VISUALIZATION OF TRIVARIATE NURBS VOLUMES

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1 ISUALIZATIO OF TRIARIATE URS OLUMES SAMUELČÍK Mat SK Abstact. I ths pap fcs patca st f f-f bcts a ts sazat. W xt appach f g cs a sfacs a ppa taat s bas z a -sp xpsss. O a ga s t saz g paatc s. Th sazat s by appxat f s by t f spaatc cs sfacs a by bay aat f s. Rg s by OpGL patf a th a s sa tchqs f hac sa ffcts taspacy. W as pst pactca tpt bas ptat f pps sts. Ky s. Gtc g URS s g OpGL Itct F-f cs a sfacs a a scb pat f athatca bacg th f f Cpt-A Gtc Dsg [2] []. I ay cass th paatc xpss th pya fcts s s. F btt pstat f bcts sph c ata xts th f f ata fct ca b app []. Th a as ay s sazat f ths f f bcts. O ss appxat f bct by pys taga shs. Dfft sazat pt s aytacg h st ptat pat s t f tsct f a g bcts. T f ths f tscts ca s sa agths t fg sbs z cppg. I sach a tyg t xt pya cs a sfacs t taat pya bcts s bas z xpss [5]. W ca as ppa xts th f f ata s a ata sps. F sps s - sp pstat - sps. Ths s ca b s ay pats f gtc sg f xap as g f th-sa fats f scatt ata tpat f pstat f s bcts [7]. I sg t s y ptat t saz bcts. W pps s basc pts h t sh ths bcts [6]. F g D bcts 2D sc s Op GL gaphcs bay bcas t s sppt ay patfs a ay haa sts [4].

2 2 Taat s Ga taat bct s f by taat fct F a th a D f that fct. Pcsy: D F TO E R TO > : I th f f gtc g pya pcs pya sp fct s st cy s. W th y ths pstat f by t f ct pts. Th ata éz ttaha s f th a g a ct t f pts a f ach pt a b ght. Th g s a pst tg b a as g f pyas s t b g pts Th a s a gat ttah ACD E hch s s f assg f bayctc cats a ct t th ghts s ttaha stct f pts E that ca b tt fg ay:. R E Lt s ha pt U f th a a t tt a bayctc cats f pt U UACtD th spct t ACD. ca f pt f ata éz ttaha R th cs Casta agth: 4 2 t R t h - a 2 4. F ths ft th aaytca xpss f ata éz ttaha ca b aat. S f bayctc cats f ay pt U f th a ha: R h t 2

3 a gaz st pyas. Rata éz ts s f th th gs a a ct t f pts a f ach pt a a b ght. Th gs a pst tgs a pst th g f bg pyas ach ct th a s gat bx ACDEFGH E a th ct t th ghts s bx stct f pts E that ca b tt fg ay: E R Ass that ha pt U f a a t a cats f U th spct t ACDEFGH s. ca f pt f éz ts R th aaytca xpss: h R a sp st pyas. F ths typ f as xsts Casta agth bt t s ss gazab. -f ata -sp URS s a a ata xts f URS cs a sfacs s gtc g. Paats that f URS a sa t th sfac cas bt th a paats t a paat ct. URS s f th : th gs th -casg t cts th-sa t f ct pts E << << << f ach ct pt a b ght p a < x< x< Th URS s g aaytcay as

4 p p S h a -sp bg fcts f patca t cts th g gs. Paats a f a < < <. sazat I ths sct scb basc sts f sazat f taat f-f bcts. cas a ag th paatc scpt f bcts ay agths ca b s pya ata sp cass. Th as xsts sp agth f ctg URS t st f z s. Cs agth s bas - agth f t st f URS bcts. Ths pcss s stat Fg b. cas taat s pst as t f th bct ha t cat s sght t t. Ths as t s appxats ss f sazat th bay. I s agths y bay f b t. a b Fg : Sph spay th spaatc cs a sfacs: a Sp URS b cps t f z s.. Ispaatc cs Ispaatc cs a cs gat f paatc fct that fs paatc. Ths c s aat f-f bct f taat fct by ag t f th paats as cstats. F xap th cas f z ts ha fg st f cs ct: 4

5 } { > < C C Ths st ctas ft b f cs s ha t pc s ft b f sap cs. Th basc chcs a cs h fy sap ta <> f as a. Ths pcss f gatg spaatc cs ca b pat a ct. Fg 2b shs st f sazat f sa cs f a cts th cas f z ttaha. a b Fg 2: a z ttaha b z ttaha th spay spaatca cs.2 Ispaatc sfacs Jst th cas f spaatc cs spaatc sfacs a gat f paatc xpss by pttg f th paats as cstat. cas ths cstat paat s f cts ta gt ft b f sfacs. W t chs s ft b f pstat sfacs ths s by sp f sapg f a ft ta. Pcsy f ata z ts chs a saz fg sfacs f ct f ct sfacs ca b t say: } { > < C C F h g t s t cssay t spaatc sfacs a cts t s ft sffct t chs y ct. 5

6 a b Fg : a URS th a ct pts b URS spay sg spaatc safcs. ay aat I ay appcats sazat f t s t cssay. Th bay f s sffct sazat t. Th bay f taat bct csst f bay spaatc cs f xta paats tgth th pat f ss h Jacb tat s z: R R R U U U R R R J U U U U R R R U U U Wh cb ths t pts ca h bay f taat bct as Fgs a..4 Taspacy It s pssb t a spaatc sfacs f th sap paats pst sct.2. I that cas y appxat bay f h b sb Fg X. F sght t t f bct ca s taspacy f gat spaatc sfacs. Rg f taspacy s sg basc fats f OpGL g g. Fg 4 shs URS sg ab taspacy. 6

7 a b Fg 4: a URS th a ct pts b URS spay sg spaatc safcs 4 Rsts & Ft W W ha pt a pps sazat pts f taat bcts. O syst ca GFg ca a saz s asy a ffct ay. A fgs pst ths pap sg ths syst. W ha tst fa ats f sazat AMD Sp syst th.6ghz a G sz f y. Us gaphcs ca as ATI Ra X7 th 28M f gaphcs y. Tab shs ths ats f as bcts a sttgs. I th ft at t fcs aytacg ths sg appxat ths a z cppg agth. W ha as s s fats xact stg f gty f pp taspacy f URS s. M Ispaatc cs Ispaatc sfacs Taspacy Sph 59 FPS 6 FPS 5 FPS Ts 6 FPS 4 FPS 5 FPS Ra 2 FPS 8 FPS 9 FPS Tab : Fa ats f sazat pts f sa URS s. Acgt Th pap as sppt by gat f EGA Gat Agcy ctact. /8/6 th tt Cpxty f Gtc Agths f Rat Rg ta Raty. Rfcs [] Jy K. Dchaa M. ay tat f taat ss Pc. f Pacfc Gaphcs '99 pp 82-9 Oct 999. [2] Fa G. Cs a Sfacs f Cpt-A Gtc Dsg - A Pactca G. IS Acac Pss st 99. [] Pg L. T W. Th URS. IS Spg-ag Y

8 [4] as. OpGL hpag [5] Dt Lass. st - éz pstat f s. Cpt A Gtc Dsg 2-: [6] Dt Lass. sazat f f-f s. I IEEE sazat pags [7] R MacCac a Kth I. Jy. F-f fats th attcs f abtay tpgy. I SIGGRAPH '96: Pcgs f th 2- aa cfc Cpt gaphcs a tact tchqs pags 8-88 Y Y USA 996. ACM Pss. RD. Mat Sačí Dpatt f App Ifatcs Facty f Mathatcs Physcs a Ifatcs Cs Usty Mysa a atsaa a: sac@fph.ba.s 8

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