same

rational cubic Bezier-like curve in different parameterizationthat is 119903(119906119898 119875

119894) = 119903(1 minus 119906119898 119875

3minus119894)

(c) Geometric Invariance The partition of unity property ofBezier-like cubic basis functions assures the invariance of theshape of the rational cubic Bezier-like curve under translationand rotation of its control points

(d) Convex Hull Property In general rational cubic curve doesnot satisfy the convex hull property for119898 isin R But if119898 isin [0 3]it is always confined to the convex hull of its defining controlpoints For 119898 = 4 and 119908 gt 0 edge 119875

11198752of the control polygon

is tangent to the curve at the midpoint of 11987511198752and the curve

(4) always lies inside the control polygon see Figure 3

Proposition 4 The rational cubic Bezier-like curve (4) repre-sents a conic segment if and only if the weight of curve (4) is

119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (6)

Proof Let 119904(119906) = sum3

119894=0119861119894119875119894119908119894be the rational cubic poly-

nomial in (4) Since the middle weights are equal thedenominator is quadratic then for 119904(119906) to be a conic by


10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)

B2(u)

(a)

10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)

B2(u)

(b)

10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)B2(u)

(c)


20

15

10

05

minus2 minus1 1 2

P0

P1 P2

P3


119894



(1198752minus 1198751) 119908 =

119898 minus 2

119898(1198753minus 1198750)

119908 = (119898 minus 2

119898)(1198753minus 1198750)

(1198752minus 1198751)

(7)

15

10

05

minus2 minus1 1 2

y-axis

x-axis




119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (8)

For119898 = 4

119908 =1

2

(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (9)


S P2P1

P3P0


2

S P2P1

P3P0




119903 (05) =11990811198751+ 11990821198752

1199081+ 1199082

(10)


119878 =11990811198751+ 11990821198752

1199081+ 1199082

(11)

From (11) we get

1199081

1199082

= 119896 (12)

where

119896 =(1198752minus 119878) sdot (119875

2minus 119878)

(119878 minus 1198751) sdot (1198752minus 119878)

(13)


1 2minus2 minus1

15

10

05

S

C

rr120572

120579

OP0

P1 P2

P3



11198752




0and 119879

1be the



1198750 (minus1 0)

1198751(minus119903cot120572

2 tan(120587 minus 120572

2))

1198752(119903cot120572

2 tan(120587 minus 120572

2))

1198753 (1 0)

(14)


119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (15)


119909 (119906) =minus1 + 2119906




(16)






119909 (119906) =(1 minus 2119906)

2 (1 minus 119906) 119906 (1 + Cos120572) minus 1

119910 (119906) = minus2119906 (1 minus 119906) Sin120572

2119906 (1 minus 119906) (1 + Cos120572) minus 1

(17)

Then

1199092(119906) + (119910 (119906) minus cot120572)2

= (1 minus 2119906

2119906(1 minus 119906)(1 + cos120572) minus 1)

2

+ (minus2119906 (1 minus 119906) sin120572

2119906 (1 minus 119906) (1 + cos120572) minus 1

minus cot120572)2

= ((1 minus 2119906)2sin2120572

+ (2119906 (1 minus 119906) sin2120572 + cos120572


)

times ( (2119906 (1 minus 119906)


= (2119906 (1 minus 119906) (1 + cos120572) minus 1)2

times ( (2119906 (1 minus 119906)


= csc2120572

= 1199032

(18)


119894 then the



01198753



120579 = 5912058736

120579 = 141205879

120579 = 131205879

120579 = 41205873

120579 = 101205879

120579 = 120587

120579 = 51205873

P0 P3

0 1 2minus1minus2



0078 (green) and 119908 = 0066 (black)

70

60

50

40

30

20

10



7 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)

B2(u)

(a)

10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)

B2(u)

(b)

10

08

06

04

02

02 04 06 08 10

B0(u)

B3(u)

B1(u)B2(u)

(c)


20

15

10

05

minus2 minus1 1 2

P0

P1 P2

P3


119894



(1198752minus 1198751) 119908 =

119898 minus 2

119898(1198753minus 1198750)

119908 = (119898 minus 2

119898)(1198753minus 1198750)

(1198752minus 1198751)

(7)

15

10

05

minus2 minus1 1 2

y-axis

x-axis




119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (8)

For119898 = 4

119908 =1

2

(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (9)


S P2P1

P3P0


2

S P2P1

P3P0




119903 (05) =11990811198751+ 11990821198752

1199081+ 1199082

(10)


119878 =11990811198751+ 11990821198752

1199081+ 1199082

(11)

From (11) we get

1199081

1199082

= 119896 (12)

where

119896 =(1198752minus 119878) sdot (119875

2minus 119878)

(119878 minus 1198751) sdot (1198752minus 119878)

(13)


1 2minus2 minus1

15

10

05

S

C

rr120572

120579

OP0

P1 P2

P3



11198752




0and 119879

1be the



1198750 (minus1 0)

1198751(minus119903cot120572

2 tan(120587 minus 120572

2))

1198752(119903cot120572

2 tan(120587 minus 120572

2))

1198753 (1 0)

(14)


119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (15)


119909 (119906) =minus1 + 2119906




(16)






119909 (119906) =(1 minus 2119906)

2 (1 minus 119906) 119906 (1 + Cos120572) minus 1

119910 (119906) = minus2119906 (1 minus 119906) Sin120572

2119906 (1 minus 119906) (1 + Cos120572) minus 1

(17)

Then

1199092(119906) + (119910 (119906) minus cot120572)2

= (1 minus 2119906

2119906(1 minus 119906)(1 + cos120572) minus 1)

2

+ (minus2119906 (1 minus 119906) sin120572

2119906 (1 minus 119906) (1 + cos120572) minus 1

minus cot120572)2

= ((1 minus 2119906)2sin2120572

+ (2119906 (1 minus 119906) sin2120572 + cos120572


)

times ( (2119906 (1 minus 119906)


= (2119906 (1 minus 119906) (1 + cos120572) minus 1)2

times ( (2119906 (1 minus 119906)


= csc2120572

= 1199032

(18)


119894 then the



01198753



120579 = 5912058736

120579 = 141205879

120579 = 131205879

120579 = 41205873

120579 = 101205879

120579 = 120587

120579 = 51205873

P0 P3

0 1 2minus1minus2



0078 (green) and 119908 = 0066 (black)

70

60

50

40

30

20

10



7 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S P2P1

P3P0


2

S P2P1

P3P0




119903 (05) =11990811198751+ 11990821198752

1199081+ 1199082

(10)


119878 =11990811198751+ 11990821198752

1199081+ 1199082

(11)

From (11) we get

1199081

1199082

= 119896 (12)

where

119896 =(1198752minus 119878) sdot (119875

2minus 119878)

(119878 minus 1198751) sdot (1198752minus 119878)

(13)


1 2minus2 minus1

15

10

05

S

C

rr120572

120579

OP0

P1 P2

P3



11198752




0and 119879

1be the



1198750 (minus1 0)

1198751(minus119903cot120572

2 tan(120587 minus 120572

2))

1198752(119903cot120572

2 tan(120587 minus 120572

2))

1198753 (1 0)

(14)


119908 = (119898 minus 2

119898)(1198753minus 1198750) sdot (1198752minus 1198751)

(1198752minus 1198751) sdot (1198752minus 1198751) (15)


119909 (119906) =minus1 + 2119906




(16)






119909 (119906) =(1 minus 2119906)

2 (1 minus 119906) 119906 (1 + Cos120572) minus 1

119910 (119906) = minus2119906 (1 minus 119906) Sin120572

2119906 (1 minus 119906) (1 + Cos120572) minus 1

(17)

Then

1199092(119906) + (119910 (119906) minus cot120572)2

= (1 minus 2119906

2119906(1 minus 119906)(1 + cos120572) minus 1)

2

+ (minus2119906 (1 minus 119906) sin120572

2119906 (1 minus 119906) (1 + cos120572) minus 1

minus cot120572)2

= ((1 minus 2119906)2sin2120572

+ (2119906 (1 minus 119906) sin2120572 + cos120572


)

times ( (2119906 (1 minus 119906)


= (2119906 (1 minus 119906) (1 + cos120572) minus 1)2

times ( (2119906 (1 minus 119906)


= csc2120572

= 1199032

(18)


119894 then the



01198753



120579 = 5912058736

120579 = 141205879

120579 = 131205879

120579 = 41205873

120579 = 101205879

120579 = 120587

120579 = 51205873

P0 P3

0 1 2minus1minus2



0078 (green) and 119908 = 0066 (black)

70

60

50

40

30

20

10



7 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119909 (119906) =(1 minus 2119906)

2 (1 minus 119906) 119906 (1 + Cos120572) minus 1

119910 (119906) = minus2119906 (1 minus 119906) Sin120572

2119906 (1 minus 119906) (1 + Cos120572) minus 1

(17)

Then

1199092(119906) + (119910 (119906) minus cot120572)2

= (1 minus 2119906

2119906(1 minus 119906)(1 + cos120572) minus 1)

2

+ (minus2119906 (1 minus 119906) sin120572

2119906 (1 minus 119906) (1 + cos120572) minus 1

minus cot120572)2

= ((1 minus 2119906)2sin2120572

+ (2119906 (1 minus 119906) sin2120572 + cos120572


)

times ( (2119906 (1 minus 119906)


= (2119906 (1 minus 119906) (1 + cos120572) minus 1)2

times ( (2119906 (1 minus 119906)


= csc2120572

= 1199032

(18)


119894 then the



01198753



120579 = 5912058736

120579 = 141205879

120579 = 131205879

120579 = 41205873

120579 = 101205879

120579 = 120587

120579 = 51205873

P0 P3

0 1 2minus1minus2



0078 (green) and 119908 = 0066 (black)

70

60

50

40

30

20

10



7 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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