Research Article Subdivision Depth Computation for Tensor Product n-ary Volumetric Models

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1 Abstrct nd Applied Anlysis Volume 2011, Article ID , 22 pges doi: /2011/ Reserch Article Subdivision Depth Computtion for Tensor Product n-ary Volumetric Models Ghulm Mustf nd Muhmmd Sdiq Hshmi Deprtment of Mthemtics, The Islmi University of Bhwlpur, Bhwlpur 63100, Pkistn Correspondence should be ddressed to Ghulm Mustf, mustf Received 22 October 2010; Revised 27 Jnury 2011; Accepted 28 Februry 2011 Acdemic Editor: Yoshikzu Gig Copyright q 2011 G. Mustf nd M. S. Hshmi. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. We offer computtionl formul of subdivision depth for tensor product n-ry n > 2 volumetric models bsed on error bound evlution technique. This formul provides nd error control tool in subdivision schemes over regulr hexhedron lttice in higher-dimensionl spces. Moreover, the error bounds of Mustf et l re specil cses of our bounds. 1. Introduction Subdivision is simple nd elegnt method to describe smooth curves nd surfces. The pproch of subdivision schemes is simple nd efficient due to its mthemticl formultion. Its ppliction rnges from industril design nd nimtion to scientific visuliztion nd simultion. Due to this, subdivision method is becoming stndrd technique nd now wellunderstood by both cdemic nd industril communities. It is n lgorithm to generte smooth curves nd surfces s sequence of successively refined control polygons. At ech subdivision level, the subdivision scheme describe the source grid mps to the subdivided grid, which results in increse in the number of points. The number of points inserted t level k 1 between two consecutive points from level k is clled rity of the scheme. In the cse when number of points inserted re 2, 3,..., nthe subdivision schemes re clled binry, ternry,..., n-ry, respectively. For more detils on n-ry subdivision schemes, we my refer to 1 4 thesis of Aspert 5 nd Ko 6. Howevertensorproducttrivriteschemes obtined from bove schemes hve been proven themselves n excellent tool for the modeling of lrgely regulr volumetric/solid models over hexhedron lttice, for exmple, mnufcturing of industril regulr block, font nimtion nd grment pressures for the biomechnicl design of functionl pprel products, nd so forth.

2 2 Abstrct nd Applied Anlysis Although subdivision hs mny ttrctive dvntges for modeling purposes, it hs received fr less ttention in volumetric modeling. One of the reson is the topologicl complexity of domin meshes in higher-dimensionl spces. Another reson is lck of proper mthemticl tools for extrordinry nlysis. Even so, it is cler tht subdivision is slowly gining interest in solid modeling community. McCrcken nd Joy 7 proposed tensor product extension of the Ctmull-Clrk scheme in the volumetric setting over hexhedron lttice. Lter on, Bjj et l. 8 further extended the scheme with the nlysis bsed on numericl experiments. In 2004, McDonnell et l. 9 present volumetric subdivision scheme for interpoltion of rbitrry hexhedrl meshes. Mustf nd Liu 10 present subdivision scheme which exhibits control over shrink-ge/size of volumetric models in The method here presented is much simpler nd esier s compred to McCrcken nd Joy 7. Cheng nd Yong 11 proposed subdivision scheme bsed on nonhexhedron lttice. Wng et l. 12 gve the ppliction of the volumetric subdivision scheme in the simultion of elstic humn body deformtion nd grment pressure. Subdivision schemes hve become importnt in recent yers becuse they provide precise nd efficient wy to describe smooth curves/surfces/volumetric models, however the little hve been done in the re of error control for tensor product n-ry volumetric models. The investigtion of error control for volumetric models rises two questions in mind. i How well the regulr hexhedron lttice pproximte to the limit volumetric model? ii How mny subdivision steps re needed to stisfy user-specified error tolernce? For given error tolernce, the subdivision levels performed on the initil control polygon, so tht the error/distnce between the resulting control polygon nd the limit volumetric models would be less thn the error tolernce is clled subdivision depth. A subdivision depth nd error bound bsed on forwrd differences of control points hve been presented by 11, 13 18, while the methods re bsed on eigennlysis. But nothing in this re hs been done for more generl tensor product n-ry volumetric models yet. In this pper, we will nswer-bove-sid questions nd present subdivision depth computtion technique bsed on error bounds for tensor product n-ry volumetric models. It is notified tht the increse in rity offers greter freedom thn offered by low rity subdivision volumetric scheme in term of coefficients. Higher rity volumetric schemes llow rnge of different behviors thn the lower rity volumetric schemes. Ko 6 notified tht subdivision curves/surfces with higher rity results in higher smoothness nd pproximtion order but smller in support, which mke it more prcticl in use. It is lso noticed tht higher rity volumetric models hve slightly lower computtionl cost thn lower rity volumetric models. This discussion motivte us to clculte error bound nd depth for higher rity subdivision volumetric models, tht is, in generl for tensor product n-ry subdivision volumetric models. Our method is generliztion of Mustf et l. 13, The pper is rrnged s follows. Section 2 is devoted for bsic definitions nd nottions. In Section 3, we hve computed subdivision depth for tensor product n-ry volumetric models. Section 4 presents pplictionsofourresultsfortensorproductn-ry volumetric models. Conclusion nd future reserch directions re given in Section 5. The typicl mthemticl proofs re plced in Appendices A nd B for improved presenttion of the pper.

3 Abstrct nd Applied Anlysis 3 2i1 2i2 i1 2i3 3i1 3i2 3i3 i1 3i4 3i5 4i3 4i2 4i1 4i4 i1 4i5 4i6 4i7 2i i i2 2i4 3i i i2 3i6 4i i i2 4i8 b c Figure 1: Solid lines show corse polygons wheres doted lines re refined polygons. c represent binry, ternry, nd quternry refinement of corse polygon of scheme 2.1 for n 2, 3, 4, respectively. 2. Preliminries In this section, first we list ll the bsic fcts bout subdivision curve, surfce nd volumetric models needed in this pper. Then we settle some nottions for fir reding nd better understnding of Section Concepts n-ary Subdivision Curve Given sequence of control points p k i R N, i Z, N > 2, where the upper index k > 0 indictes the subdivision level, n n-ry subdivision curve 5 is defined by niα α,j p k ij, α 0, 1,...,n 1, 2.1 p k1 j0 where m>0nd α,j 1, α 0, 1,...,n j0 The set of coefficients { α,j,α0, 1,...,n 1} m j0 is clled subdivision msk. Given initil vlues p 0 i R N, i Z, then in the limit k,theprocess2.1 defines n infinite set of points in R N. The sequence of control points {p k i } is relted, in nturl wy, with the dydic mesh points t k i i/n k, i Z. The process then defines scheme whereby p k1 niα replces the vlue p k iα/n for α {0,n}.Herepk1 niα is inserted t the mesh point tk1 niα 1/nn αtk i αtk i1 for α 0, 1,...,n. Lbelling of old nd new points is shown in Figure 1 which illustrtes subdivision scheme Tensor Product n-ary Subdivision Surfce Given sequence of control points p k i,j R N, i, j Z, N > 2, where the upper index k > 0 indictes the subdivision level, tensor product n-ry surfce is tensor product of 2.1 defined by p k1 niα,njβ r0 s0 α,r β,s p k ir,js, α,β 0, 1,...,n 1, 2.3

4 4 Abstrct nd Applied Anlysis 2i,2j2 2i,2j1 i,j1 2i1,2j1 2i1,2j2 i1,j1 2i2,2j2 2i2,2j1 3i,3j3 3i,3j2 3i,3j1 i,j1 3i1,3j3 3i2,3j3 i1,j1 3i3,3j3 3i3,3j2 3i3,3j1 2i,2j i,j 2i1,2j i1,j 2i2,2j 3i,3j i,j 3i1,3j 3i2,3j i1,j 3i3,3j b 4i,4j4 i,j1 4i2,4j4 i1,j1 4i4,4j4 4i,4j2 4i4,4j2 4i,4j i,j 4i2,4j c i1,j 4i4,4j Figure 2: Solid lines show one fce of corse polygons wheres doted lines re refined polygons. c cn be obtin by subdividing one fce into four, nine nd sixteen new fces by using 2.3 for n 2, 3, 4 i.e., binry, ternry nd quternry, respectively. where α,r stisfies 2.2. Given initil vlues p 0 i,j R N, i, j Z, then in the limit k,the process 2.3 defines n infinite set of points in R N. The sequence of vlues {p k i,j } is relted, in nturl wy, with the dydic mesh points i/n k,j/n k, i, j Z. The process then defines schemewherebyp k1 niα,njβ replces the vlues pk iα/n,jβ/n for α, β {0,n}. Here the vlues p k1 niα,njβ re inserted t the mesh points ni α/nk1, nj β/n k1 for α, β 0, 1,...,n. Lbelling of old nd new points is shown in Figure 2 which illustrtes subdivision scheme Tensor Product n-ary Volumetric Model Given sequence of control points p k R N, i, j, l Z, N > 2, where the upper index k > 0 indictes the subdivision level, tensor product n-ry volumetric model is tensor product of 2.3 defined by p k1 niα,njβ,nlγ r0 s0 t0 α,r β,s γ,t p k ir,js,lt, α,β,γ 0, 1,...,n 1, 2.4 where α,r stisfies 2.2. Given initil vlues p 0 R N, i, j, l Z, then in the limit k,the process 2.4 defines n infinite set of points in R N. The sequence of vlues {p k } is relted,

5 Abstrct nd Applied Anlysis 5 2i,2j2,2l2 2i2,2j2,2l2 3i,3j3,3l3 3i3,3j3,3l3 2i,2j,2l2 2i,2j,2l1 2i1,2j,2l1 2i2,2j2l 3i,3j,3l3 3i,3j,3l2 3i,3j,3l1 3i3,3j3,3l 2i,2j,2l 2i1,2j,2l 2i2,2j,2l 3i,3j,3l 3i1,3j,3l 3i2,3j,3l 3i3,3j,3l b 4i,4j4,4l4 4i4,4j4,4l4 4i,4j,4l4 4i,4j,4l2 4i4,4j4,4l 4i,4j,4l 4i2,4j,4l c 4i4,4j,4l Figure 3: Solid lines show one cube of corse polygons wheres doted lines re refined polygons. c cn be obtin by subdividing one cube into 2 3,3 3,nd4 3 new cubes by using 2.4 for n 2, 3, 4, respectively. in nturl wy, with the dydic mesh points i/n k,j/n k,l/n k, i, j, l Z. The process then defines scheme whereby p k1 niα,njβ,nlγ replces the vlues pk for α, β, γ {0,n}. iα/n,jβ/n,lγ/n Here the vlues p k1 niα,njβ,nlγ re inserted t the mesh points ni α/nk1, nj β/n k1, nl γ/n k1 for α, β, γ 0, 1,...,n. Lbelling of old nd new points is shown in Figure 3 which illustrtes subdivision scheme Subdivision Depth Given control polygon of tensor product n-ry volumetric model nd n error tolernce ɛ, if we subdivide control polygon k times so tht the error between resulting polygon nd volumetric model is smller thn ɛ, then k is clled subdivision depth of tensor product n- ry volumetric model with respect to ɛ Nottions Here, we ssume δ mx α,β,γ { } α,r β,s b γ,t, α,β,γ 0, 1,...,n 1, 2.5 r0 s0 t0

6 6 Abstrct nd Applied Anlysis where b γ,j j γ,t γ1,t, γ 0, 1,...,n 2, t0 n 2 b n 1,j 0,j b γ,j. γ0 2.6 Suppose further for α, β, γ 0, 1,...,n 1, M k α,β,γ p k1 niα,njβ,nlγ 1 n 3 {n α n β n γ p k α n β n γ p k i1,j,l βn α n γ p k i,j1,l 2.7 γn α n β p k 1 αβ n γ p k i1,j1,l αγ n β p k i1,j,l1 βγn αp k i,j1,l1 αβγpk i1,j1,l1}. Δ k,1 pk i1,j,l pk, Δk,5 pk i1,j,l1 pk 1, Δ k,2 pk i,j1,l pk, Δk,6 pk i,j1,l1 pk 1, Δ k,3 pk 1 pk, Δk,7 pk i1,j1,l1 pk i,j1,l1, 2.8 Δ k,4 pk i1,j1,l pk i,j1,l, { χ mx mx t Δ 0,t }, t 0, 1,...,7, 2.9 η 1 α,β,γ β,0 γ,0 α,t α n β n γ m 1 β,0 γ,0 ã α,s, t1 η 2 α,β,γ γ,0 β,t β n γ m 1 γ,0 α,r ã β,s, t1 n 2 n 3 r0 s1 η 3 α,β,γ γ,t t1 γ n m 1 α,r β,t ã γ,s, r0 t0 s1 η 4 α,β,γ γ,0 s1 t1 s1 α,s β,t αβ n γ m 1 γ,0 β,t ã α,s, n 3 t1 s

7 Abstrct nd Applied Anlysis 7 η 5 α,β,γ β,0 α,s γ,t αγ n β m 1 β,0 γ,t ã α,s, 2.14 where s1 t1 n 3 t1 s1 η 6 α,β,γ β,s γ,t βγ m 1 γ,t α,r ã β,s, η 7 α,β,γ s1 t1 r1 s1 t1 n 2 α,r β,s γ,t αβγ n 3 t1 r0 s1 m 1 β,r γ,t ã α,s, r1 t1 s ã α,0 α,t α, α 0, 1,...,n 1, n t1 ã α,j α,t, j > 1, α 0, 1,...,n 1. tj Furthermore, suppose ϑ mx α,β,γ { χ 7 t1 } η t α,β,γ, α,β,γ 0, 1,...,n Depth for Tensor Product n-ary Volumetric Models In this prgrph, we compute subdivision depth for tensor product n-ry volumetric model. Moreover, we show tht error bound for binry subdivision volumetric models 17 is specil cse of our bound. Here we need following lemms for Theorem 3.5. The proof of first two lemms re shown in Appendices A nd B, respectively. Lemm 3.1. Given initil control polygon p 0 p, i, j, l Z,letthevluesp k, k > 1 be defined recursively by 2.4 together with 2.2,then mx 6 δ k mx, 3.1 Δ k,t Δ 0,t where δ nd Δ k, t 1, 2,...,7,redefinedby2.5 nd 2.8, respectively.,t Lemm 3.2. Given initil control polygon p 0 p, i, j, l Z,letthevluesp k, k > 1 be defined recursively by 2.4 together with 2.2,then M k 6 α,β,γ χ δ k 7 t1 η t α,β,γ, 3.2 where δ, M k α,β,γ, χ, ηt α,β,γ respectively. for α, β, γ 0, 1,...,n 1 re defined by 2.5, 2.7, ,

8 8 Abstrct nd Applied Anlysis Lemm 3.3. Given initil control polygon p 0 p, i, j, l Z,letthevluesp k, k > 1 be defined recursively by 2.4 together with 2.2. Suppose is the piecewise liner interpolnt to the vlues p k nd P is the limit volumetric model of 2.4. Ifδ<1, then error bound between tensor product n-ry volumetric model nd its control polygon fter k-fold subdivision is P δ k 6 ϑ, δ where δ nd ϑ re defined by 2.5 nd 2.18, respectively. Proof. Let denote the uniform norm. Since the mximum difference between nd is ttined t point on the k 1th mesh, we hve } 6 mx {M k α,β,γ, α,β,γ 0, 1,...,n 1, 3.4 α,β,γ where M k α,β,γ is defined by 2.7. By3.2 nd 3.4,weget 6 ϑδ k, 3.5 where δ nd ϑ re defined by 2.5 nd 2.18, respectively. By tringle inequlity we get 3.3. This completes the proof. Remrk 3.4. Theorem 3.10 in 17 is designed to estimte error bound for binry subdivision volumetric model i.e., ech cube is divided in 8 subcubes. But for the higher rity subdivision schemes such s for n 3, 4, 5,... when ech cube is divided in 3 3, 4 3, 5 3,... subcubes error estimtes re not fesible by existing result. So estimtion of error bounds for tensor product n-ry volumetric model is quite necessry. Our Lemm 3.3 provides freedom to evlute error bound for ll rities. Here we lso mention tht Lemm 3.3 for n 2 reduces to 17, Theorem Now we offer the computtionl formul of subdivision depth for tensor product n- ry volumetric model. Theorem 3.5. Let k be subdivision depth, nd let d k be the error bound between tensor product n-ry volumetric model P nd its k-level control polygon. For rbitrry ɛ>0,if k > log δ 1 ϑ, 3.6 ɛ1 δ then d k 6 ɛ. 3.7

9 Abstrct nd Applied Anlysis 9 Proof. From 3.3,wehve d k P δ k 6 ϑ δ This implies, for rbitrry given ɛ > 0, when subdivision depth k stisfies the following inequlity: ϑ k > log δ 1, 3.9 ɛ1 δ then d k 6 ɛ This completes the proof. 4. Applictions 4.1. Error Bound nd Subdivision Depth of Tensor Product n-ary Interpolting Volumetric Models In this section, we estimte the error bound nd subdivision depth of 2b 2-point tensor product n-ry interpolting volumetric models. By tking the tensor product of 2b 2-point n-ry scheme of 4,wegetthefollowing: p k1 ni,nj,nl pk, p k1 b1 nis 1,nj,nl A s 1,t 1 p k t 1, t 1 b b1 p k1 ni,njs 2,nl p k1 ni,nj,nls 3 p k1 b1 nis 1,njs 2,nl p k1 nis 1,nj,nls 3 p k1 ni,njs 2,nls 3 p k1 nis 1,njs 2,nls 3 b1 t 2 b A s 2,t 2 p k i,t 2 j,l, b1 A s 3,t 3 p k i,j,t 3 l, t 3 b t 1 b t 2 b b1 t 1 b t 3 b b1 b1 A s 1,t 1 A s 2,t 2 p k t 1 i,t 2 j,l, b1 t 2 b t 3 b b1 t 1 b t 2 b t 3 b A s 1,t 1 A s 3,t 3 p k t 1 i,j,t 3 l, b1 A s 2,t 2 A s 3,t 3 p k i,t 2 j,t 3 l, b1 A s 1,t 1 A s 2,t 2 A s 3,t 3 p k t 1 i,t 2 j,t 3 l, 4.1

10 10 Abstrct nd Applied Anlysis Tble 1: Error bound of tensor product n-ry interpolting volumetric models: here n presents the rity of subdivision volumetric model nd k presents the subdivision level. n \ k Tble 2: Subdivision depth of tensor product n-ry interpolting volumetric models: here n presents the rity of subdivision volumetric model nd ɛ presents error tolernce. n \ ɛ where A x,y b1 m b x nm x ny 1 b 1 y n 2b1 b y! b y 1!, 4.2 s 1,s 2,s 3 1, 2,...,n 1, b 1, 2, 3,... nd n stnds for n-ry interpolting subdivision volumetric model, tht is, n 2, 3, 4,... stnds for binry, ternry, quternry nd so on, respectively. The error bounds nd subdivision depth of 4.1 re shown in Tbles 1 nd 2, respectively. In these tbles, we hve shown the error bounds nd depth of different rity interpolting subdivision volumetric models by using Lemm 3.3 nd Theorem 3.5 with χ Error Bound nd Subdivision Depth of Tensor Product n-ary Approximting Volumetric Models In this section, we estimte the error bound nd subdivision depth of tensor product 2b 2point n-ry pproximting volumetric models. By tking the tensor product of 2b 2-point n-ry scheme of 4,wegetthefollowing: p k1 nis 1,njs 2,nls 3 b1 b1 b1 t 1 b t 2 b t 3 b B s 1,t 1 B s 2,t 2 B s 3,t 3 p k t 1 i,t 2 j,t 3 l, 4.3

11 Abstrct nd Applied Anlysis 11 Tble 3: Error bound of tensor product n-ry pproximting volumetric models: here n presents the rity of subdivision volumetric model nd k presents the subdivision level. n \ k Tble 4: Subdivision depth of tensor product n-ry pproximting volumetric models: here n presents the rity of subdivision volumetric model nd ɛ presents error tolernce. n \ ɛ where B x,y b1 m b2x 1 2nm 2x 1 2ny 1 b 1 y 2n 2b1 b y! b y 1, 4.4! s 1,s 2,s 3 1, 2,...,n 1, b 1, 2, 3,...,ndn stnds for n-ry pproximting subdivision volumetric model, tht is, n 2, 3, 4,... stnds for binry, ternry, quternry, nd so on, respectively. The error bounds nd subdivision depth of 4.3 re shown in Tbles 3 nd 4, respectively. In these tbles, we hve shown the error bounds nd depth of different rity pproximting subdivision volumetric by using Lemm 3.3 nd Theorem 3.5 with χ Conclusion nd Future Work We hve computed subdivision depth bsed on error bounds for tensor product n-ry volumetric models. Furthermore, we hve shown tht error bounds for binry subdivision volumetric model 17 is specil cse of our bounds. It is noticed tht the increse in rity results grdully decrese in error, which is shown in Tbles 1, 3 nd grphiclly in Figure 4. It is noticed from Tbles 2 nd 4 tht higher rity subdivision volumetric models need less number of subdivision steps thn lower rity to stisfy user-specified error tolernce. The uthors re looking, s future work, to extend the computtionl techniques of subdivision depth for n-ry rbitrry subdivision volumetric models over rectngulr/tringulr hexhedron lttice. we will discuss them elsewhere.

12 12 Abstrct nd Applied Anlysis k k n 2 n 3 n 4 n 5 n 6 n 2 n 3 n 4 n 5 n 6 b Figure 4: Presents comprison mong the error bounds of different rity subdivision volumetric models 4.1, thtis,forn 2, 3, 4, 5, 6; b presents comprison mong the error bounds of different rity subdivision volumetric models 4.3, thtis,forn 2, 3, 4, 5, 6. Here k presents subdivision level, n presents the rity nd ɛ presents the user-specified error tolernce. Appendices A. Proof of Lemm 3.1 Proof. From 2.2, 2.4 for α, β, γ 0, 1,...,n 1, we obtin p k niα1,njβ,nlγ pk niα,njβ,nlγ s0 t0 β,s γ,t r0 b α,r ir1,js,lγ pk 1 ir,js,lγ, A.1 p k niα,njβ1,nlγ pk niα,njβ,nlγ r0 t0 α,r γ,t s0 b β,s ir,js1,lt pk 1 ir,js,lt, A.2 p k nin,njβ1,nlγ pk nin,njβ,nlγ r0 t0 0,r γ,t s0 b β,s ir1,js1,lt pk 1 ir1,js,lt, A.3 p k nin,njβ1,nln pk nin,njβ,nln m 0,r 0,t b β,s r0 t0 s0 ir1,js1,lt1 pk 1 ir1,js,lt1, A.4

13 Abstrct nd Applied Anlysis 13 p k niα,njβ,nlγ1 pk niα,njβ,nlγ r0 s0 α,r β,s t0 b γ,t ir,js,lt1 pk 1 ir,js,lt, A.5 p k nin,njβ,nlγ1 pk nin,njβ,nlγ r0 s0 0,r β,s t0 b γ,t ir1,js,lt1 pk 1 ir1,js,lt, A.6 p k niα,njn,nlγ1 pk niα,njn,nlγ m α,r 0,s b γ,t r0 s0 t0 ir,js1,lt1 pk 1 ir,js1,lt, A.7 p k nin,njn,nlγ1 pk nin,njn,nlγ m 0,r 0,s b γ,t r0 s0 t0 ir1,js1,lt1 pk 1 ir1,js1,lt, A.8 p k niα1,njn,nlγ pk niα,njn,nlγ s0 t0 0,s γ,t r0 b α,r ir1,js1,lt pk 1 ir,js1,lt, A.9 p k niα1,njβ,nln pk niα,njβ,nln s0 t0 β,s 0,t r0 b α,r ir1,js,lt1 pk 1 ir,js,lt1, A.10 p k niα,njβ1,nln pk niα,njβ,nln m α,r 0,t b β,s r0 t0 s0 ir,js1,lt1 pk 1 ir,js,lt1, A.11 p k niα1,njn,nln pk niα,njn,nln m 0,s 0,t b α,r s0 t0 r0 ir1,js1,lt1 pk 1 ir,js1,lt1, A.12 where b γ,r is defined by 2.6 nd α, β, γ 0, 1,...,n 1. Now using A.1 recursively together with nottions defined by 2.8, weget mx Δ k,1 k 6 mx β,t α,s b γ,r mx α,β,γ s0 t0 r0 Δ 0,1. A.13

14 14 Abstrct nd Applied Anlysis From 2.5 nd using the bove inequlity, we get mx Δ k,1 6 δ k mx Δ 0,1. A.14 Agin using A.2 A.4 recursively nd by utilizing 2.5 nd 2.8, wehve mx Δ k,2 6 δ k mx Δ 0,2. A.15 Further using A.5 A.8 recursively nd by utilizing 2.5 nd 2.8,wehve mx Δ k,3 6 δ k mx Δ 0,3. A.16 Similrly, using A.9 A.12 recursively together with 2.5 nd 2.8 seprtely for ech t 4, 5, 6, 7, respectively, we hve mx 6 δ k mx. A.17 Δ k,t Δ 0,t This completes the proof. B. Proof of Lemm 3.2 Proof. From 2.2 nd 2.4, p k1 ni,nj,nl pk r0 m 0,r 0,s 0,t p s0 k ir,js,lt pk. B.1 t0 By expnding innermost summtion, we get 0,t p k ir,js,lt pk 0,0 p k ir,js,l pk 0,1 p k ir,js,l1 pk t0 0,2 p k ir,js,l2 pk ir,js,l1 pk ir,js,l1 pk 0,m p k ir,js,lm pk ir,js,lm 1 pk ir,js,l1 pk, 0,t p k ir,js,lt pk 0,0 p k ir,js,l pk t0 m 0,p p k ir,js,l1 pk p1 ã 0,q p k ir,js,lq1 pk ir,js,lq, m 1 q1 B.2

15 Abstrct nd Applied Anlysis 15 where ã 0,q is defined by 2.16.Now s0 0,s t0 0,t p k ir,js,lt pk 0,0 0,s p k ir,js,l pk 0,p 0,s p k ir,js,l1 pk s0 m 1 0,s s0 q1 p1 s0 ã 0,q p k ir,js,lq1 pk ir,js,lq. B.3 Since 0,s p k ir,js,l pk s0 0,0 p k ir,j,l pk 0,p p k m 1 ir,j1,l pk p1 0,s p k ir,js,l1 pk s0 0,0 p k ir,j,l1 pk q1 ã 0,q p k ir,jq1,l pk ir,jq,l, B.4 m 1 0,p p k ir,j1,l1 pk ã 0,q p k ir,jq1,l1 pk ir,jq,l1, p1 q1 then B.3 implies m 0,s 0,t p k ir,js,lt pk s0 t0 2 0,0 p k ir,j,l pk 0,0 0,p p k m 1 ir,j1,l pk 0,0 ã 0,q p k ir,jq1,l pk ir,jq,l p1 q1 0,0 0,p p k ir,j,l1 pk 0,p 0,q p k ir,j1,l1 pk p1 p1 q1 B.5 m 1 0,p ã 0,q p k ir,jq1,l1 pk ir,jq,l1 p1 q1 m 1 0,s ã 0,q p k ir,js,lq1 pk ir,js,lq. s0 q1

16 16 Abstrct nd Applied Anlysis Substituting it into B.1, weget p k1 ni,nj,nl pk 0,0 2 0,r p k ir,j,l pk r0 0,0 p1 r0 p1 q1 r0 0,0 p1 r0 0,p 0,r p k ir,j1,l pk 0,p 0,q 0,r p k ir,j1,l1 pk 0,p 0,r p k ir,j,l1 pk B.6 m 1 0,0 0,r ã 0,q p k ir,jq1,l pk ir,jq,l r0 q1 m 1 0,p 0,r ã 0,q p k ir,jq1,l1 pk ir,jq,l1 p1 r0 q1 m 1 0,r 0,s ã 0,q p k ir,js,lq1 pk ir,js,lq. r0 s0 q1 Since 0,r p k ir,j,l pk 0,p p k m 1 i1,j,l pk r0 p1 0,r p k ir,j1,l pk 0,0 p k i,j1,l pk r0 p1 q1 m 1 0,p p k i1,j1,l pk 0,r p k ir,j,l1 pk 0,0 p k 1 pk r0 p1 0,r p k ir,j1,l1 pk 0,0 p k i,j1,l1 pk r0 ã 0,q p k iq1,j,l pk iq,j,l q1 q1, ã 0,q p k iq1,j1,l pk iq,j1,l 0,p p k m 1 i1,j,l1 pk ã 0,q p k iq1,j,l1 pk iq,j,l1, m 1 0,p p k i1,j1,l1 pk ã 0,q p k iq1,j1,l1 pk iq,j1,l1, p1 q1, B.7

17 Abstrct nd Applied Anlysis 17 then substituting these summtions into B.6 nd rerrnging, we get ni,nj,nl pk 2 0,0 0,p p k i1,j,l pk 0,0 0,p p k i,j1,l pk p k1 p1 p1 0,p p k 1 pk 0,0 0,p 0,q p k i1,j1,l pk i,j1,l p1 p1 q1 p1 q1 0,0 0,p 0,q p k i1,j,l1 pk 1 0,p 0,q p k i,j1,l1 pk 1 p1 q1 0,p 0,q 0,r p k i1,j1,l1 pk i,j1,l1 N 2 k, p1 q1 r1 B.8 where N 2 m 1 k 2 0,0 ã 0,q p k iq1,j,l pk iq,j,l q1 m 1 0,0 0,p ã 0,q p k iq1,j1,l pk iq,j1,l p1 q1 m 1 0,0 0,p ã 0,q p k iq1,j,l1 pk iq,j,l1 p1 q1 m 1 0,r 0,p ã 0,q p k iq1,j1,l1 pk iq,j1,l1 r1 p1 q1 B.9 m 1 0,0 0,r ã 0,q p k ir,jq1,l pk ir,jq,l r0 q1 m 1 0,r 0,p ã 0,q p k ir,jq1,l1 pk ir,jq,l1 r0 p1 q1 m 1 0,r 0,s ã 0,q p k ir,js,lq1 pk ir,js,lq. r0 s0 q1

18 18 Abstrct nd Applied Anlysis This implies M k 0,0,0 6 0,0 2 0,t 0,0 2 m 1 ã 0,s mx t1 s1 Δ k,1 m 1 0,0 0,t 0,0 0,r ã 0,s mx t1 t1 r0 t0 s1 r0 s1 m 1 0,t 0,r 0,t ã 0,s mx s1 t1 Δ k,2 Δ k,3 m 1 0,0 0,s 0,t 0,0 0,t ã 0,s mx s1 t1 t1 s1 Δ k,4 m 1 0,0 0,s 0,t 0,0 0,t ã 0,s mx s1 t1 t1 r0 s1 t1 s1 m 1 0,s 0,t 0,t 0,r ã 0,s mx r1 s1 t1 r1 t0 s1 Δ k,5 Δ k,6 m 1 0,r 0,s 0,t 0,r 0,t ã 0,s mx Δ k,7, B.10 where M k 0,0,0 nd Δk,t, t 0, 1,...,7redefinedby2.7 nd 2.8. Now using nottions 2.5 nd 2.9,wehve M k 0,0,0 6 δ χ{ k 0,0 2 0,t 0,0 2 m 1 ã 0,s t1 s1 m 1 0,0 0,t 0,0 0,r ã 0,s t1 t1 r0 t0 s1 r0 s1 m 1 0,t 0,r 0,t ã 0,s m 1 0,0 0,s 0,t 0,0 0,t ã 0,s s1 t1 s1 t1 t1 s1 m 1 0,0 0,s 0,t 0,0 0,t ã 0,s s1 t1 t1 r0 s1 t1 s1 m 1 0,s 0,t 0,t 0,r ã 0,s m 1 0,r 0,s 0,t 0,r 0,t ã 0,s }. r1 s1 t1 r1 t0 s1 B.11

19 Abstrct nd Applied Anlysis 19 Using for α, β, γ 0, we hve M k 0,0,0 6 δ k χ η 1 0,0,0 η2 0,0,0 η7 0,0,0 χ δ k 7 η t 0,0,0. t1 B.12 Similrly from 2.2 nd 2.4 for α 1, β γ 0, p k1 ni1,nj,nl 1 { } n 1p k n pk m i1,j,l 1,r 0,s 0,t p r0 s0 k ir,js,lt pk. B.13 t0 Now fter expnding nd rerrnging the bove summtion, we hve p k1 ni1,nj,nl 1 { } n 1p k n pk i1,j,l 2 0,0 1,p p1 1 p k i1,j,l n pk 0,0 1,p p k i,j1,l pk 0,0 1,s 0,t p k i1,j1,l pk i,j1,l p1 s1 t1 B.14 0,p p k 1 pk 1,r 0,s 0,t p k i1,j1,l1 pk i,j1,l1 p1 s1 t1 r1 s1 t1 0,0 1,s 0,t p k i1,j,l1 pk 1 0,s 0,t p k i,j1,l1 pk 1 N 3 k, s1 t1 where N 3 m 1 k 2 0,0 ã 1,q p k iq1,j,l pk iq,j,l q1 m 1 0,0 0,p ã 1,q p k iq1,j1,l pk iq,j1,l p1 q1 m 1 0,0 0,p ã 1,q p k iq1,j,l1 pk iq,j,l1 p1 q1 m 1 0,r 0,p ã 1,q p k iq1,j1,l1 pk iq,j1,l1 r1 p1 q1

20 20 Abstrct nd Applied Anlysis m 1 0,0 1,r ã 0,q p k ir,jq1,l pk ir,jq,l r0 q1 m 1 0,r 1,p ã 0,q p k ir,jq1,l1 pk ir,jq,l1 r0 p1 q1 m 1 1,r 0,s ã 0,q p k ir,js,lq1 pk ir,js,lq. r0 s0 q1 B.15 Now using nottions 2.5, nd using Lemm 3.1, wehve M k 1,0,0 6 δ χ{ k 0,0 2 1,t t1 1 m 1 n 0,0 2 ã 1,s s1 m 1 0,0 0,t 0,0 1,r ã 0,s t1 t1 r0 t0 s1 r0 s1 m 1 0,t 1,r 0,t ã 0,s m 1 0,0 1,s 0,t 0,0 0,t ã 1,s s1 t1 t1 s1 B.16 m 1 0,0 1,s 0,t 0,0 0,t ã 1,s s1 t1 s1 t1 t1 r0 s1 t1 s1 m 1 0,s 0,t 0,r 1,t ã 0,s m 1 1,r 0,s 0,t 0,r 0,t ã 1,s }. r1 s1 t1 r1 t0 s1 Using for α 1, β γ 0, we hve M k 1,0,0 6 δ k χ η 1 1,0,0 η2 1,0,0 η7 1,0,0 χ δ k 7 η t 1,0,0. t1 B.17

21 Abstrct nd Applied Anlysis 21 Hence in generl fter extensive clcultion, nd using nottions 2.2, 2.4, 2.7, nd2.8 for α, β, γ 0, 1,...,n 1, we obtin M k 6 α,β,γ χ{ δk β,0 γ,0 α,t α n β n γ m 1 β,0 γ,0 ã α,s t1 n 3 s1 γ,0 β,t β n γ m 1 γ,0 α,r ã β,s t1 n 2 r0 s1 γ,t t1 γ n m 1 α,r β,t ã γ,s r0 t0 s1 γ,0 α,s β,t αβ n γ m 1 γ,0 β,t ã α,s s1 t1 s1 t1 n 3 t1 s1 β,0 α,s γ,t αγ n β m 1 β,0 γ,t ã α,s s1 t1 n 2 n 3 t1 r0 s1 t1 s1 β,s γ,t βγ m 1 γ,r α,t ã β,s r1 s1 t1 α,r β,s γ,t αβγ n 3 m 1 β,r γ,t ã α,s }, r1 t0 s1 M k α,β,γ 6 δ k χ η 1 α,β,γ η2 α,β,γ... η7 α,β,γ χ δ k 7 t1 η t α,β,γ where η t ; t 1, 2,...,7, is defined in This completes the proof. α,β,γ, B.18 Acknowledgment This work is supported by the Indigenous Ph.D. Scholrship Scheme of Higher Eduction Commission HEC Pkistn. References 1 J.-A. Lin, On -ry subdivision for curve design. I. 4-point nd 6-point interpoltory schemes, Applictions nd Applied Mthemtics, vol. 3, no. 1, pp , J.-. Lin, On -ry subdivision for curve design. II. 3-point nd 5-point interpoltory schemes, Applictions nd Applied Mthemtics, vol. 3, no. 2, pp , G. Mustf nd F. Khn, A new 4-point C 3 quternry pproximting subdivision scheme, Abstrct nd Applied Anlysis, Article ID , 14 pges, G. Mustf nd N. A. Rehmn, The msk of 2b 4-point n-ry subdivision scheme, Computing, vol. 90, no. 1-2, pp. 1 14, N. Aspert, Non-liner subdivision of univrite signls nd discrete surfces, Ph.D. thesis, École Polytechinique Fédérle de Lusnne, Lusnne, Switzerlnd, 2003.

22 22 Abstrct nd Applied Anlysis 6 K. P. Ko, A study on subdivision scheme-drft, Dongseo University, Busn Republic of Kore, 2007, kpko/publiction/2004book.pdf. 7 R. McCrcken nd K. L. Joy, Free form deformtion with lttices of rbitrry topology, in Proceedings of the Computer Grphics, Annul Conference Series SIGGRAPH 96, pp , C. Bjj, S. Schefer, J. Wrren, nd G. Xu, A subdivision scheme for hexhedrl meshes, Visul Computer, vol. 18, no. 5-6, pp , K. T. McDonnell, Y. U. S. Chng, nd H. Qin, Interpoltory, solid subdivision of unstructured hexhedrl meshes, Visul Computer, vol. 20, no. 6, pp , G. Mustf nd X. Liu, A subdivision scheme for volumetric models, Applied Mthemtics. A Journl of Chinese Universities Series B, vol. 20, no. 2, pp , F. Cheng nd J. H. Yong, Subdivision depth computtion for Ctmull-Clrk subdivision surfces, Computer-Aided Design nd Applictions, vol. 3, no. 1 4, pp , J. M. Wng, X. N. Luo, YI. Li, X. Q. Di, nd F. You, The ppliction of the volumetric subdivision scheme in the simultion of elstic humn body deformtion nd grment pressure, Textile Reserch Journl, vol. 75, no. 8, pp , S. Hshmi nd G. Mustf, Estimting error bounds for quternry subdivision schemes, Journl of Mthemticl Anlysis nd Applictions, vol. 358, no. 1, pp , Z. Hung, J. Deng, nd G. Wng, A bound on the pproximtion of Ctmull-Clrk subdivision surfce by its limit mesh, Computer Aided Geometric Design, vol. 25, no. 7, pp , G. Mustf, F. Chen, nd J. Deng, Estimting error bounds for binry subdivision curves/surfces, Journl of Computtionl nd Applied Mthemtics, vol. 193, no. 2, pp , G. Mustf nd J. Deng, Estimting error bounds for ternry subdivision curves/surfces, Journl of Computtionl Mthemtics, vol. 25, no. 4, pp , G. Mustf, S. Hshmi, nd N. A. Noshi, Estimting error bounds for tensor product binry subdivision volumetric model, Interntionl Journl of Computer Mthemtics, vol. 83, no. 12, pp , X.-M. Zeng nd X. J. Chen, Computtionl formul of depth for Ctmull-Clrk subdivision surfces, Journl of Computtionl nd Applied Mthemtics, vol. 195, no. 1-2, pp , J. Peters nd X. Wu, The distnce of subdivision surfce to its control polyhedron, Journl of Approximtion Theory, vol. 161, no. 2, pp , H.-W. Wng nd K.-H. Qin, Estimting subdivision depth of Ctmull-Clrk surfces, Journl of Computer Science nd Technology, vol. 19, no. 5, pp , H. Wng, Y. Gun, nd K. Qin, Error estimte for Doo-Sbin surfces, Progress in Nturl Science, vol. 12, no. 9, pp , H. Wng, H. Sun, nd K. Qin, Estimting recursion depth for Loop subdivision, Interntionl Journl of CAD/CAM, vol. 4, no. 1, pp , 2004.

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