A NEW FIVE-POINT BINARY SUBDIVISION SCHEME WITH A PARAMETER

Transcription

1 Jourl of ure d Appled Mhemcs: Advces d Applcos Volume 9 Numer ges -9 Avlle hp://scefcdvcesco DOI: hp://dxdoorg/6/ms_9 A NEW FIVE-OINT BINARY UBDIVIION CHEME WITH A ARAMETER YAN WANG * d HIMING LI chool of Mhemcs d scs Hefe Norml Uversy Hefe 6 R Ch e-ml: wy@6com chool of Compuer d Iformo Hefe Uversy of Techology Hefe 9 R Ch Asrc A ew fve-po ry sudvso scheme wh prmeer s proposed By choce of ppropre prmeers some exsg sudvso schemes such s he four-po ry erpolg scheme he qurc B-sple scheme d clss of fve-po ry pproxmg scheme c e oed I s show h he resulg curves re C couous for he cer rge of he prmeer Mhemcs uec Clssfco: 6D A Keywords d phrses: ry sudvso smoohess * Correspodg Auhor Receved My cefc Advces ulshers

2 YAN WANG d HIMING LI Iroduco udvso mehod hs ecome powerful ool o cree curves d surfces my felds such s CAGD CG d geomerc modellg ecuse of s effcecy d smplcy I ccordce wh wheher he lm curve d surfce pss hrough he l corol verex sudvso c e dvded o erpolory sudvso d pproxmg sudvso The ypcl erpolory sudvso schemes clude he four-po ry erpolg scheme [ ] he sx-po ry erpolg scheme [] he hree-po erry erpolg scheme [] he fourpo erry erpolg scheme [] ec The represeve pproxmg sudvso schemes clude Ch corer cug scheme [6] d he B-sple scheme ec Geerlly speg whe he legh of he ms s equl he smoohess of he erpolory sudvso s rrely s good s he pproxmo sudvso ecuse of he lmo of he erpolo codo However erpolory sudvso c e used o cosruc he lmg curve pssg hrough ll he verces of he l polygo As such s very mpor o sudy he mehod of ufyg erpolory sudvso d pproxmo sudvso We prese ew fve-po ry sudvso scheme wh prmeer hs pper d dscuss he couy of he lm curve The res of he pper s orged s follows eco roduces he prelmres I eco fve-po ry sudvso scheme wh prmeer s roduced I ecos he couy of he lm curve s dscussed eco cocludes he pper wh umercl exmples

3 A NEW FIVE-OINT BINARY UBDIVIION relmres Gve se of l corol pos { p } le { p } e he se of corol pos level ( ) recursvely y he followg ry sudvso rules: p Defe { } p where he fe se { } s clled he ms The symol of he scheme s defed ( ) Theorem ([]) Le ry sudvso scheme e coverge The he ms ( ) ssfes p () () Theorem ([]) Le sudvso scheme wh ms { } ssfy () The here exss sudvso scheme (frs order dfferece scheme of ) whch ssfes he propery d d { } where d ( d ) ( p p ) The symol of s () ( ) ( ) Geerlly f exss d { } s he -h order dfferece scheme of wh ms ( ) ( ) he he symol of s ( ) ( ) ( ) ( )

4 YAN WANG d HIMING LI Theorem ([]) Le sudvso scheme hve ms ( ) ( ) { } d s -h order dfferece scheme ( ) exs d hve he ms ( ) ( ) { } ssfyg ( ) ( ) () If here exss eger L such h L he he sudvso scheme s C couous where ( ) [ ] : mx L L L L [ ] ( ) ( ) ( ) ( ) ( ) ( ) L Especlly whe ( ) ( ) mx L A New Fve-o Bry udvso cheme A ew -po ry sudvso scheme s defed s follows: () where ;

5 A NEW FIVE-OINT BINARY UBDIVIION Ad s eso prmeer I he cse whe we c ge he scheme y Duuc [] Whe he scheme () reduces o he qurc B-sple scheme 6 6 Whe he fve-po ry pproxmg scheme [9] c e oed moohess Alyss I hs seco we use Theorem o lye he covergece d couy of he sudvso scheme () Theorem The lmg curve geered y he sudvso scheme () s C couous he rge 9 C couous he rge 9 C couous he rge d C couous whe

6 YAN WANG d HIMING LI 6 roof The geerg polyoml for he ms of he sudvso scheme () c e wre s: ( ) I s esy o verfy h ( ) ssfes Equo () Accordg o Theorem we hve he geerg polyomls for ( ) s follows: () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6 ( ) ( ) I s esy o verfy h ( ) ( )( ) ssfy Equo () Ad whe 9 we hve mx

7 A NEW FIVE-OINT BINARY UBDIVIION Whe 9 we hve mx Whe we hve mx Whe we hve ( ) ( ) Ad whe ; whe { } mx

8 YAN WANG d HIMING LI Cosequely follows from Theorem h he proof of Theorem s compleed Fgure A ew fve-po ry sudvso scheme wh prmeer (ousde o sde: 6 )

9 A NEW FIVE-OINT BINARY UBDIVIION 9 () () (c) (d) (e) (f) Fgure A ew fve-po ry sudvso scheme wh prmeer ( )

10 9 YAN WANG d HIMING LI () () Fgure Frcl curves Numercl Exmple d Coclusos I hs seco hree exmples re depced Fgures d o llusre our scheme I Fgures d for he sme l corol polygos g dffere vlues of prmeer we c ge dffere erpolo d pproxmo curves wh dffere couy I he process of mppg we fd whe eso prmeer e some specl vlues he lm curve wll produce frcl ehvour Fgure shows wo exmples of pproxmo of frcl lm curves produced y pplyg our sudvso scheme fer sx sudvso seps sed o he sme l corol polygo The curves Fgure () d () re oed wh d respecvely I hs pper ew fve-po ry sudvso scheme s preseed d lyed We dscussed he uform covergece d couy of hs sudvso scheme y usg geerg polyoml mehod The exmples llusre h our proposed scheme gves gre flexly o geomerc desgers whe desgg smooh curves ccordg o her ow requremes y choosg ppropre prmeers Oe of our furher wor wll e med he sudy of he codos of he scheme s covexy preservo whe les cer egve ervl Exedg he curves sudvso scheme o surfces s lso our eresed opc

11 A NEW FIVE-OINT BINARY UBDIVIION 9 Acowledgemes Ths wor s suppored y he pecl Fuds for Youg cholrs of Hefe Norml Uversy uder Gr No QN pecl hs lso go o he referees Refereces [] Duuc Ierpolo hrough erve scheme Jourl of Mhemcl Alyss d Applcos () (96) - DOI: hps://doorg/6/-x(6)9-6 [] N Dy D Lev d J A Gregory A -po erpolory sudvso scheme for curve desg Compuer Aded Geomerc Desg () (9) -6 DOI: hps://doorg/6/6-96()9-x [] A Wessm A 6-o Ierpolory udvso cheme for Curve Desg Mser s Thess Tel-Avv Uversy 99 [] M F Hss d N A Dodgso Terry d Three o Uvre udvso chemes : Aler Cohe Je-Lous Merre d Lrry L chumer (Edors) Curve d urfce Fg: -Mlo Nshoro ress Brewood () 99- [] M F Hss I Ivrssms N A Dodgso d M A A erpolg -po C erry sory sudvso scheme Compuer Aded Geomerc Desg 9() () - DOI: hps://doorg/6/6-96()-x [6] G M Ch A lgorhm for hgh-speed curve geero Compuer Grphcs d Imge rocessg () (9) 6-9 DOI: hps://doorg/6/6-66x()9- [] N Dy d D Lev udvso schemes geomerc modellg Ac Numerc () - DOI: hps://doorg//9699 [] H heg Ye d H ho A clss of four-po sudvso schemes wh wo prmeers d s properes Jourl of Compuer-Aded Desg d Compuer Grphcs 6() () - ( Chese) [9] J T X hug d L hg A ew four-po shpe-preservg C sudvso scheme Compuer Aded Geomerc Desg () () -6 DOI: hps://doorg/6/cgd g