Subdivision of Uniform ωb-spline Curves and Two Proofs of Its C k 2 -Continuity

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1 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 Subdvso of Uform ωb-sple Curves ad Two Proofs of Its C -Cotuty Jg Tag, Me-e Fag, * ad Guozhao Wag Abstract: ωb-sples have may optmal propertes ad ca reproduce pletful commoly-used aalytcal curves. I ths paper, we further propose a o-statoary subdvso method of herarchcally ad effcetly geeratg ωb-sple curves of arbtrary order of ωb-sple curves ad prove ts C -cotuty by two ds of methods. The frst method drectly prove that the sequece of cotrol polygos of subdvso of order coverges to a C -cotuousωb-sple curve of order. The secod oe s based o the theores upo subdvso mass ad asymptotc equvalece etc., whch s more coveet to be further exteded to the case of surface subdvso. Ad the problem of approxmato order of ths o-statoary subdvso scheme s also dscussed. The a uform ωb-sple curve has both perfect mathematcal represetato ad effcet geerato method, whch wll beeft the applcato of ωb-sples. Keywords: ωb-sple, subdvso, C -cotuty, asymptotc equvalece, approxmato order. Itroducto Polyomal B-sples ad URBS are mportat modelg tools CAD/CAM. But polyomal B-sples are ot able to exactly represet ofte-used cocs (except for parabola), trgoometrc fuctos ad hyperbolc fuctos etc. URBS ca represet cocs, but ts rato form results complcated computatos about dfferetal ad tegral. The all ds of B-le sples are proposed [Fag ad Wag (008); Zhag (996); Vasov ad Sattayatham (999); Maar ad Pe a (00)]. I paper [Wag, Che ad Zhou (004)], we further ufed these B-le sples to ωb-sples, whch are costructed over cos t,s t,, t,..., t,.... ω ca be o-egatve real umber ad pure magary umber. If tag the value of ω as a costat 0, or, we wll get usual polyomal B-sples, trgoometrc polyomal B-sples ad hyperbolc polyomal B-sples respectvely. ωb-sples hert most of optmal propertes from polyomal B-sples, cludg the subdvso property. Due to optmal propertes of these B-le sples, may applcatos are studed recet years [Maa, Pelos ad Speleers (0); Xu, Su, Xu et al. (07)]. I ths paper, we perfect the subdvso School of Computer Sceces ad Educatoal Softwares, Guagzhou Uversty, Guagzhou, Cha. Departmet of Mathematcs, Zhejag Uversty, Hagzhou, Cha. * Correspodg Author: Me-e Fag. Emal: fme@gzhu.edu.c. CMES. do:0.970/cmes

2 64 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 method ad theory of ωb-sples order to apply them better the future. Subdvso s a stadard techque of recursvely geeratg smooth curves/surfaces from a tal polygo/mesh. Please see paper [Cha (974); Doo ad Sab (978); Catmull ad Clar (978); Dy (99); Stam (00); Jea, Shumugaraj ad Das(00); Jea, Shumugaraj ad Das (00); Adersso, Lars-Er, Stewart et al. (00); Cot ad Roma (0); Cot, Cotroe ad Sauer (07)] for more detals. Ths d of modelg method s popularly appled geometrc modellg ad D amato because of ts umercal stablty, smple mplemet ad sutablty for arbtrary topology. But most of subdvso curves ad surfaces lac exactly mathematcal represetatos, whch are the fudametal of all ds of dfferetal/tegral computatos. So subdvso methods whch have sple bacgrouds are very terestg. Subdvso models wth sple bacgrouds clude all merts metoed above. For example, Doo-Sab method [Doo ad Sab (978)], Catmull-Clar method [Catmull ad Clar (978)], ad the subdvso method proposed paper [Stam (00)] respectvely have ther sple bacgrouds of B-sples of degree, cubc B-sples, polyomal B-sples of arbtrary order. These subdvso methods are all statoary,.e, ther subdvso rules persst uchaged each level of subdvso. Whle statoary subdvso ca ot geerate ωb-sple curves wth frequecy parameters. I ths paper, we troduce a parameter relatve to the frequecy parameter to buld a ostatoary subdvso method wth the bacgroud of ωb-sples. The ths d of modelg method has the merts of both subdvso ad ωb-sples. Cocretely, we cosder the subdvso of uform ωb-sples wth uform ot tervals ad ω tag a certa costat. At frst, we derve the defto of uform ωb-sple bases ad curves accordg to the correspodg deftos paper [Wag, Che ad Zhou (004)]. Defto. (uform ωb-sple bases) Let T be a gve uform ot sequece { t = } +=, be the legth of uform ot tervals, refers to the order of sples.ω be a gve frequecy parameter, where ω ca tae value as a o-egatve real umber (ω 0, ) ths case) or a pure magary umber whose magary part s postve. () t, costructed by the followg formula are called uform ωb-sple bases the spa of cos,s,,,..., t t t t for t,( + ) ). We frst defe uform ωb-sple basc fuctos of order = as follows. s ( t ), t ( + s ), s (( + ) t) 0, ( t) =, s ( + ) t ( + ), () 0, otherwse, ad ( t) = ( t ), 0,. I formula (.), whe ω = 0, we compute t by the L Hosptal rule about ω.

3 Subdvso of Uform ωb-sple Curves ad Two Proofs 65 For ( t) are defed recursvely by,, t t = s ds (), t, Defto. (uform ωb-sple curves) LetP,, ( t ) = ωb-sple bases of order correspodg to the partto the parameter axs t. be uform T t + = The P ( t ) = ( t, ) P,(( ) t ( + ), ) = = = of s called a uform ωb-sple curve of order correspodg to the ot vector T. P =,..., are cotrol pots. ωb-sple curves ca reproduce cocs, trgoometrc ad hyperbolc curves. They also have may useful propertes for geometry modellg, cludg those herted from commo B-sple curves ad some specal merts. Please refer to paper [Fag ad Wag (008)] for detals. But we ca see that the basc fuctos eed to be recursvely computed by tegrato from ther defto, whch results low effcecy of evaluato. I ths paper, we devote to buld a hgh-effcecy subdvso method of geeratg ωb-sple curves. The rest of ths paper s orgazed as follows. I Secto, we derve the relato formula of cotrol pots betwee two represetatos of the same uform ωb-sple curve of order respectvely wth the orgal ot tervals ad ther bsectos. The the explct subdvso rule s costructed based o ths. By ths d of subdvso rule of order, a sequece of cotrol polygos geerates from the orgal cotrol polygo of a uform ωb-sple curve of order. We drectly prove that the lmt of ths sequece coverges to the C -cotuous uform ωb-sple curve Secto. But ths d of proof method s hard to be appled the correspodg proof of the cotuty for the case of surface subdvso. So Secto 4, we recosder the proof from the aspect of subdvso mass ad provde a more geeral proof of the cotuty of subdvso whch wll be easer to be exteded to the case of surface subdvso. Because our proposed surface scheme s o-statoary, we use the theores of asymptotc equvalece betwee o-statoary subdvso ad the correspodg statoary subdvso wth the rule lmt status to complete the proof. The approxmato order of the proposed subdvso scheme s also dscussed. Secto 5 maes a cocluso.

4 66 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 The subdvso method of uform ωb-sple curves Accordg to Defto. ad Defto., we fd that a uform ωb-sple curve ca also be equvaletly represeted by aother uform ωb-sple curve wth ot tervals after bsecto. Theorem. Let, (, t ) ad (,, ) t = ( = 0,,,... ) ad t ( 0,,,... ) The a uform ωb-sple curve of order ( ) defed by =, ( + ) p t P, t, t,, = ad also be defed by + =, ( + ) p t P, t, t,. = where ( + ) P = P + P, >, t represet bases wth ots = = respectvely. ( 4 cos( ) ) P ( + ) + P ( + ), s odd; 4 cos( ) p = () P + ( 4 cos( ) ) P ( + ), s eve. 4 cos( ) Proof. Accordg to the meags of (, ) ad (, ) t t, t s easy to obta,, ther represetato formula by formula (). Ad the relato formula betwee two bases ca be deduced as below: (, t ) =,, (, t ) +, (, t ) + (, t ), cos cos Furthermore by the above formula, formula (.) ad the recursve formula (), we get

5 Subdvso of Uform ωb-sple Curves ad Two Proofs 67 t ds t (, ) = (,s ), t, = + t t ( ), t t, ( ) ( ) ( ) s ds ( ) 4 cos = (, t ) + -,, (, t ) 4 cos 4 cos 4 cos + (, t ) +,, (, t ). 4 cos 4 cos I the followg, we prove the cocluso by ducto. ( ) Whe =, t,( + ), we ow = The ( ) P 4 cos ( ) ( 4 cos ( ) ) 4 cos ( ) +, =, = = P =, ( t ), t,,0,,., (4 cos ) + P P, t (, t = Whe =4, = P + P + +, P,, ( ) P P + + ( + ) 4 cos( ) P + P +, ( ) ( ) 4 cos 4 cos 4 cos 4 cos 4 cos, ( t), s odd, s eve.

6 68 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 P t P s ds t (,,4 ) =, (, ) = = = = We ow t t t t t = =,, (, ) (, ) t = P, (, s ) ds t = = P P s ds s ds P,,4 ( t ) + (, t,4 ) =, t = 0, = 0,, the we get,4 P (, t ) = P (, t ) +,4,4 P,4 (, t ) = = = 4 That s P = ( P + P + ) ( P P + ),4 (, t ) = + =. ow assume that the cocluso holds for ( 4),.e. + + P, (, t ) =, (, t ) = = P + P P The for +, we have + =. P + P

7 Subdvso of Uform ωb-sple Curves ad Two Proofs 69 P t = P s t ds t (,, + ), (, ) = = = t = + = (, ) + t P + P + = t = = t P, s ds t ( (, s ) ds t, ) ( + ) + P + P + = + = ( + ) + + = P, t. =, +,, (, ) (, t) s ds So the cocluso holds for +. Based o ths, a uform ωb-sple curve ca be geerated by cotuously usg formula (4) from ts tal cotrol polygo. Let u cos ( ) =, we get the followg defto of geeratg uform ωb-sple curves by subdvso (ωbs for short). Defto. (ωbs scheme) Let P P P P =,,... be the tal cotrol polygo ad u be the teso parameter. The subdvso rule of ωbs curves of order ( ) S s defed as: = ( + ) S : P P P, P,... P, + P = P + P, >, ( 4u ) P ( + ) + P ( + ), s odd ; 4u P = P + ( 4u ) P (5) ( + ) 4u, s eve ; Usg the subdvso rule S,the teratve process of ωbs s descrbed as below. (4)

8 70 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 Table : The tme report of geeratg ωbs curves ad ωb-sple curves from the same cotrol polygo the type of curve = =4 =5 =0 ωbs s s s 0.085s ωb-sple 0.04 s s.0794 s s. Gve a tal cotrol polyle P, a teso parameter u (u 0) ad the subdvso order ;. Determe a subdvso tmes 0 advace;. = 0; S : P = S P, = + ; If = 0 else ed go to step 6; u = 6. Output P ; P = P ; + u ; go to step ; From the above defto, we ca see that the parameter u updates each level of subdvso. So ωbs method s a o-statoary subdvso method. The updatg formula u = + u u : = s derved from the half-agle cose formula because ( ) cos. Fg. llustrates the proposed subdvso rules ad a example.

9 Subdvso of Uform ωb-sple Curves ad Two Proofs 7 (a) (b) (c) (d) (e) (f) Fgure : The subdvso rules (a)(b)(c) ad a example (d)(e)(f). P I Fg. (a), 7 polyle P 4. = 0 = 0 s computed by formula (5) from the tal cotrol P P. = 0 = 0 Smlarly, P ca be computed by formula (5) fromp whe 4. I Fg. (c), the blac poly les are respectvely the results after oe level ad two levels of subdvso from the tal cotrol poly le whe =5, u=. The red curve s the results after sx levels of subdvso whch ca be see as the approxmato of the lmt curve. The gree ad purple curves respectvely correspod to the cases of =5, u= ad =5, u=0.5. I Fg. (d), the profle of a dustral model whch cossts of three peces of crcular arcs (red), some le segmets ad some cushog curves. I Fg. (e), the cotrol polygo of the profle s computed accordg to the ωb-sple represetato proposed paper [Fag ad Wag (008)]. I Fg. (f), the profle s reproduced by I Fg. (b), 4 6 s computed by formula (5) from 7

10 7 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 subdvdg the cotrol polygo accordg to the proposed method ths paper, =, u = cos 0. wth Comparg Defto. wth Defto. ad., we ca see that ωbs curves oly clude lear computatos, whch s much smpler ad more effcet tha those recursve tegral computatos cluded the defto of uform ωb-sple bases. Ths s very mportat for real-tme rederg ad herarchcally dsplayg curves ad surfaces. Tag the cotrol polygo llustrated Fg. (e) wth cotrol pots as a example, Tab. shows the comparso of the effcecy of both methods to reder the curve joted wth the same umber (about 00) of pots. Apparetly, the effcecy of rederg ωbs curves s much faster tha rederg ωb-sple curves. Ad wth the crease of order, the dfferece betwee them becomes bgger ad bgger. From Theorem., we ow ωbs curve s derved by the ot terpolato method of uform ωb-sple curves. The sequece of cotrol polygos formed by cotuous bsectos of ot tervals wll coverge to smooth ωb-sple curves, whch are C - cotuous. That s to say, ωb-sple curve s the lmt curve of ωbs curve wth the same cotrol polygo whe the subdvso level teds to fty. I the ext two sectos, we prove that ωbs curves are also C cotuous usg two provg methods. Oe proof of C -cotuty of ωbs curves Theorem. shows how the ew cotrol polygo ca be obtaed from the old cotrol polygo after a roud of subdvso. We have the followg theorem. Theorem. Let ( 0 B P ; ) ( t ) p ( t ) P 0 ( t ) = = be a uform ωb-sple =, 0 ( 0) curve of order whose cotrol polygo s P = S P P, P,.., P = ad the legth of ot terval s. Let ( ) 0 ( ) 0,,,,,...,, = = ( + ) + S P S S P P P P 0,,, S P = P, P,..., P. + The 0 0 S P = B ( P ) ( t ) lm ;. Proof. By Theorem. ad smple ducto o, we have ( ) 0 0 B S P ; ( t ) = B ( P ; ) ( t ). 0 0 Let M = max P P +, the,, P P M + cos Furthermore, we get. where

11 Subdvso of Uform ωb-sple Curves ad Two Proofs 7,, P P... M. + cos cos By Defto., we ow 0 whe ω s a real umber. Ad clearly cos cosh (,,..., = = ) whe ω s a pure magary umber. The we have M, s real ad 0 ; cos,, P + M, s magary ad 0. cosh P Because lm = the frst case, ad cosh the secod case, cos we have lm P P = 0 both cases. Therefore,, + lm P P = 0 (6) for ay,, + j From the covex hull property, for ay ( + ),..., + +,,...,, + j + +. ( 0 ; ) ( 0 ) B P t les wth the covex hull of Together wth (6), we coclude that ( ) 0 0 = lm S ;. P B P t -cotuous uform ωb- Theorem. ωbs curves of order ( ) coverge to sple curves. t 0,, we ow that P, P,..., P + for some.,,, + C

12 74 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 Proof. Based o Defto. ad Theorem., we ca coclude that ωbs curves of C order ( ) coverge to uform ωb-sple curves of order whose -cotuty are obvous accordg to the defto of ωb-sple bass fuctos ad paper [Wag, Che ad Zhou (004)]. So the cocluso holds. 4 Aother proof of C -cotuty of ωbs curves The proof Secto s smple. But ths proof method s dffcult to be exteded to the case of surface subdvso, especally o-tesor product surface subdvso. So we C provde aother proof method for -cotuty of ωbs curves based o those theores upo subdvso mass, whch wll be advatageous to be appled the proof of our further surface subdvso. From the steps of ωbs descrbed Defto., we ca see that the teso parameter s chagg wth the subdvso level, so ωbs s a o-statoary subdvso scheme. For coveece of provg ts cotuty, we troduce the correspodg otos of the mas of ωbs at frst. Gve a set of cotrol pots P 0 P 0 : j j =, a o-statoary subdvso scheme for curves defes recursvely ew sets of cotrol pots P = P : j formally by j ( ) + P = S P Each pot P + the ew pots, =0,,... s defed by a lear combato of pots P. The rule defg +. j j P + s deoted by m : m : = such that P = m P (7) The sequece m depedet of,.e. of coeffcets s called the subdvso mas at level. If m assumed that oly a fte umber of coeffcets m s = m, the correspodg scheme s sad to be statoary. It s m are o-zero so that chages a cotrol pot affect oly ts local eghborhood. The specfc rules of ωbs are gve as follows. Frst cosder the case =. For a gve teso parameteru 0 0,the ωbs scheme of order geerates a ew set of the cotrol pots by the rule ( S ) :

13 Subdvso of Uform ωb-sple Curves ad Two Proofs 75 4u ( S P ) P P = + + 4u 4u 4u ( S P ) P P = u 4u At each subdvso level, the parameters u are also updated as,. (8) u + = + u. (9) u = Sce u 0,we see that the sequece 0 0 coverges to as.whe u ( e u for ) s mootoe ad bouded such that t =.., = all, the 0 scheme becomes statoary ad t s the well-ow Cha s algorthm [Cha (974)]. I vew of (4.), the o-statoary mas m m for 4u 4u =,,,. 4u 4u 4u 4u P + ca be wrtte as ( ) Further, based o the rule (8), the scheme S for > s defed recursvely by ( ) ( ) ( ) ( S P + ) = ( S P ) + ( S P ) () j j j + It ca be easly checed that the support of the mas m s deed the same as the oe of the classcal B-sple of order [Stam (00)]. It s dffcult to drectly prove the cotuty of a d of o-statoary subdvso C scheme. So we prove -cotuty of ωbs curves accordg to the theorems cludg asymptotc equvalece proposed paper [Dy ad Lev (995)]. Here we cte the oto of asymptotc equvalece betwee two schemes defed paper [Dy ad Lev (995)]. Defto 4. A o-statoary scheme wth the mass m s sad to be asymptotcally equvalet to a statoary scheme wth the mas m f + m Lemma 4. Let m.assume that S m ( ) (0) () be a o-statoary subdvso scheme wth the mas ( ) S s asymptotcally equvalet to a statoary scheme S wth

14 76 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 the mas m. If S s q C wth q ad = 0 q m m, the the o-statoary scheme ( ) q S s also C. Theorem 4. The ωbs scheme of order wth the mas (0) geerates C -cotuous lmt curves. Proof. For a gve parameter 0 0 u u, we ca duce from (9) that = mootoe ad bouded sequece. To be more precse, f 0 u, the sequece 0 u = s a s mootocally creasg to. Otherwse, t s mootocally decreasg to. Hece, t s mmedate that m : =,,, m coverges to Ths s the mas of the Cha s corer cuttg algorthm ad t geerates C lmt curve (see Cha [Cha (974); Dy ad Lev (995)]). ow, to estmate the C - smoothess of the proposed scheme of order, t s ecessary to estmate the dfferece betwee m m ad m. From (9), we see that 4u m = max,. 4u 4 4u 4 If u =,the u =,. e., m = m.thus, what follows, we assume that u.i 0 0 fact, some elemetary calculato easly reveals that m - m = 4 -u The, followg Lemma 4., we oly eed to show that u m - m = 0 (4) for the proof of thec -smoothess of the proposed scheme of order. To ths ed, for smple otato, we use the abbrevato -u u U : = m - m = ()

15 Subdvso of Uform ωb-sple Curves ad Two Proofs 77 -u + Here, by (9), t s clear that u =u - +. Hece, U = u - U -u u + + = U u -u + -u u - = u = u u + u - + ( + u ) Cosequetly, sce u coverges to as, t follows that U + lm = U. Thus, we have Followg the D Alembert crtera for covergece of postve seres ad vew of (), the clam (4) s proved. We are ow ready to prove the smoothess of the proposed scheme of order >. I the followg aalyss, we wll see that t s coveet to represet a subdvso rule wth the mas : m j j j m z = m z Sce m terms of the symbol s ftely supported, the symbol m ( z ) s fact a Lauret polyomal. The followg lemma s obtaed from Dy et al. [Dy ad Lev (995)]. Lemma 4. Let S be a o-statoary subdvso scheme assocated wth the a symbol of the form a ( z ) = ( + z ) b ( z ) ad the o-statoary scheme correspodg to q scheme s C +. S a S b C + q s wth q C.The the Theorem 4. The ωbs scheme of order geerate -cotuous lmt curves. Proof. We prove ths theorem by mathematcal ducto for. The case = deed holds mmedately by Theorem 4.. For the case >, we use the otato

16 78 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08,, j m = m : j for the mas of the ωbs scheme of order at level. By costructo, the mas, m ca be teratvely obtaed by usg the equato ( + ),,, = + j j j m m m (5) whch s a mmedate cosequece of the relato (). Accordgly, the Lauret polyomal assocated to the mas where m ( z) m, m ca be wrtte as z + m z : = z, >. s the symbol of the ωbs scheme of order wth the mas, m By Theorem 4., the scheme assocated to the Lauret polyomal m ( z) (0). s C. Hece, applyg Lemma 4. ductvely, we ca coclude that the proposed scheme of order s C. The approxmato order of the proposed o-statoary subdvso s also mportat. I the followg, we dscuss ths problem. Theorem 4. shows that t s of approxmato order -, where refers to the order of the correspodg ωb-sples. [ ] Theorem 4. For the ωbs scheme S of order, the approxmato order of ths o-statoary subdvso s -. Proof.Based o Lemma 4., the proposed o-statoary subdvso scheme s asymptotcally equvalet to a statoary scheme S. S coverges to ωb-sples of order, wth a costat frequecy sequece, whch ca reproduce polyomals of order -. Accordg to the results cocluded paper [Cot, Dy, Ma et al. (05); Cot, Roma ad Yoo (06)], we ow a o-statoary subdvso mples approxmato order - (- refers to the degree of ωb-sples) asymptotc smlarty to statoary scheme s assumed. So based o the above, the cocluso of Theorem 4. s proved. 5 Cocluso I ths paper, we proposed the subdvso scheme for uform ωb-sple curves. The a uform ωb-sple curve has both perfect mathematcal represetato ad effcet geerato method. We also provde two proofs of C -cotuty ωbs curves of - order two dfferet aspects ad dscuss ts approxmato order. The frst method s drect ad smple. The secod d of proof s based o subdvso mass ad some correspodg theores, whch wll be advatageous to prove the correspodg coclusos of surface subdvso. I the future, we wll exted the subdvso scheme to the case of surfaces wth tesor product form ad further arbtrary topology as well. I addto, we wll apply ωb-sples ad especally the subdvso scheme the all ds of applcatos relatve to fte elemet method (FEM) ad sogeometrc aalyss (IGA) to mprove the accuracy durg modelg ad aalyss [Wag, She, Zou et al. (08);

17 Subdvso of Uform ωb-sple Curves ad Two Proofs 79 Guo ad ar (07); Xu, Su, Xu et al. (07)]. Acowledgemet: The wor descrbed ths artcle s partally supported by the atoal atural Scece Foudato of Cha (67764, ) ad the atural Scece Foudato of Zhejag Provce (LY7F0005). Refereces Adersso, L. E.; Stewart,. F. (00): Itroducto to the mathematcs of subdvso surfaces. Socety for Idustral ad Appled Mathematcs, Phladelpha. Catmull, E.; Clar, J. (978): Recursvely geerated B-sple surfaces o arbtrary topologcal meshes. Computer Aded Desg, vol. 0, o. 6, pp Cha, G. M. (974): A algorthm for hgh speed curve geerato. Computer Graphcs ad Image Processg, vol., o. 4, pp Cot, C.; Dy,.; Ma, C.; Mazure, M. L. (05): Covergece of uvarate ostatoary subdvso schemes va asymptotc smlarty. Computer Aded Geometrc Desg, vol. 7, o. 6, pp. -8. Cot, C.; Roma, L. (0): Algebrac codtos o o-statoary subdvso symbols for expoetal polyomal reproducto. Joural of Computatoal ad Appled Mathematcs, vol. 6, o. 4, pp Cot, C.; Roma, L.; Yoo, J. (06): Approxmato order ad approxmate sum rules subdvso. Joural of Approxmato Theory, vol. 07, o., pp Cot, C.; Cotroe, M.; Sauer, T. (07): Covergece of level-depedet hermte subdvso schemes. Appled umercal Mathematcs, vol. 6, o., pp Doo, D.; Sab, M. (978): Behavour of recursve subdvso surfaces ear extraordary pots. Computer Aded Desg, vol. 0, o. 6, pp Dy,. (99): Subdvso schemes computer-aded geometrc desg. I: Advaces umercal aalyss: Volume II: Wavelets, subdvso algorthms ad radal bass fuctos. Claredo Press, Oxford. Dy,.; Lev, D. (995): Aalyss of asymptotcally equvalet bary subdvso schemes. Joural of Mathematcl Aalyss ad Applcato, vol. 9, o., pp Fag, M.; Wag, G. (008): ωb-sples. Scece Cha Seres F: Iformato Sceces, vol. 5, o. 8, pp Guo, Y.; ar, J. (07): RETRACTED: Smulato of Dyamc D crac propagato wth the materal pot method. Computer Modelg Egeerg & Sceces, vol., o. 4, pp Jea, M. K.; Shumugaraj, P.; Das, P. C. (00): A o-statoary subdvso scheme for geeralzg trgoometrc sple surfaces to arbtrary meshes. Computer Aded Geometrc Desg, vol. 0, o., pp Jea, M. K.; Shumugaraj, P.; Das, P. C. (00): A subdvso algorthm for trgoometrc sple curves. Computer Aded Geometrc Desg, vol. 9, o., pp

18 80 Copyrght 08 Tech Scece Press CMES, vol.5, o., pp.6-80, 08 Maar, E.; Pe a, J. M. (00): A bass of C-Bezer sples wth optmal propertes. Computer Aded Geometrc Desg, vol. 9, o. 4, pp Ma, C.; Pelos, F.; Speleers, H. (0): Local herarchcal h-refemets IgA based o geeralzed B-Sples. Iteratoal Coferece o Mathematcal Methods for Curves & Surfaces, vol. 877, o. 4, pp Stam, J. (00): O subdvso schemes geeralzg uform B-sple surfaces of arbtrary degree. Computer Aded Geometrc Desg, vol. 8, o. 5, pp Vasov, B. K.; Sattayatham, P. (996): GB-sples of arbtrary order. Joural of Computatoal ad Appled Mathematcs, vol., o., pp Wag, C.; She, Q.; Zou, Y.; L, T.; Feg X. (08): Stffess degradato characterstcs cestructve testg ad fte-elemet aalyss of prestressed cocrete T- beam. Computer Modelg Egeerg & Sceces, vol. 4, o., pp Wag, G.; Che, Q.; Zhou, M. (004): UAT B-sple curves. Computer Aded Geometrc Desg, vol., o., pp Xu, G.; Su,.; Xu, J.; Hu, K.; Wag, G. (07): A ufed approach to costruct geeralzed B-Sples for sogeometrc applcatos. Joural of Systems Scece & Complexty, vol. 0, o. 4, pp Zhag, J. (996): C-curves: a exteso of cubc curves. Computer Aded Geometrc Desg, vol., o., pp