Research Article Construction and Analysis of...

Research ArticleConstruction and Analysis of Binary Subdivision Schemes forCurves and Surfaces Originated from Chaikin Points

Rabia Hameed and Ghulam Mustafa

Department of Mathematics The Islamia University of Bahawalpur Bahawalpur Pakistan

Correspondence should be addressed to Ghulam Mustafa ghulammustafaiubedupk

Received 23 August 2016 Revised 24 October 2016 Accepted 30 October 2016

Academic Editor Jacques Liandrat

Copyright copy 2016 R Hameed and G Mustafa 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

We present a new variant of Lane-Riesenfeld algorithm for curves and surfaces both Our refining operator is the modificationof ChaikinDoo-Sabin subdivision operator while each smoothing operator is the weighted average of the foursixteen adjacentpoints Our refining operator depends on two parameters (shape and smoothing parameters) So we get new families of univariateand bivariate approximating subdivision schemes with two parameters The bivariate schemes are the nontensor product schemesfor quadrilateral meshes Moreover we also present analysis of our families of schemes Furthermore our schemes give cubicpolynomial reproduction for a specific value of the shape parameterThenonuniform setting of our univariate and bivariate schemesgives better performance than that of the uniform schemes

1 Introduction and Related Work

Subdivision is a process of generating curvessurfaces byiteratively refining a set of control points according to somespecific refinement rules The set of these specific rules arecalled the subdivision scheme For stationary linear anduniform subdivision schemes these refinement rules aresame at each refinement level So the set of new control pointsat next refinement level can be generated by computing theaffine combination of the set of control points at previousrefinement level A general compact form of linear uniformand stationary binary 119898-variate (119898 = 1 2) subdivisionscheme 119878119886 which maps a polygon 119891119896minus1 = 119891119896minus1120572 120572 isin Z119898 to arefined polygon 119891119896 = 119891119896120572 120572 isin Z119898 is defined as

119891119896120572 = sum120573isinZ119898

119886120572minus2120573119891119896minus1120573 120572 isin Z119898 (1)

The symbol of above subdivision scheme is given by theLaurent polynomial

119886 (119911) = sum120572isinZ119898

119886120572119911120572 119911 isin (C 0)119898 (2)

where 119886 = 119886120572 120572 isin Z119898 is called the mask of subdivisionschemeThe detailed information about refinement rules andLaurent polynomial can be found in [1ndash3]

Surface modeling via subdivision is very important topicin computer graphics and computer aided geometric designand it has been studied by several authors see surveys [4ndash10]and references therein

The construction of univariate subdivision schemes bydifferent variants of Lane-Riesenfeld algorithm has beenstudied by few authors in [8 11ndash13] and construction ofbivariate subdivision schemes by a variant of Lane-Riesenfeldalgorithm has been studied by Romani [8] In fact the Lane-Riesenfeld algorithm is the combination of two operatorsOne operator refines the initial points and other operatorsmooths these refining points for a fixed number of timesTherefore it is also named as Refining-Smooth algorithm (RS-algorithm) Moreover more smoothing steps provide limitcurvessurfaces of wider support as well as of higher smooth-ness

Lane and Riesenfeld [7] proposed an algorithm forgenerating (119899+1)-degree B-spline curve which is the simplestform of the RS-algorithm In Lane-Riesenfeld algorithm theycombined the symbols of linear B-spline scheme and odd

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2016 Article ID 1092476 15 pageshttpdxdoiorg10115520161092476

2 International Journal of Analysis

dkminus112i+1

dkminus112i

fkminus1i

Ckminus112i minus fkminus1

iminus1

Ckminus112i+1 minus fkminus1

i+1


i+1


iminus1

(a) Our refining stage

fkminus1i

dkminus112i+1

dkminus112i

1

2(fkminus1

i minus fkminus1iminus1 )

1

2(f kminus1

i minus fkminus1i+1 )

1

2(Ckminus11

2i+1 minus Ckminus112i )

1

2(Ckminus11

2i minus Ckminus112i+1 )

(b) Romanirsquos refining stage

Figure 1 The geometrical difference between our refining stage and Romanirsquos refining stage

stencil of linear B-spline scheme as refining and smoothingoperators respectively Therefore the symbol of 119899th familymember of Lane-Riesenfeld schemes gives symbol of (119899 + 1)-degree B-spline schemeTheir family gives119862119899 continuity andlinear polynomial reproduction

In literature there are few variants which are appliedon the Lane-Riesenfeld algorithm For example Cashmanet al [12] also proposed a family of univariate subdivisionschemes by using the RS-algorithm which is based onDubuc Deslauriers 4-point interpolatory scheme [14] Theycombined the symbols of 4-point interpolatory subdivisionscheme and odd stencil of 4-point interpolatory subdivisionscheme as refining and smoothing operators respectivelyTheir univariate family gives cubic polynomial reproductionbut by increasing smoothing stages continuity of their familymay or may not be increased

Ashraf et al [11] proposed a family of univariate subdivi-sion schemes by using similar technique to that used byCash-man et al [12] on Dubuc Deslauriers 6-point interpolatoryscheme [14]They also combined the symbol and odd stencilrsquossymbol of Dubuc Deslauriers 6-point interpolatory scheme[14] as refining and smoothing operators respectively Theirunivariate family gives quintic polynomial reproduction butlevel of continuity of their family does not increase in generalby increasing the smoothing stages

Mustafa et al [13] also proposed a family of univariatesubdivision schemes by using the symbol of 4-point inter-polatory subdivision scheme [15] as the refining operatorand the symbol of even stencil of the 4-point approximatingscheme [16] as the smoothing operator Their univariatefamily gives cubic polynomial reproduction but level ofcontinuity of their family also does not increase in generalby increasing the smoothing stages

Romani [8] proposed families of univariate and bivariatesubdivision schemes by using RS-algorithm in which therefining operator is based on a perturbation of Chaikinrsquos

corner cutting subdivision scheme [17] and the smoothingoperator takes the average of two adjacent vertices as in Lane-Riesenfeld algorithm Their univariate family gives linearpolynomial reproduction and continuity of their 119899th familymember is 119862119899 Moreover if 120596 = 116 their 119899th familymember becomes (119899 + 3)th member of Hormann and Sabinrsquos[16] family and it gives cubic polynomial reproduction

There are few variants of Lane-Riesenfeld algorithm butall those algorithms started with two binary schemes bytaking the symbol of first scheme as the refining operator andthe odd or even symbol of second scheme as the smoothingoperator Our approach is different in the sense that we donot use symbol of any existing subdivision scheme as refiningoperator and also we choose appropriate smoothing operatorby ourselves

In this paper we propose a new RS-algorithm which isalso a variant of Lane-Riesenfeld algorithm that comparedwith the one proposed by Romani we modify refiningand smoothing operators both There are two main differ-ences between our RS-algorithm andRomanirsquos RS-algorithmwhich are as follows

(i) Both refining operators depend on Chaikin pointscomputed around initial points but the differencebetween our refining operator and Romanirsquos refiningoperator is that Romani draws two vectors at everyinitial control point and changes position of bothChaikin points which have been computed aroundcorresponding initial point by using these two vec-tors while we draw one vector at every Chaikinpoint and change the position of each Chaikin pointby using each corresponding vector which we havedrawn on it (see Figure 1)

(ii) Difference between both smoothing operators is thatRomanirsquos smoothing operator takes average of two

International Journal of Analysis 3

adjacent points while our smoothing operator takesweighted average of the four adjacent points

Our refining and smoothing operators are different fromthe operators proposed by RomaniThe framework proposedin this paper provides families of schemes where all membersare approximating while the first member of Romanirsquos familyis interpolatory and all the othermembers are approximatingMoreover the flexibility of our shape parameter allows us toadjust the value of shape parameter locally according to thegeometry of initial polygon

The article is organized as follows In Section 2 wepresent the framework for the construction of univariate andbivariate families of subdivision schemes and their graphicaland mathematical comparisons with Romanirsquos families InSection 3 we present some basic properties of these familiesof subdivision schemes that include discussion on basic limitfunctions analysis of continuity polynomial generation andreproduction of the schemes In Section 4 applications ofsome members of the family and comparisons with Romanischemes are presented Conclusions are given in Section 5

2 Construction and Comparison

The Refining-Smooth subdivision scheme is the schemewhich consists of two operators One operator refines theinitial points while the other operator smooths the refinedpoints for the required number of times If we denote refiningoperator by 119877 smoothing operator by 119878 and the subdivisionoperator by 119878119886119897 then

119891119896 = 119878119886119897119891119896minus1 (3)

where

119891119896minus11 = 119877119891119896minus1119891119896 = 119891119896minus1119899+1 = 119878119891119896minus1119899 119899 = 1 2 119897 (4)

In fact refining operator 119877 maps a polygon 119891119896minus1 =119891119896minus1120572 120572 isin Z119898 to a refined polygon119891119896minus11 = 119891119896minus11120572 120572 isin Z119898where119898 = 1 2 that is

119891119896minus11120572 = sum120573isinZ119898


and smoothing operator 119878 maps a polygon 119891119896minus1119899 =119891119896minus1119899120572 120572 isin Z119898 to a refined polygon119891119896minus1119899+1 = 119891119896minus1119899+1120572 120572 isinZ119898 which is defined as

119891119896minus1119899+1120572 = sum120573isinZ119898

119904120573119891119896minus1119899120573 120572 isin Z119898 119899 = 1 2 119897 (6)

The symbols of refining and smoothing operators are

119903 (119911) = sum120572isinZ119898

119903120572119911120572119904 (119911) = sum

120572isinZ119898119904120572119911120572

119911 isin (C 0)119898 (7)

with

119903 (1) = 2119898119904 (1) = 1

where 1 isin R119898

(8)

In RS-algorithm combination of one refining operator and 119897smoothing operators is called the subdivision operatorMore-over we denote proposed univariate and bivariate families ofsubdivision schemes by 119878119886119897 and 119878119887119897 respectively

The construction procedure for our families of schemes isgiven below

21 Univariate Family Here we denote the Chaikin points by119862119896minus112119894 and119862119896minus112119894+1 that are computed around the point119891119896minus10119894 =119891119896minus1119894 as shown in Figure 1 In this figure black solid circlesshow control points at (119896minus1)th subdivision level blue crossesshow the Chaikin points and blue solid circles show refiningpoints Red vectors are involved in the insertion of even point119891119896minus112119894 and green vectors are involved in the insertion of oddpoint 119891119896minus112119894+1 In this figure we have shown the refinement ofonly two Chaikin points which have been inserted aroundpoint 119891119896minus1119894

In our framework we compute vectors 119889119896minus112119894 and 119889119896minus112119894+1by taking the linear combinations of two vectors as shown inFigure 1(a)

119889119896minus112119894 = (119897 + 1) (119862119896minus112119894 minus 119891119896minus1119894+1 )+ (119897 minus 1) (119862119896minus112119894 minus 119891119896minus1119894minus1 )

119889119896minus112119894+1 = (119897 + 1) (119862119896minus112119894+1 minus 119891119896minus1119894minus1 )+ (119897 minus 1) (119862119896minus112119894+1 minus 119891119896minus1119894+1 )

(9)

where in Figure 1(b) vectors are computed as

119889119896minus112119894 = (119897 + 3) (119862119896minus112119894 minus 119862119896minus112119894+1 )+ (119897 minus 12 ) (119891119896minus1119894 minus 119891119896minus1119894minus1 )

119889119896minus112119894+1 = (119897 + 3) (119862119896minus112119894+1 minus 119862119896minus112119894 )+ (119897 minus 12 ) (119891119896minus1119894 minus 119891119896minus1119894+1 )

(10)

with 119897 isin NIn our framework we compute refining point by adding

each Chaikin point to the corresponding vector that is

119891119896minus112119894 = 119862119896minus112119894 + 120596119889119896minus112119894 119891119896minus112119894+1 = 119862119896minus112119894+1 + 120596119889119896minus112119894+1

(11)


fk2i

1

48minus

19

24120596

23

48+

19

24120596

23

48+

19

24120596

1

48minus

19

24120596

(a) Edge rule

minus1

6120596

1

6minus

1

2120596

2

3+

4

3120596

1

6minus

1

2120596 minus

1

6120596

fk2i+1

(b) Vertex rule

Figure 2 Schematic overview of the refinement rules when we apply one time refining operator for 119897 = 1 and one time smoothing operatorSolid circles show the points at (119896 minus 1)th subdivision level while circles show the points at 119896th subdivision level after applying one refiningand one smoothing stage

where 120596 isin R While in Romanirsquos framework refining pointsare computed as

119891119896minus112119894 = 119891119896minus1119894 + 2120596119889119896minus112119894 119891119896minus112119894+1 = 119891119896minus1119894 + 2120596119889119896minus112119894+1

(12)

Now by substituting (9) in (11) we get the followingsubdivision scheme which will refine the course polygon119891119896minus10 into refined polygon 119891119896minus11

119891119896minus112119894 = 1205721119891119896minus1119894minus1 + 1205722119891119896minus1119894 + 1205723119891119896minus1119894+1 119891119896minus112119894+1 = 1205723119891119896minus1119894minus1 + 1205722119891119896minus1119894 + 1205721119891119896minus1119894+1

(13)

where

1205721 = 14 (1 + 2120596 (2 minus 119897))

1205722 = 34 (1 + 2120596119897)

1205723 = minus120596 (119897 + 1) (14)

Now the second step in our frame work is to smooththe refined polygon 119891119896minus11 = 119891119896minus11119894 119894isinZ by applying 119897-timessmoothing operatorThat is smoothing operator maps 119891119896minus11119894into 119891119896119894 = 1198911198960119894 = 119891119896minus1119897+1119894 by using the following rule

119891119896minus1119899+1119894 = 112119891119896minus1119899119894minus1 + 5

12119891119896minus1119899119894 + 512119891119896minus1119899119894+1

+ 112119891119896minus1119899119894+2

(15)

where 119899 = 1 2 119897

By applying one time refining operator and 119897-timessmoothing operator we get the new family of subdivisionschemes with symbol

119886119897 (119911) = 119911minus119897minuslceil1198972rceilminus2 ( 112 (119911 + 1) (1199112 + 4119911 + 1))

119897

sdot (minus120596 (119897 + 1) + 14 1 minus 2120596 (119897 minus 2) 119911

+ 34 (1 + 2120596119897) 1199112 +

34 (1 + 2120596119897) 1199113

+ 14 1 minus 2120596 (119897 minus 2) 1199114 minus 120596 (119897 + 1) 1199115)

(16)

where lceil rceil is a ceiling functionThe first member of our proposed family is a primal 5-

point relaxed subdivision scheme with two refinement rulesThe brief description of these refinement rules is given below

(i) Edge splitting rule for every edge in the coarsepolygon a new edge point is calculated by using theaffine combination of 4 neighboring control points asshown in Figure 2(a)

(ii) Vertex updating rule for every vertex in the coarsepolygon a new vertex point is calculated by using theaffine combination of 5 neighboring control points asshown in Figure 2(b)

Remark 1 For 120596 = 0 symbol (16) takes the form

119886119897 (119911) = 119911minus119897minuslceil1198972rceilminus1 (1199112 + 4119911 + 13 )119897 (119911 + 1)119897+3

4119897+1 (17)

Now we shall construct family of bivariate nontensorproduct subdivision schemes for quadrilateral meshes bychoosing Doo-Sabin points as origin

22 Bivariate Family Let us denote the Doo-Sabin pointsfor one face of quadrilateral mesh by 119862119896minus1121198942119895 119862119896minus112119894+12119895 119862119896minus1121198942119895+1


minus fkminus1i+1j+1

fkminus1i+1jminus

fkminus1ij

Dkminus112i2j

Dkminus112i+12j

minus fkminus1ij+1

Dkminus112i+12j+1Dkminus11

2i2j+1

fkminus1ij+1 fkminus1

i+1j+1

fkminus1i+1j

Ckminus112i+12j+1

Ckminus112i2j+1 minus fkminus1

ij+1

Ckminus112i2j minus fkminus1

i+1j+1Ckminus11

2i+12j

Ckminus112i+12j minus fkminus1

i+1j


ij

Ckminus112i+12j+1 minus fkminus1

ijCkminus112i2j+1

Figure 3 The geometrical construction of the refining stage for quadrilateral meshes

and 119862119896minus112119894+12119895+1 that are computed from the initial points 119891119896minus1119894119895 119891119896minus1119894+1119895 119891119896minus1119894119895+1 and 119891119896minus1119894+1119895+1 as shown in Figure 3 In this figureblack solid circles show initial points at (119896 minus 1)th subdivisionlevel blue cross symbols show Doo-Sabin points and bluesolid circles show refining points at (119896 minus 1)th level red graygreen and orange vectors are involved in the insertion of therefining points 119891119896minus1121198942119895 119891119896minus112119894+12119895 119891119896minus1121198942119895+1 and 119891119896minus112119894+12119895+1 respec-tively

We compute vectors 119863119896minus1121198942119895 119863119896minus112119894+12119895 119863119896minus1121198942119895+1 and119863119896minus112119894+12119895+1 by taking the linear combination of two vectorsthat are drawn on the Doo-Sabin points 119862119896minus1121198942119895 119862119896minus112119894+12119895119862119896minus1121198942119895+1 and 119862119896minus112119894+12119895+1 respectively as shown in Figure 3

119863119896minus1121198942119895 = (119897 + 1) (119862119896minus1121198942119895 minus 119891119896minus1119894+1119895+1)+ (119897 minus 1) (119862119896minus1121198942119895 minus 119891119896minus1119894119895 )

119863119896minus112119894+12119895 = (119897 + 1) (119862119896minus112119894+12119895 minus 119891119896minus1119894119895+1)+ (119897 minus 1) (119862119896minus112119894+12119895 minus 119891119896minus1119894+1119895)


119863119896minus112119894+12119895+1 = (119897 + 1) (119862119896minus112119894+12119895+1 minus 119891119896minus1119894119895 )+ (119897 minus 1) (119862119896minus112119894+12119895+1 minus 119891119896minus1119894+1119895+1)

(18)

where 119897 isin NSimilarly to the curve case we compute refining points as


119891119896minus1121198942119895+1 = 119862119896minus1121198942119895+1 + 120596119863119896minus1121198942119895+1119891119896minus112119894+12119895+1 = 119862119896minus112119894+12119895+1 + 120596119863119896minus112119894+12119895+1

(19)

where 120596 isin RNow by substituting (18) in (19) we get the subdivision

scheme which will refine the course mesh fkminus10 into refinedmesh fkminus11

119891119896minus1121198942119895 = 1205731119891119896minus1119894119895 + 1205732119891119896minus1119894+1119895 + 1205732119891119896minus1119894119895+1 + 1205733119891119896minus1119894+1119895+1119891119896minus112119894+12119895 = 1205732119891119896minus1119894119895 + 1205731119891119896minus1119894+1119895 + 1205733119891119896minus1119894119895+1 + 1205732119891119896minus1119894+1119895+1119891119896minus1121198942119895+1 = 1205732119891119896minus1119894119895 + 1205733119891119896minus1119894+1119895 + 1205731119891119896minus1119894119895+1 + 1205732119891119896minus1119894+1119895+1

119891119896minus112119894+12119895+1 = 1205733119891119896minus1119894119895 + 1205732119891119896minus1119894+1119895 + 1205732119891119896minus1119894119895+1 + 1205731119891119896minus1119894+1119895+1

(20)

where 1205731 = (116)9 + 2(119897 + 8)120596 1205732 = (116)3(2120596119897 + 1)and 1205733 = (116)1 minus 2(7119897 + 8)120596

Now we smooth the refined mesh fkminus11 = 119891119896minus11119894119895 119894119895isinZ byapplying 119897-time smoothing operator That is we map 119891119896minus11119894119895into 119891119896119894119895 = 1198911198960119894119895 = 119891119896minus1119897+1119894119895 by using the following smoothingoperator

119891119896minus1119899+1119894119895 = 112

112119891119896minus1119899119894minus1119895minus1 +

512119891119896minus1119899119894119895minus1 +

512119891119896minus1119899119894+1119895minus1

+ 112119891119896minus1119899119894+2119895minus1 +

512

112119891119896minus1119899119894minus1119895 +

512119891119896minus1119899119894119895

+ 512119891119896minus1119899119894+1119895 +

112119891119896minus1119899119894+2119895 +

512

112119891119896minus1119899119894minus1119895+1

+ 512119891119896minus1119899119894119895+1 +

512119891119896minus1119899119894+1119895+1 +

112119891119896minus1119899119894+2119895+1


minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

288

minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

2881

12120596 +

1

9

1

12120596 +

1

9

5

6120596 +

4

9

(a) Vertex rule

minus1

16120596 +

1

288

1

72

1

72

minus1

16120596 +

1

288minus

1

16120596 +

1

288minus

3

16120596 +

23

288minus

3

16120596 +

23

288

minus3

16120596 +

23

288minus

3

16120596 +

23

288

1

2120596 +

23

72

1

2120596 +

23

72

minus1

16120596 +

1

288

(b) Edge rule

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

1

2120596 +

23

72

1

2120596 +

23

72

1

72

1

72

(c) Edge rule

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

minus19

384120596 +

23

2304

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

(d) Face rule

Figure 4 Schematic overview of the refinement rules for 119897 = 1 Solid circles show points at (119896 minus 1)th subdivision level where solid squaresshow the new points at level 119896

+ 112

112119891119896minus1119899119894minus1119895+2 +

512119891119896minus1119899119894119895+2 +

512119891119896minus1119899119894+1119895+2

+ 112119891119896minus1119899119894+2119895+2

(21)

where 119899 = 1 2 119897 By applying one time refining operator119877 and 119897-times smoothing operator 119878 we get the family ofbivariate primaldual subdivision schemes for quadrilateralmeshes with symbol

119887119897 (1199111 1199112) = (11991111199112)minus1minus119897minuslceil1198972rceil ( 1144 (1199112 + 1)

sdot (11991122 + 41199112 + 1) (1199111 + 1) (11991121 + 41199111 + 1))119897

sdot [ 116 9 + 2 (119897 + 8) 120596 11991111199112 +

116 3 (1 + 2120596119897) 119911311199112

+ 116 3 (1 + 2120596119897) 119911111991132 +

116 1 minus 2 (7119897 + 8) 120596

sdot 1199113111991132 + 116 3 (1 + 2120596119897) 1199112 +

116 9 + 2 (119897 + 8) 120596

sdot 119911211199112 + 116 1 minus 2 (7119897 + 8) 120596 11991132 +

116 3 (1 + 2120596119897)

sdot 1199112111991132 + 116 3 (1 + 2120596119897) 1199111 +

116 1 minus 2 (7119897 + 8) 120596

sdot 11991131 + 116 9 + 2 (119897 + 8) 120596 119911111991122 +

116 3 (1 + 2120596119897)

sdot 1199113111991122 + 116 1 minus 2 (7119897 + 8) 120596 +

116 3 (1 + 2120596119897) 11991121

+ 116 3 (1 + 2120596119897) 11991122 +

116 9 + 2 (119897 + 8) 120596 1199112111991122]

(22)

Remark 2 The bivariate family for 120596 = 0 correspondingto the symbol (22) reduces to the tensor product univariatefamily This tensor product family of schemes can also beobtained from the symbol defined by (16) If we apply onetime refining operator (20) for 119897 = 1 and one time smoothingoperator (21) we get the primal approximating subdivisionscheme with four refinement rules The schematic overviewof these refinement rules is given in Figure 4

Remark 3 For 1199111 = 1 and 1199112 = 119911 or 1199111 = 119911 and 1199112 = 1 ourfamily of bivariate schemes reduce to the family of univariateschemes with the symbol

119887119897 (119911) = 12119887119897 (1 119911) =

12119887119897 (119911 1) = 119911minus(119897+1)minuslceil1198972rceil

sdot (1 + 119911)119897+12119897+2 (1199112 + 4119911 + 16 )119897

[1 minus 2 (119897 + 2) 120596 1199112

+ 2 1 + 2 (119897 + 2) 120596 119911 + 1 minus 2 (119897 + 2) 120596]

(23)


This family of schemes is different from our proposedunivariate family of schemes

3 Analysis of Univariate andBivariate Schemes

This section deals with the analysis of proposed families ofschemes The analysis includes support width and smooth-nesscontinuity via Laurent polynomialmethodThe capacityof generating and reproducing polynomials has also beendiscussed mathematically

31 Support of Basic Limit Function Support of basic limitfunction of a subdivision scheme is equal to the number ofspans of the curve affected when one control point is movedfrom its initial place

A convergent subdivision scheme 119878119886119897 defines a basic limitfunction 120601119897 = 119878infin119886119897 120575119894 with 120575119894 being the initial data such that

120575119894 = 1 for 119894 = 00 otherwise

(24)

Support of basic limit function of our univariate family ofschemes depends on parameter 120596

(i) For 120596 = 0 isin Ω119897 the numbers of nonzero coefficientsin the symbol of refining operator (13) and smoothingoperator (15) are 6 and 4 respectively So by [18]support size of basic limit function of the schemes 119878119886119897for the general symbol 119886119897(119911) is 3119897 + 5 and its supportwidth is [minus(3119897 + 5)2 (3119897 + 5)2]

(ii) For 120596 = 0 the number of nonzero coefficients in thesymbol of both refining and smoothing stages is 4therefore support size of basic limit function of theschemes 119878119886119897 for the general symbol 119886119897(119911) is 3(119897+1) andits support width is [minus(32)(119897 + 1) (32)(119897 + 1)]

A convergent bivariate subdivision scheme 119878119887119897 defines abasic limit function 120601119897 = 119878infin119887119897 120575119894119895 where 120575119894119895 is the initial datasuch that

120575119894119895 = 1 for 119894 = 119895 = 00 otherwise

(25)

Support of basic limit of our bivariate family of schemes is3(119897+1)times3(119897+1) and its support width is [minus(32)(119897+1) (32)(119897+1)] times [minus(32)(119897 + 1) (32)(119897 + 1)]Basic limit functions for the first three family members

of univariate family for three different values of shapeparameters are presented in Figure 5 whereas basic limitfunctions for the first family member of bivariate family forthree different values of shape parameter are presented inFigure 6 From these figures we see that the support widthfor positive value is less than the support width for negativevalue of the parameter involved in the bivariate family

32 Continuity Analysis Here we present the smoothness ofproposed families of schemes by using the Laurent polyno-mial method For this purpose we have used the methodsthat are given ([3] Theorems 411 413 429 and 430)

Theorem 4 The subdivision scheme 1198781198861 is convergent for minus78 lt 120596 lt 1338 generates 1198621 limiting curve for minus310 lt 120596 lt16 and generates 1198623 limiting curve for 120596 = 0Proof Since the scheme with the symbol (119911(1 + 119911))1198861(119911) iscontractive for minus78 lt 120596 lt 1338 therefore the scheme 1198781198861is convergent for minus78 lt 120596 lt 1338 Similarly the schemewith the symbol 2(119911(1 + 119911))21198861(119911) is contractive for minus310 lt120596 lt 16 this implies that the scheme 1198781198861 is 1198621 continuous forminus310 lt 120596 lt 16Moreover the schemewith symbol 4(119911(1+119911))31198861(119911) is contractive for 120596 = 0 accordingly the scheme 1198781198861is 1198622 continuous for 120596 = 0 Furthermore the scheme withsymbol 6(119911(1 + 119911))41198861(119911) is contractive for 120596 = 0 hence thescheme 1198781198861 is 1198623 continuous for 120596 = 0Theorem 5 (i) The subdivision scheme 1198781198862 is convergent forminus53100 lt 120596 lt 726 generates 1198621 and 1198622 limiting curves forminus53100 lt 120596 lt 726 and minus1127 lt 120596 lt 19108 respectivelyand generates 1198624 limiting curve for 120596 = 0

(ii) The subdivision scheme 1198781198863 is convergent for minus17233322 lt 120596 lt 5851954 generates 1198621 1198622 and 1198623 limitingcurves for minus114271 lt 120596 lt 2691154 minus114271 lt 120596 lt 2691154 and minus1958 lt 120596 lt 61358 respectively and generates1198625 limiting curve for 120596 = 0

(iii) The subdivision scheme 1198781198864 is convergent for minus502111121 lt 120596 lt 20997808 generates119862111986221198623 and1198624 limitingcurves for minus502111121 lt 120596 lt 20997808 minus26357128 lt120596 lt 8543993 minus26357128 lt 120596 lt 8543993 and minus7032404 lt 120596 lt 3912384 respectively and generates 1198626 limitingcurve for 120596 = 0Proof Proof of this theorem is same as Theorem 4

Theorem 6 The subdivision scheme 1198781198871 is convergent forminus4136 lt 120596 lt 1121 generates 1198621 limiting surface for minus12 lt120596 lt 1945 and generates 1198623 limiting surface for 120596 = 0Proof Since the schemes with symbols (1199111(1 + 1199111))1198871(1199111 1199112)and (1199112(1 + 1199112))1198871(1199111 1199112) are contractive for minus4136 lt 120596 lt1121 therefore the scheme 1198781198871 is convergent for minus4136 lt120596 lt 1121 Similarly the schemes with symbols 2(1199111(1 +1199111))21198871(1199111 1199112) and 2(1199112(1+1199112))21198871(1199111 1199112) are contractive forminus12 lt 120596 lt 518 and the scheme with symbol 2(1199111(1 +1199111))(1199112(1 + 1199112))1198871(1199111 1199112) is contractive for minus1715 lt 120596 lt1945 this implies that the scheme 1198781198871 is 1198621 for minus12 lt120596 lt 1945 Moreover the schemes with symbols 4(1199111(1 +1199111))31198871(1199111 1199112) 4(1199112(1+1199112))31198871(1199111 1199112) 4(1199111(1+1199111))2(1199112(1+1199112))1198871(1199111 1199112) and 4(1199111(1 + 1199111))(1199112(1 + 1199112))21198871(1199111 1199112) arecontractive for 120596 = 0 accordingly the scheme 1198781198871 is 1198622 for120596 = 0 Furthermore the schemes with symbols 6(1199111(1 +1199111))41198871(1199111 1199112) 6(1199112(1+1199112))41198871(1199111 1199112) 6(1199111(1+1199111))3(1199112(1+1199112)) times 1198871(1199111 1199112) 6(1199111(1 + 1199111))(1199112(1 + 1199112))31198871(1199111 1199112) and


(a) 120596 = minus18 (b) 120596 = 0

(c) 120596 = 18

Figure 5 Basic limit functions of subdivision schemes 119878119886119897 for three values of shape parameter 120596 Cross symbols show control points whilesolid lines dashed dotted lines and dash lines represent curves obtained by subdivision schemes 1198781198861 1198781198862 and 1198781198863 respectively

6(1199111(1+1199111))2(1199112(1+1199112))21198871(1199111 1199112) are contractive for120596 = 0thus the scheme 1198781198871 is 1198623 for 120596 = 0Theorem 7 The subdivision scheme 1198781198872 is convergent forminus1407512444 lt 120596 lt 11911600 generates 1198621 and 1198622limiting surfaces for minus54634928 lt 120596 lt 8011388 andminus1116 lt 120596 lt 10892716 respectively and generates 1198624limiting surface for 120596 = 0Proof Proof is same as Theorem 6

When the initial control points are taken at the sameintervals from some polynomial and if the new data pointsafter applying subdivision scheme are located on the polyno-mial of same degree then we say that the subdivision schemegenerates polynomials up to that certain degree (degreeof polynomial) But if the new data points after applyingsubdivision scheme are located on the same polynomial thenwe say that the subdivision scheme reproduces polynomialsup to that certain degree See [16] for details We will discussthe polynomial generation and reproduction of our familiesin the upcoming subsections

33 Generation Degree Let Ω119897 and Ω1015840119897 be the range of 120596 forwhich the subdivision schemes 119878119886119897 and 119878119887119897 are convergentrespectively

Here we compute the polynomial generation of ourproposed family of schemes by using the conditions given in[1] (Conditions 1198851 and 119885119896)Theorem8 The subdivision scheme 119878119886119897 generates polynomials

(i) up to degree 119897 for 120596 isin Ω119897(ii) up to degree 119897 + 2 for 120596 = 0

Proof Since 119886119897(1) = 2 119886119897(minus1) = 0 and for 120596 isin Ω119897119863(119899)119886119897(119911)|119911=minus1 = 0 119899 = 1 2 3 119897 Then by [1] (Condition119885119896) 119878119886119897 generates polynomial up to degree 119897 For 120596 = 0119863(119899+2)119886119897(119911)|119911=minus1 = 0 119899 = 1 2 3 119897 Therefore again by[1] (Condition 119885119896) 119878119886119897 generates polynomial up to degree119897 + 2Theorem 9 The subdivision scheme 1198781198871 generates

(i) linear polynomials for 120596 isin Ω10158401(ii) cubic polynomials for 120596 = 0


minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2

(a) 120596 = minus13minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13

Figure 6 Basic limit functions of subdivision scheme 1198781198871 for three values of shape parameter 120596

Proof It is clear that

1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)

Let 1198991 1198992 isin N0 = N cup 0 and 120596 isin Ω10158401 so119863(1198991 1198992)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=minus1 = 0119863(1198991 1198992)1198871 (1199111 1199112)100381610038161003816100381610038161199111=minus11199112=1 = 0119863(11989911198992)1198871 (1199111 1199112)100381610038161003816100381610038161199111=minus11199112=minus1 = 0

(27)

where 1198991 +1198992 le 1 Therefore 1198781198871 generates linear polynomialsfor 120596 isin Ω10158401

Again let 120596 = 0 then119863(1198991 1198992)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=minus1 = 0119863(1198991 1198992)1198871 (1199111 1199112)100381610038161003816100381610038161199111=minus11199112=1 = 0119863(1198991 1198992)1198871 (1199111 1199112)100381610038161003816100381610038161199111=minus11199112=minus1 = 0

(28)

where 1198991 + 1198992 le 3 Therefore 1198781198871 generates cubic polynomialsfor 120596 = 0Theorem 10 The subdivision scheme 1198781198872 generates

(i) quadratic polynomials for 120596 isin Ω10158402(ii) quartic polynomials for 120596 = 0

Proof Proof is same as Theorem 9

34 Reproduction Degree Now we evaluate the polynomialreproduction of proposed family of schemes by using themethods given in [1] (Corollaries 27 and 23)


(a) Initial polygon (b) 120596 = minus110

(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110

Figure 7 In (b)ndash(f) solid lines dashed dotted lines and dash lines show the curves obtained by subdivision schemes 1198781198861 1198781198862 and 1198781198863 respectively

Theorem 11 The subdivision schemes 119878119886119897 reproduce polynomi-als as follows

(i) If 120596 isin Ω119897 119878119886119897 reproduces linear polynomials(ii) If 120596 = 23216 and 119897 = 2 119878119886119897 reproduces quadratic

polynomials(iii) If120596 = (9+7119897)(48+84119897) and 119897 gt 2 119878119886119897 reproduces cubic

polynomials

Proof For 119901 isin N the parametric value 120591 is given below

120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)

For 120596 isin Ω119897 we have 119863(1)119886119897(119911)|119911=1 = 2120591 Combining thiscondition withTheorem 8 we get the required result


510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13

Figure 8 (b)ndash(f) are limit surfaces obtained by applying one time our refining and one time smoothing operators for different values of shapeparameter 120596 after 3 iterations

Similarly for 120596 = 23216 and 119897 = 2 119878119886119897 reduces to thesubdivision scheme with mask

1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)

Since119863(2)1198862(119911)|119911=1 = minus36120596+103 then it is equal to 2120591(120591minus1)for 120596 = 23216 isin Ω2 Now by combining this condition withTheorem 8 we get the required result

Again for 120596 = (9 + 7119897)(48 + 84119897) isin Ω119897 and 119897 gt 2 we have119863(2)119886119897(119911)|119911=1 = 2120591(120591 minus 1) and119863(3)119886119897(119911)|119911=1 = 2120591(120591 minus 1)(120591 minus 2)Once more by combining this condition withTheorem 8 weget the result

Theorem 12 The subdivision scheme 1198781198871 reproduces linearpolynomials for 120596 isin Ω10158401Proof Since

1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)

Now by combing these results with Theorem 9 the requiredresult has been proved


510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12

Figure 9 (b)ndash(f) are limit surfaces obtained by applying one time our refining and two times our smoothing operators for different valuesof shape parameter 120596 after 3 iterations

Theorem 13 The subdivision scheme 1198781198872 reproduces linearpolynomials for 120596 isin Ω10158402 and quadratic polynomials for 120596 =2348Proof Since

1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)

Also for 120596 = 2348 we have

412059111205912 = 119863(11)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 141205911 (1205911 minus 1) = 119863(20)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = minus141205912 (1205912 minus 1) = 119863(02)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = minus1

(35)

Now by combing these results withTheorem 10 the requiredresult has been proved

Remark 14 If we take120596 = 0 the symbols (16) and (22) reduceto the symbol of ldquounivariate family of subdivision schemesrdquoand ldquobivariate family of subdivision schemesrdquoThese familiesproduce 119862119897+2 limiting curves and surfaces respectively Alsothese families generate polynomials up to degree 119897 + 2 whilepolynomial reproduction of these families is linear


1

2 3

4

5

67

8

(a) Proposed uniform scheme 1198781198862 120596 = 181

2 3

4

5

67

8

(b) Uniform Romanirsquos scheme 1198781198862120596 120596 =18

1

2 3

4

5

67

8

(c) Nonuniform proposed scheme 1198781198862 120596119894 = [132 18 18 132 132 18 18132]

1

2 3

4

5

67

8

(d) Nonuniform Romanirsquos scheme 1198781198862120596119894 120596119894 = [132 18 18 132 132 18 18132]

Figure 10 Solid lines represent initial polygons while dash lines represent limit curves obtained by subdivision schemes proposed ((a) and(c)) by us and ((b) and (d)) by Romani

4 Applications and Comparisons

In this section we have shown the capacity of parameter 120596 tocontrol shapes in different ways The optical presentation ofour proposed families of univariate and bivariate (nontensorproduct for quadrilateral meshes) subdivision schemes alsohas been shown In Figure 7 we have shown limit curves thatare obtained by our proposed family of univariate subdivisionschemes at five values of shape parameter In Figures 8 and 9we have also shown limit surfacesThese surfaces are obtainedby 1198781198871 (ie scheme with one time smoothing operator) and1198781198872 (ie scheme with two times smoothing operator) onquadrilateral meshes respectively From these figures weobserve that shapes of limit curvessurfaces are affected byshape parameter and number of applied smoothing stagesboth

Tables 1 and 2 show comparison between our univariateand bivariate families with Romanirsquos [8] univariate andbivariate families respectively From these tables it is clearthat our schemes give better smoothness than Romanirsquosschemes for 120596 = 0 however in this case the support size islarger than Romanirsquos schemes and the reproduction degree isonly one

The geometrical construction of our refining opera-tor allows us to extend our subdivision schemes to the

nonuniform setting according to the geometry of the initialpolygonmesh by defining local parameter 120596119894 Sometimeschoosing a variable shape parameter 120596119894 for each edge of theinitial polygonmesh gives better result than choosing a con-stant shape parameter 120596 for an entire initial polygonmeshFigures 10(a) and 10(b) demonstrate the visual effects of theproposed scheme 1198781198862 and the scheme proposed by Romani1198781198862120596 respectively for 120596 = 18 Similarly Figures 11(a) and11(b) display the visual performance of the proposed scheme1198781198872 and Romanirsquos scheme 1198781198872120596 respectively for 120596 = 18 whileFigures 10(c) and 11(c) show the performance of nonuniformsetting of the schemes 1198781198862 and 1198781198872 respectively Also Figures10(d) and 11(d) represent the performance of nonuniformsetting of Romanirsquos schemes 1198781198862120596119894 and 1198781198872120596119894 respectivelyThese figures show that the limit curvessurfaces generated bynonuniform schemes are better than the limit curvessurfacesobtained from uniform schemes

5 Conclusion

In this paper we have presented univariate and bivariatefamilies of approximating subdivision schemes dependingon two parameters These families are called the Refining-Smooth families where refining operator is the operator in


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5

(b) Romanirsquos uniform scheme 1198781198872120596 120596 = 18

minus2

0

2

4

6

8

10

12

14

minus10123456780

5

(c) Nonuniform proposed scheme 1198781198872 120596119894 = 18 at blue initial pointsand 120596119894 = 0 at red initial points

minus202468101214

minus10123456780

5

(d) NonuniformRomanirsquos scheme 1198781198872120596119894 120596119894 = 18 at blue initial pointsand 120596119894 = 0 at red initial points

Figure 11 Solid red lines represent initial mesh while red and blue bullets show initial points

Table 1 Comparison between our univariate family 119878119886119897 and Romanirsquos univariate family 119878119886119897120596 via different properties Let S SS RD and GDbe the integer smoothness support size reproduction degree and generation degree respectively

RS-schemes S SS RD GD119878119886119897 for 120596 isin Ω119897 119862119897 3119897 + 5 1 119897119878119886119897 for 120596 = 0 isin Ω119897 119862119897+2 3119897 + 3 1 119897 + 21198781198862 for 120596 = 23216 isin Ω2 1198622 11 2 2119878119886119897 for 120596 = (9 + 7119897)(48 + 84119897) isin Ω119897 and 119897 gt 2 119862119897 3119897 + 5 3 119897119878119886119897120596 for 120596 isin Ω119897 119862119897 119897 + 5 1 119897119878119886119897120596 for 120596 = 116 119862119897 119897 + 5 3 119897 + 2

which each refining point 119891119896minus11119894 moves around correspond-ing Chaikin point119862119896minus11119894 in curve case and each refining point119891119896minus11119894119895 moves around corresponding Doo-Sabin point 119862119896minus11119894119895in surface case for different values of shape parameter 120596 isinR and smoothing parameter 119897 isin N while each smoothingoperator is the weighted average of the four adjacent points119891119896minus1119899119894 119894 isin Z in curve case and each smoothing operator is

the weighted average of sixteen adjacent neighboring points119891119896minus1119899119894119895 119894 119895 isin Z in surface case where 119899 = 1 2 119897We also have analyzed some basic properties of our

proposed families Our families of schemes give cubic poly-nomial reproduction for specific values of shape parameter120596 Polynomial generation and continuity of our schemes are119897 for specific range of parameter 120596 while for 120596 = 0 they are119897 + 2 However for 120596 = 0 our schemes reproduce only linear


Table 2 Comparison between our bivariate family 119878119887119897 and Romanirsquosbivariate family 119878119887119897120596 via different properties Let S SS RD and GDbe the integer smoothness support size reproduction degree andgeneration degree respectively

RS-schemes with 119897 le 2 S SS RD GD119878119887119897 for 120596 isin Ω1015840119897 119862119897 (3119897 + 3) times (3119897 + 3) 1 119897119878119887119897 for 120596 = 0 isin Ω1015840119897 119862119897+2 (3119897 + 3) times (3119897 + 3) 1 119897 + 21198781198872 for 120596 = 2348 isin Ω10158402 1198621 9 times 9 2 2119878119887119897120596 for 120596 isin Ω1015840119897 119862119897 (119897 + 5) times (119897 + 5) 1 119897119878119887119897120596 for 120596 = 116 119862119897 (119897 + 5) times (119897 + 5) 3 119897 + 2

polynomials Nonuniform setting of the proposed familymembers gives better performance than that of uniformsetting of the schemes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by NRPU of HEC Pakistan (P no3183)

References

[1] M Charina and C Conti ldquoPolynomial reproduction of multi-variate scalar subdivision schemesrdquo Journal of Computationaland Applied Mathematics vol 240 pp 51ndash61 2013

[2] N Dyn ldquoInterpolatory subdivision schemes and analysis ofconvergence and smoothness by the formalism of Laurent poly-nomialsrdquo in Tutorials onMultiresolution in GeometricModelingA Iske E Quak and M S Floater Eds pp 51ndash68 SpringerBerlin Germany 2002

[3] N Dyn and D Levin ldquoSubdivision schemes in geometric mod-ellingrdquo Acta Numerica vol 11 pp 73ndash144 2002

[4] E Catmull and J Clark ldquoRecursively generated B-spline sur-faces on arbitrary topological meshesrdquo Computer-Aided Designvol 10 no 6 pp 350ndash355 1978

[5] D Doo andM Sabin ldquoBehaviour of recursive division surfacesnear extraordinary pointsrdquo Computer-Aided Design vol 10 no6 pp 356ndash360 1978

[6] M-E Fang W Ma and G Wang ldquoA generalized surfacesubdivision schemeof arbitrary orderwith a tensionparameterrdquoComputer-Aided Design vol 49 pp 8ndash17 2014

[7] J M Lane and R F Riesenfeld ldquoA theoretical development forthe computer generation and display of piecewise polynomialsurfacesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 2 no 1 pp 35ndash46 1980

[8] L Romani ldquoA Chaikin-based variant of Lane-Riesenfeld algo-rithm and its non-tensor product extensionrdquo Computer AidedGeometric Design vol 32 pp 22ndash49 2015

[9] J Stam ldquoOn subdivision schemes generalizing uniformB-splinesurfaces of arbitrary degreerdquoComputer AidedGeometric Designvol 18 no 5 pp 383ndash396 2001

[10] D Zorin and P Schroder ldquoA unified framework for primaldualquadrilateral subdivision schemesrdquo Computer Aided GeometricDesign vol 18 no 5 pp 429ndash454 2001

[11] P Ashraf G Mustafa and J Deng ldquoA six-point variant onthe Lane-Riesenfeld algorithmrdquo Journal of AppliedMathematicsvol 2014 Article ID 628285 7 pages 2014

[12] T J Cashman K Hormann and U Reif ldquoGeneralized Lane-Riesenfeld algorithmsrdquo Computer Aided Geometric Design vol30 no 4 pp 398ndash409 2013

[13] G Mustafa P Ashraf and M Aslam ldquoBinary univariate dualand primal subdivision schemesrdquo SEMA Journal vol 65 no 1pp 23ndash35 2014

[14] G Deslauriers and S Dubuc ldquoSymmetric iterative interpolationprocessesrdquo Constructive Approximation vol 5 no 1 pp 49ndash681989

[15] N Dyn D Levin and J A Gregory ldquoA 4 -point interpolatorysubdivision scheme for curve designrdquoComputer Aided Geomet-ric Design vol 4 no 4 pp 257ndash268 1987

[16] K Hormann andM A Sabin ldquoA family of subdivision schemeswith cubic precisionrdquo Computer Aided Geometric Design vol25 no 1 pp 41ndash52 2008

[17] GM Chaikin ldquoAn algorithm for high-speed curve generationrdquoComputer Graphics and Image Processing vol 3 no 4 pp 346ndash349 1974

[18] I P Ivrissimtzis M A Sabin and N A Dodgson ldquoOn thesupport of recursive subdivisionrdquoACM Transactions on Graph-ics vol 23 no 4 pp 1043ndash1060 2004

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


dkminus112i+1

dkminus112i

fkminus1i


iminus1


i+1


i+1


iminus1

(a) Our refining stage

fkminus1i

dkminus112i+1

dkminus112i

1

2(fkminus1

i minus fkminus1iminus1 )

1

2(f kminus1

i minus fkminus1i+1 )

1

2(Ckminus11

2i+1 minus Ckminus112i )

1

2(Ckminus11

2i minus Ckminus112i+1 )

(b) Romanirsquos refining stage

Figure 1 The geometrical difference between our refining stage and Romanirsquos refining stage

stencil of linear B-spline scheme as refining and smoothingoperators respectively Therefore the symbol of 119899th familymember of Lane-Riesenfeld schemes gives symbol of (119899 + 1)-degree B-spline schemeTheir family gives119862119899 continuity andlinear polynomial reproduction

In literature there are few variants which are appliedon the Lane-Riesenfeld algorithm For example Cashmanet al [12] also proposed a family of univariate subdivisionschemes by using the RS-algorithm which is based onDubuc Deslauriers 4-point interpolatory scheme [14] Theycombined the symbols of 4-point interpolatory subdivisionscheme and odd stencil of 4-point interpolatory subdivisionscheme as refining and smoothing operators respectivelyTheir univariate family gives cubic polynomial reproductionbut by increasing smoothing stages continuity of their familymay or may not be increased

Ashraf et al [11] proposed a family of univariate subdivi-sion schemes by using similar technique to that used byCash-man et al [12] on Dubuc Deslauriers 6-point interpolatoryscheme [14]They also combined the symbol and odd stencilrsquossymbol of Dubuc Deslauriers 6-point interpolatory scheme[14] as refining and smoothing operators respectively Theirunivariate family gives quintic polynomial reproduction butlevel of continuity of their family does not increase in generalby increasing the smoothing stages

Mustafa et al [13] also proposed a family of univariatesubdivision schemes by using the symbol of 4-point inter-polatory subdivision scheme [15] as the refining operatorand the symbol of even stencil of the 4-point approximatingscheme [16] as the smoothing operator Their univariatefamily gives cubic polynomial reproduction but level ofcontinuity of their family also does not increase in generalby increasing the smoothing stages

Romani [8] proposed families of univariate and bivariatesubdivision schemes by using RS-algorithm in which therefining operator is based on a perturbation of Chaikinrsquos

corner cutting subdivision scheme [17] and the smoothingoperator takes the average of two adjacent vertices as in Lane-Riesenfeld algorithm Their univariate family gives linearpolynomial reproduction and continuity of their 119899th familymember is 119862119899 Moreover if 120596 = 116 their 119899th familymember becomes (119899 + 3)th member of Hormann and Sabinrsquos[16] family and it gives cubic polynomial reproduction

There are few variants of Lane-Riesenfeld algorithm butall those algorithms started with two binary schemes bytaking the symbol of first scheme as the refining operator andthe odd or even symbol of second scheme as the smoothingoperator Our approach is different in the sense that we donot use symbol of any existing subdivision scheme as refiningoperator and also we choose appropriate smoothing operatorby ourselves

In this paper we propose a new RS-algorithm which isalso a variant of Lane-Riesenfeld algorithm that comparedwith the one proposed by Romani we modify refiningand smoothing operators both There are two main differ-ences between our RS-algorithm andRomanirsquos RS-algorithmwhich are as follows

(i) Both refining operators depend on Chaikin pointscomputed around initial points but the differencebetween our refining operator and Romanirsquos refiningoperator is that Romani draws two vectors at everyinitial control point and changes position of bothChaikin points which have been computed aroundcorresponding initial point by using these two vec-tors while we draw one vector at every Chaikinpoint and change the position of each Chaikin pointby using each corresponding vector which we havedrawn on it (see Figure 1)

(ii) Difference between both smoothing operators is thatRomanirsquos smoothing operator takes average of two







119891119896 = 119878119886119897119891119896minus1 (3)

where



119891119896minus11120572 = sum120573isinZ119898



119891119896minus1119899+1120572 = sum120573isinZ119898

119904120573119891119896minus1119899120573 120572 isin Z119898 119899 = 1 2 119897 (6)


119903 (119911) = sum120572isinZ119898

119903120572119911120572119904 (119911) = sum

120572isinZ119898119904120572119911120572

119911 isin (C 0)119898 (7)

with

119903 (1) = 2119898119904 (1) = 1


(8)







(9)




(10)




(11)


fk2i

1

48minus

19

24120596

23

48+

19

24120596

23

48+

19

24120596

1

48minus

19

24120596

(a) Edge rule

minus1

6120596

1

6minus

1

2120596

2

3+

4

3120596

1

6minus

1

2120596 minus

1

6120596

fk2i+1

(b) Vertex rule




(12)



(13)

where

1205721 = 14 (1 + 2120596 (2 minus 119897))

1205722 = 34 (1 + 2120596119897)

1205723 = minus120596 (119897 + 1) (14)



12119891119896minus1119899119894 + 512119891119896minus1119899119894+1

+ 112119891119896minus1119899119894+2

(15)

where 119899 = 1 2 119897



119897


+ 34 (1 + 2120596119897) 1199112 +

34 (1 + 2120596119897) 1199113


(16)







4119897+1 (17)





fkminus1i+1jminus

fkminus1ij

Dkminus112i2j

Dkminus112i+12j

minus fkminus1ij+1


2i2j+1


i+1j+1

fkminus1i+1j

Ckminus112i+12j+1


ij+1


i+1j+1Ckminus11

2i+12j


i+1j


ij


ijCkminus112i2j+1








(18)




(19)





(20)

where 1205731 = (116)9 + 2(119897 + 8)120596 1205732 = (116)3(2120596119897 + 1)and 1205733 = (116)1 minus 2(7119897 + 8)120596


119891119896minus1119899+1119894119895 = 112


512119891119896minus1119899119894119895minus1 +

512119891119896minus1119899119894+1119895minus1

+ 112119891119896minus1119899119894+2119895minus1 +

512

112119891119896minus1119899119894minus1119895 +

512119891119896minus1119899119894119895

+ 512119891119896minus1119899119894+1119895 +

112119891119896minus1119899119894+2119895 +

512

112119891119896minus1119899119894minus1119895+1

+ 512119891119896minus1119899119894119895+1 +

512119891119896minus1119899119894+1119895+1 +

112119891119896minus1119899119894+2119895+1


minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

288

minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

2881

12120596 +

1

9

1

12120596 +

1

9

5

6120596 +

4

9

(a) Vertex rule

minus1

16120596 +

1

288

1

72

1

72

minus1

16120596 +

1

288minus

1

16120596 +

1

288minus

3

16120596 +

23

288minus

3

16120596 +

23

288

minus3

16120596 +

23

288minus

3

16120596 +

23

288

1

2120596 +

23

72

1

2120596 +

23

72

minus1

16120596 +

1

288

(b) Edge rule

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

1

2120596 +

23

72

1

2120596 +

23

72

1

72

1

72

(c) Edge rule

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

minus19

384120596 +

23

2304

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

(d) Face rule


+ 112

112119891119896minus1119899119894minus1119895+2 +

512119891119896minus1119899119894119895+2 +

512119891119896minus1119899119894+1119895+2

+ 112119891119896minus1119899119894+2119895+2

(21)



sdot (11991122 + 41199112 + 1) (1199111 + 1) (11991121 + 41199111 + 1))119897

sdot [ 116 9 + 2 (119897 + 8) 120596 11991111199112 +

116 3 (1 + 2120596119897) 119911311199112

+ 116 3 (1 + 2120596119897) 119911111991132 +

116 1 minus 2 (7119897 + 8) 120596

sdot 1199113111991132 + 116 3 (1 + 2120596119897) 1199112 +

116 9 + 2 (119897 + 8) 120596

sdot 119911211199112 + 116 1 minus 2 (7119897 + 8) 120596 11991132 +

116 3 (1 + 2120596119897)

sdot 1199112111991132 + 116 3 (1 + 2120596119897) 1199111 +

116 1 minus 2 (7119897 + 8) 120596

sdot 11991131 + 116 9 + 2 (119897 + 8) 120596 119911111991122 +

116 3 (1 + 2120596119897)

sdot 1199113111991122 + 116 1 minus 2 (7119897 + 8) 120596 +

116 3 (1 + 2120596119897) 11991121

+ 116 3 (1 + 2120596119897) 11991122 +

116 9 + 2 (119897 + 8) 120596 1199112111991122]

(22)



119887119897 (119911) = 12119887119897 (1 119911) =


sdot (1 + 119911)119897+12119897+2 (1199112 + 4119911 + 16 )119897

[1 minus 2 (119897 + 2) 120596 1199112

+ 2 1 + 2 (119897 + 2) 120596 119911 + 1 minus 2 (119897 + 2) 120596]

(23)







120575119894 = 1 for 119894 = 00 otherwise

(24)





120575119894119895 = 1 for 119894 = 119895 = 00 otherwise

(25)









(a) 120596 = minus18 (b) 120596 = 0

(c) 120596 = 18










minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2


0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13



1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)


(27)



(28)







(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119891119896 = 119878119886119897119891119896minus1 (3)

where



119891119896minus11120572 = sum120573isinZ119898



119891119896minus1119899+1120572 = sum120573isinZ119898

119904120573119891119896minus1119899120573 120572 isin Z119898 119899 = 1 2 119897 (6)


119903 (119911) = sum120572isinZ119898

119903120572119911120572119904 (119911) = sum

120572isinZ119898119904120572119911120572

119911 isin (C 0)119898 (7)

with

119903 (1) = 2119898119904 (1) = 1


(8)







(9)




(10)




(11)


fk2i

1

48minus

19

24120596

23

48+

19

24120596

23

48+

19

24120596

1

48minus

19

24120596

(a) Edge rule

minus1

6120596

1

6minus

1

2120596

2

3+

4

3120596

1

6minus

1

2120596 minus

1

6120596

fk2i+1

(b) Vertex rule




(12)



(13)

where

1205721 = 14 (1 + 2120596 (2 minus 119897))

1205722 = 34 (1 + 2120596119897)

1205723 = minus120596 (119897 + 1) (14)



12119891119896minus1119899119894 + 512119891119896minus1119899119894+1

+ 112119891119896minus1119899119894+2

(15)

where 119899 = 1 2 119897



119897


+ 34 (1 + 2120596119897) 1199112 +

34 (1 + 2120596119897) 1199113


(16)







4119897+1 (17)





fkminus1i+1jminus

fkminus1ij

Dkminus112i2j

Dkminus112i+12j

minus fkminus1ij+1


2i2j+1


i+1j+1

fkminus1i+1j

Ckminus112i+12j+1


ij+1


i+1j+1Ckminus11

2i+12j


i+1j


ij


ijCkminus112i2j+1








(18)




(19)





(20)

where 1205731 = (116)9 + 2(119897 + 8)120596 1205732 = (116)3(2120596119897 + 1)and 1205733 = (116)1 minus 2(7119897 + 8)120596


119891119896minus1119899+1119894119895 = 112


512119891119896minus1119899119894119895minus1 +

512119891119896minus1119899119894+1119895minus1

+ 112119891119896minus1119899119894+2119895minus1 +

512

112119891119896minus1119899119894minus1119895 +

512119891119896minus1119899119894119895

+ 512119891119896minus1119899119894+1119895 +

112119891119896minus1119899119894+2119895 +

512

112119891119896minus1119899119894minus1119895+1

+ 512119891119896minus1119899119894119895+1 +

512119891119896minus1119899119894+1119895+1 +

112119891119896minus1119899119894+2119895+1


minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

288

minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

2881

12120596 +

1

9

1

12120596 +

1

9

5

6120596 +

4

9

(a) Vertex rule

minus1

16120596 +

1

288

1

72

1

72

minus1

16120596 +

1

288minus

1

16120596 +

1

288minus

3

16120596 +

23

288minus

3

16120596 +

23

288

minus3

16120596 +

23

288minus

3

16120596 +

23

288

1

2120596 +

23

72

1

2120596 +

23

72

minus1

16120596 +

1

288

(b) Edge rule

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

1

2120596 +

23

72

1

2120596 +

23

72

1

72

1

72

(c) Edge rule

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

minus19

384120596 +

23

2304

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

(d) Face rule


+ 112

112119891119896minus1119899119894minus1119895+2 +

512119891119896minus1119899119894119895+2 +

512119891119896minus1119899119894+1119895+2

+ 112119891119896minus1119899119894+2119895+2

(21)



sdot (11991122 + 41199112 + 1) (1199111 + 1) (11991121 + 41199111 + 1))119897

sdot [ 116 9 + 2 (119897 + 8) 120596 11991111199112 +

116 3 (1 + 2120596119897) 119911311199112

+ 116 3 (1 + 2120596119897) 119911111991132 +

116 1 minus 2 (7119897 + 8) 120596

sdot 1199113111991132 + 116 3 (1 + 2120596119897) 1199112 +

116 9 + 2 (119897 + 8) 120596

sdot 119911211199112 + 116 1 minus 2 (7119897 + 8) 120596 11991132 +

116 3 (1 + 2120596119897)

sdot 1199112111991132 + 116 3 (1 + 2120596119897) 1199111 +

116 1 minus 2 (7119897 + 8) 120596

sdot 11991131 + 116 9 + 2 (119897 + 8) 120596 119911111991122 +

116 3 (1 + 2120596119897)

sdot 1199113111991122 + 116 1 minus 2 (7119897 + 8) 120596 +

116 3 (1 + 2120596119897) 11991121

+ 116 3 (1 + 2120596119897) 11991122 +

116 9 + 2 (119897 + 8) 120596 1199112111991122]

(22)



119887119897 (119911) = 12119887119897 (1 119911) =


sdot (1 + 119911)119897+12119897+2 (1199112 + 4119911 + 16 )119897

[1 minus 2 (119897 + 2) 120596 1199112

+ 2 1 + 2 (119897 + 2) 120596 119911 + 1 minus 2 (119897 + 2) 120596]

(23)







120575119894 = 1 for 119894 = 00 otherwise

(24)





120575119894119895 = 1 for 119894 = 119895 = 00 otherwise

(25)









(a) 120596 = minus18 (b) 120596 = 0

(c) 120596 = 18










minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2


0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13



1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)


(27)



(28)







(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










fk2i

1

48minus

19

24120596

23

48+

19

24120596

23

48+

19

24120596

1

48minus

19

24120596

(a) Edge rule

minus1

6120596

1

6minus

1

2120596

2

3+

4

3120596

1

6minus

1

2120596 minus

1

6120596

fk2i+1

(b) Vertex rule




(12)



(13)

where

1205721 = 14 (1 + 2120596 (2 minus 119897))

1205722 = 34 (1 + 2120596119897)

1205723 = minus120596 (119897 + 1) (14)



12119891119896minus1119899119894 + 512119891119896minus1119899119894+1

+ 112119891119896minus1119899119894+2

(15)

where 119899 = 1 2 119897



119897


+ 34 (1 + 2120596119897) 1199112 +

34 (1 + 2120596119897) 1199113


(16)







4119897+1 (17)





fkminus1i+1jminus

fkminus1ij

Dkminus112i2j

Dkminus112i+12j

minus fkminus1ij+1


2i2j+1


i+1j+1

fkminus1i+1j

Ckminus112i+12j+1


ij+1


i+1j+1Ckminus11

2i+12j


i+1j


ij


ijCkminus112i2j+1








(18)




(19)





(20)

where 1205731 = (116)9 + 2(119897 + 8)120596 1205732 = (116)3(2120596119897 + 1)and 1205733 = (116)1 minus 2(7119897 + 8)120596


119891119896minus1119899+1119894119895 = 112


512119891119896minus1119899119894119895minus1 +

512119891119896minus1119899119894+1119895minus1

+ 112119891119896minus1119899119894+2119895minus1 +

512

112119891119896minus1119899119894minus1119895 +

512119891119896minus1119899119894119895

+ 512119891119896minus1119899119894+1119895 +

112119891119896minus1119899119894+2119895 +

512

112119891119896minus1119899119894minus1119895+1

+ 512119891119896minus1119899119894119895+1 +

512119891119896minus1119899119894+1119895+1 +

112119891119896minus1119899119894+2119895+1


minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

288

minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

2881

12120596 +

1

9

1

12120596 +

1

9

5

6120596 +

4

9

(a) Vertex rule

minus1

16120596 +

1

288

1

72

1

72

minus1

16120596 +

1

288minus

1

16120596 +

1

288minus

3

16120596 +

23

288minus

3

16120596 +

23

288

minus3

16120596 +

23

288minus

3

16120596 +

23

288

1

2120596 +

23

72

1

2120596 +

23

72

minus1

16120596 +

1

288

(b) Edge rule

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

1

2120596 +

23

72

1

2120596 +

23

72

1

72

1

72

(c) Edge rule

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

minus19

384120596 +

23

2304

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

(d) Face rule


+ 112

112119891119896minus1119899119894minus1119895+2 +

512119891119896minus1119899119894119895+2 +

512119891119896minus1119899119894+1119895+2

+ 112119891119896minus1119899119894+2119895+2

(21)



sdot (11991122 + 41199112 + 1) (1199111 + 1) (11991121 + 41199111 + 1))119897

sdot [ 116 9 + 2 (119897 + 8) 120596 11991111199112 +

116 3 (1 + 2120596119897) 119911311199112

+ 116 3 (1 + 2120596119897) 119911111991132 +

116 1 minus 2 (7119897 + 8) 120596

sdot 1199113111991132 + 116 3 (1 + 2120596119897) 1199112 +

116 9 + 2 (119897 + 8) 120596

sdot 119911211199112 + 116 1 minus 2 (7119897 + 8) 120596 11991132 +

116 3 (1 + 2120596119897)

sdot 1199112111991132 + 116 3 (1 + 2120596119897) 1199111 +

116 1 minus 2 (7119897 + 8) 120596

sdot 11991131 + 116 9 + 2 (119897 + 8) 120596 119911111991122 +

116 3 (1 + 2120596119897)

sdot 1199113111991122 + 116 1 minus 2 (7119897 + 8) 120596 +

116 3 (1 + 2120596119897) 11991121

+ 116 3 (1 + 2120596119897) 11991122 +

116 9 + 2 (119897 + 8) 120596 1199112111991122]

(22)



119887119897 (119911) = 12119887119897 (1 119911) =


sdot (1 + 119911)119897+12119897+2 (1199112 + 4119911 + 16 )119897

[1 minus 2 (119897 + 2) 120596 1199112

+ 2 1 + 2 (119897 + 2) 120596 119911 + 1 minus 2 (119897 + 2) 120596]

(23)







120575119894 = 1 for 119894 = 00 otherwise

(24)





120575119894119895 = 1 for 119894 = 119895 = 00 otherwise

(25)









(a) 120596 = minus18 (b) 120596 = 0

(c) 120596 = 18










minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2


0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13



1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)


(27)



(28)







(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











fkminus1i+1jminus

fkminus1ij

Dkminus112i2j

Dkminus112i+12j

minus fkminus1ij+1


2i2j+1


i+1j+1

fkminus1i+1j

Ckminus112i+12j+1


ij+1


i+1j+1Ckminus11

2i+12j


i+1j


ij


ijCkminus112i2j+1








(18)




(19)





(20)

where 1205731 = (116)9 + 2(119897 + 8)120596 1205732 = (116)3(2120596119897 + 1)and 1205733 = (116)1 minus 2(7119897 + 8)120596


119891119896minus1119899+1119894119895 = 112


512119891119896minus1119899119894119895minus1 +

512119891119896minus1119899119894+1119895minus1

+ 112119891119896minus1119899119894+2119895minus1 +

512

112119891119896minus1119899119894minus1119895 +

512119891119896minus1119899119894119895

+ 512119891119896minus1119899119894+1119895 +

112119891119896minus1119899119894+2119895 +

512

112119891119896minus1119899119894minus1119895+1

+ 512119891119896minus1119899119894119895+1 +

512119891119896minus1119899119894+1119895+1 +

112119891119896minus1119899119894+2119895+1


minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

288

minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

2881

12120596 +

1

9

1

12120596 +

1

9

5

6120596 +

4

9

(a) Vertex rule

minus1

16120596 +

1

288

1

72

1

72

minus1

16120596 +

1

288minus

1

16120596 +

1

288minus

3

16120596 +

23

288minus

3

16120596 +

23

288

minus3

16120596 +

23

288minus

3

16120596 +

23

288

1

2120596 +

23

72

1

2120596 +

23

72

minus1

16120596 +

1

288

(b) Edge rule

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

1

2120596 +

23

72

1

2120596 +

23

72

1

72

1

72

(c) Edge rule

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

minus19

384120596 +

23

2304

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

(d) Face rule


+ 112

112119891119896minus1119899119894minus1119895+2 +

512119891119896minus1119899119894119895+2 +

512119891119896minus1119899119894+1119895+2

+ 112119891119896minus1119899119894+2119895+2

(21)



sdot (11991122 + 41199112 + 1) (1199111 + 1) (11991121 + 41199111 + 1))119897

sdot [ 116 9 + 2 (119897 + 8) 120596 11991111199112 +

116 3 (1 + 2120596119897) 119911311199112

+ 116 3 (1 + 2120596119897) 119911111991132 +

116 1 minus 2 (7119897 + 8) 120596

sdot 1199113111991132 + 116 3 (1 + 2120596119897) 1199112 +

116 9 + 2 (119897 + 8) 120596

sdot 119911211199112 + 116 1 minus 2 (7119897 + 8) 120596 11991132 +

116 3 (1 + 2120596119897)

sdot 1199112111991132 + 116 3 (1 + 2120596119897) 1199111 +

116 1 minus 2 (7119897 + 8) 120596

sdot 11991131 + 116 9 + 2 (119897 + 8) 120596 119911111991122 +

116 3 (1 + 2120596119897)

sdot 1199113111991122 + 116 1 minus 2 (7119897 + 8) 120596 +

116 3 (1 + 2120596119897) 11991121

+ 116 3 (1 + 2120596119897) 11991122 +

116 9 + 2 (119897 + 8) 120596 1199112111991122]

(22)



119887119897 (119911) = 12119887119897 (1 119911) =


sdot (1 + 119911)119897+12119897+2 (1199112 + 4119911 + 16 )119897

[1 minus 2 (119897 + 2) 120596 1199112

+ 2 1 + 2 (119897 + 2) 120596 119911 + 1 minus 2 (119897 + 2) 120596]

(23)







120575119894 = 1 for 119894 = 00 otherwise

(24)





120575119894119895 = 1 for 119894 = 119895 = 00 otherwise

(25)









(a) 120596 = minus18 (b) 120596 = 0

(c) 120596 = 18










minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2


0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13



1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)


(27)



(28)







(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

288

minus7

24120596 +

1

288

1

12120596 +

1

9

minus7

24120596 +

1

2881

12120596 +

1

9

1

12120596 +

1

9

5

6120596 +

4

9

(a) Vertex rule

minus1

16120596 +

1

288

1

72

1

72

minus1

16120596 +

1

288minus

1

16120596 +

1

288minus

3

16120596 +

23

288minus

3

16120596 +

23

288

minus3

16120596 +

23

288minus

3

16120596 +

23

288

1

2120596 +

23

72

1

2120596 +

23

72

minus1

16120596 +

1

288

(b) Edge rule

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

minus3

16120596 +

23

288

minus3

16120596 +

23

288

minus1

16120596 +

1

288

minus1

16120596 +

1

288

1

2120596 +

23

72

1

2120596 +

23

72

1

72

1

72

(c) Edge rule

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

43

384120596 +

529

2304

minus19

384120596 +

23

2304

minus5

384120596 +

1

2304

minus5

384120596 +

1

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

minus19

384120596 +

23

2304minus

19

384120596 +

23

2304

(d) Face rule


+ 112

112119891119896minus1119899119894minus1119895+2 +

512119891119896minus1119899119894119895+2 +

512119891119896minus1119899119894+1119895+2

+ 112119891119896minus1119899119894+2119895+2

(21)



sdot (11991122 + 41199112 + 1) (1199111 + 1) (11991121 + 41199111 + 1))119897

sdot [ 116 9 + 2 (119897 + 8) 120596 11991111199112 +

116 3 (1 + 2120596119897) 119911311199112

+ 116 3 (1 + 2120596119897) 119911111991132 +

116 1 minus 2 (7119897 + 8) 120596

sdot 1199113111991132 + 116 3 (1 + 2120596119897) 1199112 +

116 9 + 2 (119897 + 8) 120596

sdot 119911211199112 + 116 1 minus 2 (7119897 + 8) 120596 11991132 +

116 3 (1 + 2120596119897)

sdot 1199112111991132 + 116 3 (1 + 2120596119897) 1199111 +

116 1 minus 2 (7119897 + 8) 120596

sdot 11991131 + 116 9 + 2 (119897 + 8) 120596 119911111991122 +

116 3 (1 + 2120596119897)

sdot 1199113111991122 + 116 1 minus 2 (7119897 + 8) 120596 +

116 3 (1 + 2120596119897) 11991121

+ 116 3 (1 + 2120596119897) 11991122 +

116 9 + 2 (119897 + 8) 120596 1199112111991122]

(22)



119887119897 (119911) = 12119887119897 (1 119911) =


sdot (1 + 119911)119897+12119897+2 (1199112 + 4119911 + 16 )119897

[1 minus 2 (119897 + 2) 120596 1199112

+ 2 1 + 2 (119897 + 2) 120596 119911 + 1 minus 2 (119897 + 2) 120596]

(23)







120575119894 = 1 for 119894 = 00 otherwise

(24)





120575119894119895 = 1 for 119894 = 119895 = 00 otherwise

(25)









(a) 120596 = minus18 (b) 120596 = 0

(c) 120596 = 18










minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2


0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13



1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)


(27)



(28)







(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















120575119894 = 1 for 119894 = 00 otherwise

(24)





120575119894119895 = 1 for 119894 = 119895 = 00 otherwise

(25)









(a) 120596 = minus18 (b) 120596 = 0

(c) 120596 = 18










minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2


0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13



1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)


(27)



(28)







(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) 120596 = minus18 (b) 120596 = 0

(c) 120596 = 18










minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2


0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13



1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)


(27)



(28)







(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus5

0

5

minus6minus4

minus20

24

0

02

04

06

08

1

04

minus20

2


0

5

minus6minus4

minus20

24

0

02

04

06

08

1

(b) 120596 = 0

0

5

0

minus5minus5

0

50

02

04

06

08

1

(c) 120596 = 13



1198871 (1 1) = 41198871 (1 minus1) = 01198871 (minus1 1) = 0

1198871 (minus1 minus1) = 0(26)


(27)



(28)







(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(c) 120596 = minus120 (d) 120596 = 0

(e) 120596 = 120 (f) 120596 = 110





polynomials


120591 = 0 if 119897 = 2119901 minus 112 if 119897 = 2119901 (29)



510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










510

1520

2530

35

510

152025

3035012

(a) Initial mesh

510

1520

2530

35

510

1520

2530

35012

(b) 120596 = minus13

510

1520

2530

35

510

1520

253035

012

(c) 120596 = minus110

510

1520

2530

35

510

1520

2530

35012

(d) 120596 = 0

510

1520

2530

35

510

1520

2530

35012

(e) 120596 = 110

510

1520

2530

35

510

1520

2530

35012

(f) 120596 = 13



1198862 = [minus 2310368 minus

532592

minus 1372592

28110368

18075184

9071296

9071296

18075184

28110368

minus 1372592 minus

532592 minus

2310368]

(30)




1205911 = 14119863(10)1198871 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 = 01205912 = 1

4119863(01)1198871 (1199111 1199112)10038161003816100381610038161003816100381610038161199111=11199112=1 = 0

(31)

then

41205911 = 119863(10)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 041205912 = 119863(01)1198871 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 0 (32)



510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










510

1520

2530

35

510

1520

2530

35012

(a) Initial mesh

510

1520

2530

35

51015

2025

3035012

(b) 120596 = minus12

510

1520

2530

35

51015

2025

3035012

(c) 120596 = minus135

1015

2025

3035

5101520

2530

35012

(d) 120596 = 0

510

1520

253035

5101520

2530

35012

(e) 120596 = 135

1015

2025

3035

51015

202530

35012

(f) 120596 = 12



1205911 = 14119863(10)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

1205912 = 14119863(01)1198872 (1199111 1199112)

10038161003816100381610038161003816100381610038161199111=11199112=1 =12

(33)

then

41205911 = 119863(10)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 241205912 = 119863(01)1198872 (1199111 1199112)100381610038161003816100381610038161199111=11199112=1 = 2 (34)



(35)




1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2 3

4

5

67

8


2 3

4

5

67

8


1

2 3

4

5

67

8


1

2 3

4

5

67

8








5 Conclusion



minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus2

0

2

4

6

8

10

12

14

minus10123456780

5


minus202468101214

minus10123456780

5












Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Construction and Analysis of...

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