Conversion of Binary Space Partitioning Trees. to Boundary Representation. Abstract. Binary Space Partitioning Trees (BSP-Trees) have been proposed

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1 Conversion o Bry Spe Prtitiong Trees to Bounry Representtion Jo~o Com 1 n Brue Nylor 2 1 Computer Siene Deprtment, Stnor University, USA? 2 Sptil Ls, In.?? Astrt. Bry Spe Prtitiong Trees (BSP-Trees) hve een propose s n lterntive wy to represent polytopes se on the sptil suivision prigm. Algorithms tht onvert rom Bounry Representtion (BRep) to BSP-Trees hve een propose, ut none is known to perorm the opposite onversion. In this pper we present suh n lgorithm, tht tkes s put BSP-Tree representtion or polytope n proues BRep s put. The iulty esigng suh lgorithm omes rom the t tht the ormtion the ounry is not expliitly represente the BSP-Tree. The solution we present volves reursive trversl o the tree to ompute lower imensionl ormtion, long with glug lgorithm tht ome the onvex regions ene y the BSP-Tree, removg ternl etures. A new t struture is propose ( Topologil BSP-Tree), tht ugments the tritionl BSPtree with topologil poters n is use to store termeite results use the reonstrution o the BRep. 1 Introution Bounry representtion (BRep) is wiely use representtion o soli geometry, se on the esription o n ojet y its ounry, s olletion o es, eges n verties[?]. On the other hn, Bry Spe Prtitiong Trees (BSP-Trees) onsist o onvex hierrhil eompositions o the spe, where the ojet is represente y the union o onvex regions. The si opertion or the onstrution o this eomposition onsists o prtition o the unerlyg spe y given hyperplne. This prtition is represente s ry tree, where eh noe is ientie y hyperplne, n the let n right sutrees o the noe represent the two hlspes ote the prtition. The reursive pplition o this opertion retes onvex hierrhil eompositions o the spe. In Soli Moelg, BSP-Trees hve een use long with BReps, n mny lgorithms hve een propose, like the onversion rom BRep to BSP-Tree [?], or the one tht omputes oolen opertion with BSP-Trees [?]. In this pper we onsier the prolem o onvertg BSP-Tree representtion o polytope to BRep. This prolem is similr to the onversion o CSG to BRep (lso lle the Bounry Evlution), euse the BSP-Tree n? e-mil: om@s.stnor.eu, we: e-mil: nylor@sptil-ls.om

2 e rst onverte to CSG y omputg the union o ll the onvex regions tht the BSP-Tree enes, n pplyg over the result ny o the Bounry Evlution lgorithms propose the literture([?], [?]). However, this pproh oes not exploit the sptil suivision ormtion tht the BSP-Tree enoes, whih my le to more eient lgorithms. In orer to use this ormtion, we propose n lgorithm tht onverts the BSP- Tree to BRep y workg iretly the struture o the BSP-Tree. In Fig e e () () () Fig. 1. () BSP-Tree, () Prtition ue y the BSP-Tree. The resultg ojet is ompose o 3 ierent regions (4, 5, 6), orrespong to the IN noes () Computg the ounry requires glug ierent regions n removg ternl elements we illustrte the prolem o reonstrutg the ounry o n ojet ote y prtition ue y the BSP-Tree. In this exmple the union o regions 4, 5 n 6 (Fig. 1) orm the ojet. Lookg t the exmple we ientiy some iulties the esign o the onversion lgorithm: { The BSP-Tree oes not hve lower imensionl ormtion: Intersetions nee to e ompute to ot the es, eges n verties on the ounry. { The BSP-Tree oes not hve jent-topologil ormtion. { The BSP-Tree represents onvex eomposition o solis s onvex regions, n the ounry my e prtitione multiple epenent omponents tht re not miml. In the ollowg setions we present how we solve eh o the ove prolems, n give the lgorithm to ompute BRep rom BSP-Tree. Initilly we review

3 some si onepts BSP-Trees. The lgorithm tht omputes the BRep rom BSP-Tree is isusse next, where we esrie the Topologil BSP-Tree (TBSP-Tree), n uxiliry t struture tht stores termeite results n topologil reltions mong them. 2 Review o BSP-Trees 2.1 Bsi onepts Bry Spe Prtitiong trees (BSP-Trees) re sptil serh strutures use mny ierent spets o Computer Grphis n Geometri Moelg. Applitions Soli Moelg [?] [?] [?] [?], Visiility Orergs [?] [?] [?] n Imge Representtions [?], mong others, n e oun the literture. In orer to esrie the onepts o BSP-Trees, it is lwys nie to rst expl the reltion it hs with Bry Serh Trees. Bry Serh Trees hve een use Computer Siene mny wys, ut mostly s t struture to elerte serh queries se symoli vlues. A geometri terprettion o this t struture is hierrhy o ry prtitions o the rel le, where the prtitioner is pot n eh prtition ote represents n tervl. The prolem with this terprettion is tht it oes not iretly generlize to higher imensions, s pots o not prtition suh spes. The misleg ormtion with this terprettion is tht the prtitioner tht ws suppose to e pot is t hyperplne. In generl, or D-imensionl spe the prtitioner orrespons to the hyperplne to tht spe ( (D-1)-D element), n the prtition hs the sme imension o the unerlyg spe. BSP-Trees n Prtition Trees [?] use this nlogy to exten the onepts o Bry Serh Trees to higher imensionl spes, ut we hoose to work here with BSP-Trees s they hve more nturl orresponene with the representtion o solis thn Prtition Trees. One vntge o BSP-Trees is the ilityto ome serh struture with representtion sheme one unique t struture. The use o BSP-Trees s soli representtion sheme revels one pplition where this omtion is well exploite. Solis re represente usg BSP-Trees y union o onvex regions, whih re ientie y ssoitg ttriutes to the leves o the tree. These regions my e either sie or sie the soli, n re ene y the hyperplnes tht re the pth rom the root to the le noe. In orer to preisely ene the region, the leves hve ssoite n ttriute tht ites tht the region is sie or sie the soli. 2.2 Forml enitions Mny lgorithms BSP-Trees re etter exple i we represent the m onepts ormlly. In this setion we present some enitions n properties tht re gog to e use the pper.

4 Denition1. Hyperplnes n Hlspes: The hyperplne is the si element to reursively prtition the spe, n it is esrie y the ollowg eqution: h = (x 1 ::: x ) j 1 x 1 + ::: + x + +1 = 0g The hyperplne seprtes the spe two hlspes, the positive n the negtive hlspe. Eh o them is expresse s: h + = (x 1 ::: x ) j 1 x 1 + ::: + x + +1 > 0g h ; = (x 1 ::: x ) j 1 x 1 + ::: + x + +1 < 0g The norml o the hyperplne is ene y thevetor ( 1 2 ::: ). The positive hlspe orrespons to the one tht lies the iretion o the norml. Denition2. BSP-Tree Noes n Leves: A BSP-Tree noe represents the ormtion o the ry prtition eg perorme on the spe. It onsists o prtitiong hyperplne, n let n right hilren tht pot to BSP-Tree representtions o the positive n negtives hlspes. The hyperplne tht enes the noe n is enote with H(n). A BSP-Tree le onts ttriutes ssoite to given region. It onts the lels IN (i the region is sie the soli), or OUT (i the region is sie the soli), ut my lso ont itionl ttriutes, like olor or ensity. Denition3. Region Pth: A Region Pth orrespons to the pth tht les rom the root o the tree to nother noe o the BSP-Tree. This pth represents one given prtition o the spe, n is represente y n orere list o noes L. RP (n) = L = root ::: prent(prent(n)) prent(n) ng Denition4. Region: A region o given noe represents the geometri terprettion o the prtition ene y the region pth RP(n). It orrespons to the tersetion o ll positive hlspes the region pth, n n e ormulte s: R(n) = \H () j H () 2 RP (n) =+ ;g Denition5. Su-hyperplne: A su-hyperplne s o hyperplne h t given noe n is ene y the tersetion o the hyperplne h with the region R(n) ene the noe. s = SUB(h n) = h \ R(n)g Denition6. Projete Hyperplne: A projete hyperplne p is one imension lower hyperplne tht is ote y projetg the tersetion o two hyperplnes h1 n h2 orthogonlly to one o the oorte xis.

5 Denition7. Pth Prtil Orerg p :Leth 1 n h 2 e two hyperplnes ommon region pth p rom the root o the tree. We eneh 1 p h 2 i h 1 is the let sutree o h 2. Otherwise, h 2 p h 1. 3 Computg the BRep rom BSP-Tree The BSP-Tree represents solis s the union o onvex regions, n the ounry ssoite with eh o these regions is ene y the tersetion o hlspes the tree. Eh onvex region is ientie y two ttriutes: le noe l lele s IN (i.e. not empty), n region pth RP(l) ssoite with l. The ounry o this region is ote y the tersetion o the hyperplnes its orrespong region pth. In the simple se where the BSP-Tree onsists o sgle onvex region we nee to onsier only one region pth, n to ot the ounry we ompute the tersetion o ll hyperplnes the region pth (Fig. 2). R R = Fig. 2. Simple BSP-Tree genertg only one onvex region In more omplite BSP-Tree, with more thn one onvex region, it is likely tht some o the region pths ssoite with these regions will shre noes the tree. In Fig. 3, region pths o the regions R1, R2, R3 n R4 shre the noes n, n the tersetion o H() n H() is on the ounry o ll these regions. In t, ny noe the tree my ontriute to the ounry o ll regions tht onts the noe its region pth. A relte onsequene o this t is tht when omputg given region we my not nee to ompute the tersetions o ll hyperplnes the region pth, s some my not ontriute to the region (reunnts to the region), The shrg o noes mong ierent regions implies tht the omputtion o tersetions to n the ounry o regions my require repete omputtions. This t revels struture ommon Dynmi Progrmmg (DP) prolems. In eh noe o the tree we n prtition the prolem o omputg the ounry to smller prolems, orrespong to the ounry the positive

6 e e e R1 R3 R2 R4 R1 = e- R2 = R3 = e- R4 = Fig. 3. Complex BSP-Tree genertg more thn one onvex region n negtive hlspes o the noe. In orer to ot suh ounries, we nee to ompute tersetions mong hyperplnes eh o the sutrees (seprte suprolems) n nestor hyperplnes (shre suprolems), whih ienties the DP struture. One wy to voi re-omputtion o ommon tersetions shre y suprolems is to store them termeiry t strutures. The wy o representg these tersetions, here lle lower imensionl ormtion, is esrie etils the next setion. 3.1 Storg lower imensionl elements One metho o representg lower imensionl elements BSP-Tree relies on usg BSP-Trees o lower imension, whih results struture lle multi imensionl BSP-Tree. In 1990, Nylor [?] propose ut i not elorte the use o pure BSP-Tree moel to represent solis. The proposl ws se the extension o the stnr BSP-Tree moel to represent expliitly the multi-imensionl ormtion ene y the struture o the tree. In 1991, Vneek [?] use similr ies n propose the BRep-Inex, whih onsiste o multi-imensionl BSP-Tree (lle MSP) tthe to BRep, with the gol o provig eient ess to the BRep strutures. The BRep-Inex is onstrute suh wy tht there is orresponene etween (0,1,2)- noes o the MSP with the verties, eges n es o the BRep. In this pper, we represent the lower imensionl ormtion ote ierent k o multi-imensionl BSP-Tree. In the BRep Inex o Vneek the topologil ormtion is store the BRep struture n not the MSP. In the t struture propose this pper, the topologil ormtion is store multi imensionl BSP-Tree, ugmente with itionl topologil

7 lower(,-) lower(,-) lower(,-) Topologil Lks Lower Dimension Lks Fig. 4. Topologil BSP-Tree (TBSP-Tree) exmple 2D poters onnetg elements topologilly jent. We ll this t struture Topologil BSP-Tree (TBSP-Tree). The motivtion or retg suh struture omes rom the t tht orer to ot lower imensionl ormtion we must ompute the tersetion o hyperplnes, whih gives ert ormtion the wy tht elements re topologilly relte. In the exmple o Fig. 4, when omputg the tersetion o the les n we ot pot tht we nee to sert to the lower imensionl BSP-Trees ssoite with n. In other wors, is topologilly jent to y. In orer to preserve this ormtion, we onnet the noes we ot this tersetion with topologil poters. Besies the ition o these topologil poters, the TBSP-tree is ierent rom the MSP o Vneek euse we keep not only one, ut two poters to lower imensionl imensionl BSP-Trees, orrespong to the suivisions orme over oth sies o the hyperplne. One reson or this hoie lues the simplition o the rementl lgorithm to uil the topologil BSP-Tree, whih requires prtil orerg mong the hyperplnes, tht n e simplie i we proess the suivisions oth sies seprtely.

8 3.2 Nvigtg the TBSP-Tree The TBSP-Tree stores topologil ormtion the tersetions ompute pre-orer trversl o the tree. This is hieve y retg opies o the sme tersetion n onnetg them with topologil poters. In generl, or eh tersetion o twohyperplnes h 1 n h 2 wekeep three opies the TBSP- Tree. The noe higher lote the tree reeives one opy, store the lower imensionl tree eg use the urrent region pth. The other two opies re store the two lower imensionl trees ssoite with the other noe. These lst two opies re onnete to the rst opy rete y topologil poter. In the se where h 1 p h 2, the rst opy is store the negtive lower imensionl tree o h 2, euse h 1 is the let sutree o h 2. The other two opies re store the positive n negtive lower imensionl trees o h1. Usg the ormtion store the TBSP-Tree we re le to reover the si elements o the Bounry Representtion. This n e illustrte y lookg t the prtil TBSP-Tree representtion or ue Fig. 5. The es o the ue re ene y the hyperplnes H(), H() n H(), whih re represente3dbsp-tree y the noes, n. The tersetion H() \ H() is represente the lower imensionl trees ssoite with n s n respetively. The eges tht elong to the e H() re ene the lower imensionl trees ssoite with noe, whih this se hs one empty tree, s the prtition ours only one o the sies o. In orer to trverse the es o the ue iretly rom the TBSP-Tree we mke use o the ollowg opertors: { TreePrent(noe) n TreeSon(noe,sie): Tritionl poters trees or prents n sons. { DimensionPrent(noe) n DimensionSon(noe,sie): Poters tht reet n iene reltion the imension o the spe. DimensionSon(noe,sie) returns poter to the lower imensionl BSP-Tree ssoite with the noe the spei sie, n DimensionPrent(Noe) returns n upperimensionl BSP-Tree ssoite with the noe. { TopologilCopy(noe,sie): The pplition o this opertors returns the topologil opy o noe t given sie. In Fig. 5, TopologilCopy (,-) returns the opy ;,whih orrespons to the sme tersetion, ut store the negtive lower imensionl BSP-Tree o. { TopologilNeighor(noe): In 1D it is possile to ene n jeny reltion. The Topologil neighor o noe orrespons to the next noe seng orer the rel le. This orer is ene terms o the projete hyperplne norml, n reets our onvention or the orienttion o loops the representtion o the ounry. In Fig. 5, the 1D-BSP-Trees re oriente y the hyperplne norml to eh eges. The iretion o the hyperplne norml is ene y the projete hyperplne n the urrent sie o the lower imensionl tree eg use.

9 e e e e e e e e 3D-BSP-Tree e 2D-BSP-Tree e e 1D-BSP-Tree e e e Fig. 5. Topologil BSP-Tree (TBSP-Tree) exmple 3D These opertors llow us to ess the ounry elements or the ue. In Fig. 6 we show the proeure to visit the elements o e (VisitFe). The pplition o this proeure to the le noe IN (next to noe e) will visit eges (e, e), (e, ), (, ) n(, e), whih orrespon to the eges o the e. 3.3 Inrementlly Computg the TBSP-Tree For every noe visite the trversl o the tree, we isover more ormtion how the BSP-Tree prtitions the spe. When new noe is rehe, we my ompute the tersetion o the hyperplne tht enes the noe gst ll hyperplnes the region pth o the noe. These tersetions re use to upte the lower imensionl BSP-Trees o ll the noes this region pth.

10 proeure VisitFe(1D-BSPTree noe1d) // The strtg noe orrespons to le ontg // n IN ttriute the 1D-BSP-Tree urrentnoe1d = strtnoe1d = DimensionPrent(noe1D) o nextnoe1d = TopologilNeighor (urrentnoe1d) VisitEge (urrentnoe1d, nextnoe1d) urrentnoe1d = TopologilCopy (urrentnoe1d) while (urrentnoe1d!= strtnoe1d) en VisitFe Fig. 6. Proeure to visit e usg the TBSP-Tree Every tersetion ote is projete onto one o the oorte hyperplnes eore sertion to the lower imensionl trees, orrespong to the onept ene eore o projete hyperplne. One wy to unerstn this opertion is to rememer the wy tht Gussin Elimtion (GE) solves ler systems. In GE, the solution o ler system volves rst tringultion step, where eh o the olumns uner the pivot (the igonl element) is elimte (reple y zeros). In t, t every elimtion step the olumns o the mtrix, whih n e terprete s the oeients tht ene hyperplne, re projete to one o its imensions. This is extly the sme opertion we re perormg here to ot the projete hyperplnes. In orer to ompute ll lower imensionl ormtion ue y the BSP-Tree on spei hyperplne we ompute the tersetions with the other hyperplnes, n projet these tersetions iretion orthogonl to one o the oorte xis. This hs the sme eet o sweepg with zeros olumn GE. For simpliity,rom now on when we reer to tersetions we re t reerrg to the projete hyperplnes o the tersetions. The upte step o the lower imensionl trees is perorme ter n tersetion is oun, onsistg o the sertion o projete hyperplnes lower imensionl trees. This sertion opertion is guie y prtitiong opertion, whih volves the lssition o hyperplne gst prtitiong hyperplne. Depeng on the result o this lssition, the tersetion etween the hyperplnes is ompute n the hyperplne eg serte is prtitione to two su-hyperplnes. The su-hyperplnes re reursively serte to the let n right sutrees o the prtitioner noe. However, we exploit one unreognize property o the BSP-Trees tht gurntees tht we nee to perorm this sertion to only one sutree o the noe. Suppose tht the BSP-Tree esrie Fig. 7 we wnt to ompute the lower imensionl ormtion ssoite with noe. Perormg pre-orer trversl o the tree we visit this orer noes n the tree. In this se tersets, thereore we ompute the tersetion n sert s noe

11 () () () Fig. 7. Simple se the rementl onstrution o the TBSP-Tree. The sttus o the lower imensionl BSP-Tree ssoite with ter the visit o noes, n is illustrte (), () n () to the negtive lower imensionl BSP-Tree o, s p. The next noe visite is, n the sertion o to the lower imensionl tree o volves prtition gst, to eie whih sieo we nee to sert. Usg the the pth prtil orerg ene y the tree, we oserve tht p, n thereore we must hve tht p, whih mens tht nees to e serte to the let sutree o. But it is esy to ormulte se where the sertion opertion n not e guie y suh prtil orerg. In Fig. 8, oes not terset. This is only possile euse is prllel to, s is hil o. Clerly, one o the sutrees o (let or right) oes not nee to e teste gst ( this se the right sutree). Thereore, even i noes n terset we o not nee to prtition one gst the other, s is shielg one o them rom tersetg. A more omplite exmple volves the se where some noe (not hil) oes not terset. Consier the se Fig. 9 where n e terset, ut oes not terset. The sertion o to the lower imensionl BSP-Tree ssoite with requires rst prtition gst. Usg s ove the prtil orer p we propgte the result to the let sutree o. Nowwe nee to prtition gst e, ut the pth prtil orerg is not ene etween e n. Note tht the t tht oes not terset tells us tht one o its sutrees oes not nee to e teste or tersetion gst, euse s eore n its nestors shiel one o its sutrees rom tersetg. In this se, n shiel rom tersetg Usg suh results, we my eie whih sie to sert given hyperplne when we perorm the prtitiong opertion. We prtition the hyperplne y

12 e e e () () () Fig. 8. First se o Prtil Orerg not ene. The sttus o the lower imensionl BSP-Tree ssoite with ter the visit o noes, n e is illustrte (), () n () e e e () () () Fig. 9. Seon se o Prtil Orerg not ene. The sttus o the lower imensionl BSP-Tree ssoite with ter the visit o noes, n e is illustrte (), () n () omputg the tersetion with the prtitioner, n propgte the result to the sie onsistent with the prtil orerg ene y the urrent pth. When le noe is rehe, we evlute the su-hyperplne prtitione, n i it is not emptywe rete new noe with unene let n right ttriutes, n upte

13 ll the 1D-BSP-Trees o the most reent prtitioner hyperplnes oun long the wy. One iulty this rementl onstrution is tht we only isover the ttriutes (IN or OUT) whenwe visit le noe. At thispot, we nee to propgte this ttriute to ll lower imensionl BSP-Trees the urrent region pth n rete le noes with the ttriute just isovere. In t this iulty is prtiulr se o the sertion tht we re perormg the other ses, only tht this se we re not sertg hyperplne ut n ttriute, tht n e lolize the tree usg the prtil orerg s ove. Another importnt opertion tht nees to e perorme when omputg tersetions n sertg them to lower imensionl BSP-Trees is to keep trk o the ple where we sert the ierent opies o n tersetion. For stne, the sme tersetion h 12 o h 1 n h 2 nees to e serte to the lower imensionl BSP-Tree ssoite with h 1 n with h 2.Topologil poters re rete to onnet the opies o this tersetion, whih will llow the reonstrution o the BRep next step. The proeure to ompute the lower imensionl ormtion is esrie Fig. 10. proeure ComputeLowerDimension(BSPTree *urrent, BSPTree *pth) or eh hyperplne h p the urrent pth i urrent hyperplne h is le // We upte the lower imensionl trees with the ttriute Insert(DimensionSon(h p, sie(h p )), urrent.ttriute, pth) else Compute tersetion h p etween h p n urrent h i h p is not empty // We sert the tersetion to the lower BSP-Trees // o h n hp n lk them with topologil lks Insert(DimensionSon(h, +), h p, pth) Insert(DimensionSon(h, -), h p, pth) Insert(DimensionSon(h p, sie(h p )), h p, pth) eni eni enor ComputeLowerDimension(urrent.let, pth [ urrent + ) ComputeLowerDimension(urrent.let, pth [ urrent ; ) en ComputeLowerDimension Fig. 10. Proeure to Compute Lower Dimensionl Inormtion

14 3.4 Usg the TBSP-Tree to reonstrut the Bounry The rementl onstrution o the TBSP-tree uilt ormtion topologil reltions mong the elements ll imensions. A noe the tree hs ll the ormtion neessry to reonstrut the ounry when oth lower imensionl trees hve een ompletely ompute. In orer to extrt the ounry we perorm glug opertion the imension o the emeg spe, whih remove eges ternl to es n es ternl to solis. In Fig. 11 we hve n exmple where the prtitions oth sies o the hyperplne h nee to e glue together to remove ternl elements. The glug opertion is responsile or omg the results rom the two sutrees suh wy tht the result is vli representtion or the ounry ene the sutrees o the noe. In orer to glue the ormtion the positive n negtive hlspes o noe, we perorm the symmetri ierene o the lower imensionl BSP-Trees oth sies o the hyperplne, whih is oolen opertion tht n e exeute y tree mergg lgorithm or BSP-Trees[?]). The result o the symmetri ierene opertion is tree whose IN regions re elements o the ounry o the ojet. h h+ h- Fig. 11. Glug opposite es to ot the ounry Note tht the glug proess to remove ternl etures entils reursion on imension tht rst glues lower imensionl etures. In other wors, orer to

15 glue two es we rst glue the elements ternl to eh o these es seprtely, whih will orrespon to the removl o ternl eges o e. The lgorithm itilly perorms two glug opertions the positive n negtive lower imensionl trees, n symmetri ierene opertion tht omes the results ote. The lgorithm or glug TBSP-Trees is imension-epenent, n the pseuo-oe is esrie Fig. 12. proeure GlueBSPTree(BSPTree noe, Dimension im) i imension > 1 tree1 = GlueBSPTree(DimensionSon(noe,+), im-1) tree2 = GlueBSPTree(DimensionSon(noe,-), im-1) eni return SymmetriDierene(tree1, tree2) en GlueBSPTree Fig. 12. Proeure to glue TBSP-Trees The glug proess n e implemente suh wy tht the ounry reltions o the ierent imensione elements is preverse. In Fig. 13 we illustrte the glug opertion or n exmple 2D. The omplite ses rise noes n, where we hve prtitions ue y the BSP-Tree oth sies o the hyperplnes. In Fig. 13 we show the prtition ue y the tree, n the ormtion store the TBSP-Tree is illustrte y the yles o eges n verties eh ell. When perormg the symmetri ierene we not only remove ternl etures, ut we lso jo the yles oth sies o the hyperplne. In Fig. 13 we show the result o the pplition o the glug opertion to the noe. The orrespong etures re ientie y perormg the symmetri ierene opertion or oth lower imensionl trees one imension lower, n the yles re joe together when we hve mutully ient reltions. In this se, the ege generte y the hyperplne is remove, n the yle o eges o oth opies o re joe together. In similr wy Fig. 13 shows the result o the glug opertion or noe. It is importnt to emonstrte tht the ounry ormtion ote is t miml. This is hieve ue to the t tht ll ternl etures re remove y the reursion imension perorme y the glug lgorithm. This is n importnt onsequene, euse the onvex eomposition rete y the BSP-Tree eomposition my generte rgmenttion o the ounry when non-onvex ojets re represente. The glug opertion, s propose, provies n elegnt solution to the prolem o reonstrutg the miml BRep rom BSP-Tree representtion o polytope.

16 e e g g () e e g g () () Fig. 13. Exmple o the glug Proeure 2D 4 Conlusions In this pper, we hve presente n lgorithm to onvert BSP-Tree representg polytope to BRep representtion. The storge o lower imensionl ormtion TBSP-tree llowe us to reonstrut the ormtion neessry to reover the ounry o the ojet. The TBSP-Tree is rementlly ompute urg pre-orer visit o the tree, whih omputes ll lower imension ormtion oth sies o eh noe. A glug step is perorme when ll ormtion noe is isovere. whih volves reursion the imension o the spe to remove ternl etures n glue the ounry representtion eh sie o the noe. As onsequene o this proess, the ounry representtion ote t the en orrespons to miml BRep. We elieve tht the importne o the BSP-Tree representtion n e extene y hvg suh lgorithm ville. BSP-Trees n BReps re oth representtion o polytopes use Soli Moelg, eh one with its own vntges. For exmple, BSP-Trees re more eient where visiility orergs or oolen opertions nee to e ompute, wheres some other pplitions,

17 like topologil eormtions, we woul preer to use the BRep. By hvg oth onversion lgorithms ville we n exploit the vntges o eh moel more eiently. Another pplition where the onversion lgorithm propose this pper n e use reers to the prolem o ng ner-optiml BSP-Tree representtion o polytope. The BSP-Tree representtion is not unique, n mny ierent trees my represent given polytope. The prolem o ng miml tree is even more importnt when we perorm suessive oolen opertions y pplyg tree mergg lgorithms, whih my generte trees tht nee to e re-struture. Unlike Bry Serh Trees, where we hve lgorithms to keep tree lne, the lng o tree higher imensions is muh morei- ult prolem. By pplyg the onversion esrie here we ot miml BRep, whih ggregtes the geometri ormtion expresse y the tree. The urther onversion rom BRep to BSP-Tree my generte muh more lne tree, s the topologil ormtion o the BRep gives etter heuristis the onstrution o the tree. Flly, the omtion o topologil ormtion the BSP-Tree, whih resulte the TBSP-Tree, shows promisg pplitions Soli Moelg. In uture work we pln to exten BRep lgorithms, like topologil eormtions, to work iretly with TBSP-Trees. 5 Aknowlegments The rst uthor woul like to thnk Chrles Loop or the proposl o the prolem o onvertg BSP-Trees to BRep, s well s orienttion the rst ttempts to solve the prolem Leonis Guis or helpul isussions n grnt rom Brzilin Ageny CNPq uner proess numer /92.9. Reerenes 1. H. Fuhs, Z. M. Keem, n B. F. Nylor. On visile sure genertion y priori tree strutures. Computer Grphis (SIGGRAPH '80 Proeegs), 14(3):124{133, July Dn Goron n Shuhong Chen. Front-to-k isply o BSP trees. IEEE Computer Grphis n Applitions, 11(5):79{85, Septemer M. Mntyl. An Introution to Soli Moelg. Computer Siene Press, Rokville, M, J. Mtousek. Eient prtition trees. Disrete Comput. Geom., 8:315{334, Brue Nylor. Bry spe prtitiong trees s n lterntive representtion o polytopes. Computer-Aie Design, 22(4):250{252, My Brue Nylor. SCULPT n tertive soli moelg tool. In Proeegs o Grphis Intere '90, pges 138{148, My Brue Nylor, John Amnties, n Willim Thiult. Mergg BSP trees yiels polyherl set opertions. In Forest Bskett, eitor, Computer Grphis (SIG- GRAPH '90 Proeegs), volume 24, pges 115{124, August 1990.

18 8. Brue F. Nylor. Prtitiong tree imge representtion n genertion rom 3D geometri moels. In Proeegs o Grphis Intere '92, pges 201{212, My A. A. G. Requih n H. B. Voelker. Boolen opertions soli moelg: Bounry evlution n mergg lgorithms. Pro. IEEE, 73(1):30{44, Jnury Jroslw R. Rossign n Herert B. Voelker. Ative zones CSG or elertg ounry evlution, reunny elimtion, tererene etetion, n shg lgorithms. ACM Trnstions on Grphis, 8(1):51{87, Willim C. Thiult n Brue F. Nylor. Set opertions on polyher usg ry spe prtitiong trees. In Mureen C. Stone, eitor, Computer Grphis (SIGGRAPH '87 Proeegs), volume 21, pges 153{162, July Enri Torres. Optimiztion o the ry spe prtition lgorithm (BSP) or the visuliztion o ynmi senes. In C. E. Vnoni n D. A. Due, eitors, Eurogrphis '90, pges 507{518. North-Holln, Septemer G. Vneek, Jr. Brep-ex: multiimensionl spe prtitiong tree. Internt. J. Comput. Geom. Appl., 1(3):243{261, 1991.