Geometric constraints within Feature Hierarchies

Transcription

1 Gomtri onstrints witin Ftur Hirris Mr Sitrm Jin-Jun Oun Yon Zou Am Arr April 8, 005 Astrt W stuy t prolm o nlin nrl D n D vritionl onstrint rprsnttion to us in onjuntion wit tur irry rprsnttion, wr som o t turs my us prourl or otr non-onstrint s rprsnttions. W tr t lln to rquirmnt on onstrint omposition loritms or omposition-romintion (DR) plnnrs us y most vritionl onstrint solvrs, ormliz t tur irry inorportion prolm or DR-plnnrs, lriy its rltionsip to otr prolms, n provi n iint loritmi solution. T nw loritms v n implmnt in t nrl, D n D opnsour omtri onstrint solvr FRONTIER vlop t t Univrsity o Flori. Kywors: Vritionl omtri onstrint solvin, Ftur-s n ssmly molin, Conptul sin, Cylil n D omtri onstrint systms, Domposition o omtri onstrint systms, Unronstrin n Ovronstrin systms, Usr nvition o solution onormtions, Prmtri onstrint solvin, Dr o From nlysis, Constrint rps. Introution n Motivtion Dsinrs in it intuitiv to us sptil tur irry rprsnttion, wi inlus: prourl istory or n lmost linr squn o ttmnts, xtrusions, swps; or CSG Booln oprtions su s intrstions; or prmtri onstrints, wil itionlly prmittin B-rp n otr rprsnttions o som turs. (W us t FEMEX n otr stnr initions o tur irry, [5], [] n r onrn primrily wit t onptul sin st). Wil sinrs itionlly pprit t xprssivnss o vritionl onstrints, toy s CAD systms lrly rstrit vritionl onstrint rprsnttions to D ross stions. Tis prsists spit t nrl onsnsus tt vots juiious us o D vritionl onstrints or t intuitiv xprssion n mintnn o rtin omplx n yli rltionsips tt otn our twn turs, prts or sussmlis. To rtiy tis sitution, it woul sirl i tur irry oul simultnously inorport D n D vritionl onstrints. In tis ppr, w not su rprsnttions s mix rprsnttions. S Fiur. Prvious work on su mix rprsnttions n lssii into two ro typs. T irst typ, su s [6, 7, 9], itts unii rprsnttion lnu wi is n mlmtion o vritionl onstrints wit otr rprsnttion lnus su s CSG n Brp. T son typ, su s [] wrstls wit tronous ppro, usin mny srvrs, on or rprsnttion lnu, so tt t pproprit on n ll wn rquir. Bot ppros, wil ily nrl in sop, v tir rwks. Our ppro in tis ppr s nrrowr ous: ow to rly nl vritionl onstrint solvr to l wit tur intrtions t ny lvl o tur irry, prmittin t turs to inpnntly mnipult. Rursivly, ts turs oul tmslvs rprsnt usin otr rprsnttions, or in similr, mix mnnr, usin onstrints to rlt t su-turs. Tis is nturl rprsnttion, sin rrlss o t wy in wi t turs t ny ivn lvl r rprsnt, Univrsity o Flori, Work support in prt y NSF Grnt CCR , NSF Grnt EIA orrsponin utor: sitrm@is.ul.u

2 S S S 6 L R S S 5 R M D E A C B G H S 5 S S 6 S S S 5L 5R L 5M M F J I S 7 S 8 S9 S 6R 6M K L, S 0 6L Fiur : Lt: Smll D onstrint rp (vrtis r points wit os n s r istn onstrints rmovin o) tt is not trinl omposl, in t, s no DR-pln o siz. Mil: Aritrrily lr non-trinl-omposl onstrint rp; E S i onsists o jnt trinls (lso o vrtis rprsntin points n s rprsntin istn onstrints) in t mniition on rit. onstrints twn turs t tt lvl oul spii twn primitiv omtri ojts or nls lonin to t turs. S Fiur. On min lln to ivin tis typ o rprsnttion is t ollowin omptin pir o rquirmnts on vritionl onstrint solvrs. Etivly rprsntin t intrtion o inpnnt turs, spilly in D, woul rquir t us o irly nrl, yli onstrint systms, wi r otn (onsistntly) ovronstrin. To l wit n rsolv su onstrint systms t iint nrtion o los-to-optiml omposition n romintion (DR) pln (ormlly in in Stion o nrl D or D onstrint rps is n. Ts my not mnl to omposition into trinulr or otr ix pttrns wi mny DR-plnnrs us. S or xmpl Fiur. DR-plns r wily us in omtri onstrint solvin n r ruil in orr to l wit t trtility ottlnk in onstrint solvin: minimizin t siz o simultnous polynomil qution systms, try ontrollin t pnn on xponntil tim lri-numri solvrs wi r prtilly rippl wn lin wit vn mortly siz systms. Etiv DR-plns r us lso or nvitin t solution sp o t onstrint systm [0], optimizin t lri omplxity o susystms snt to n lri-numri solvr [], or lin wit xpliit inonsistnis (ovronstrin systms) [5], impliit or omtri runnis n onstrint pnns [], [], miuitis (unronstrin systms) n or iint upts n onlin solvin [8, 9]. In orr or vritionl onstrints to us in onjuntion wit tur irry rprsnttion, t DR-plns o omtri onstrint systms soul now m to inorport n input, tur omposition rprsntin t unrlyin tur, prt or sussmly irry. Tis is prtil orr, typilly rprsnt s Dirt Ayli Grp or. S Fiurs,,. Ain, tis inorportion o n ritrry input, onptul sin omposition into t DR-pln is only possil i t DR-plnnin pross is suiintly nrl. In prtiulr, t inorportion soul insnsitiv to t orr in wi t lmnts in t onstrint rp r onsir urin t DRplnnin pross, t so-ll Cur-Rossr proprty. Howvr tis proprty wil nssry, is not suiint s xplin in Stion ). T ppr [] irst ormliz t onpt o DR-pln s wll s svrl prormn msurs o DR-plnnrs som o wi r rlvnt to tis ppr. It itionlly ivs tl o omprisons wi sows tt mny o t prvious DR-plnnrs..[, 5, 6, ], [, 5,, ], [, 7, 0, 8], (or ny ovious moiitions o tm) woul inrntly il to inorport vn tr-lik input sin ompositions or tur irris. W xplin in Stion wt t iiulty is in t s o DR-plnnrs tt pn on omposition s on ix pttrns. Inorportion o input ompositions is ruil lso in orr to ptur sin intnt, or ssmly orr n llow inpnnt n lol mnipultion o turs, prts, sussmlis or susystms witin tir lol oorint systms. It is lso ruil or ilittin solution sp nvition [0] in

3 mnnr tt rlts onptul sin intnt. In ition, it is ruil or proviin t usr wit tur rprtoir, t ility to pst into skt lry rsolv turs n onstrint susystms tt r spii in notr rprsnttion or llowin rtin turs or sir susystms to solv usin otr, simplr, mtos su s trinl-s omposition or prmtri onstrint solvin. Somtims t usr woul prr to spiy priority t t vrtis o t wi itts t orr o rsolution o t turs, prts, sussmlis or susystms. Tis ours lso wn on turs n only in or nrt s on notr, s in prourl or istory s rprsnttions. Also, not tt prmtri onstrint solvin n in t iv s spil s, wr t orr is omplt, totl orr. Mor nrlly, tis n us in t CAD ts mintnn o multipl prout viws s in [6], [8], or xmpl t sin viw n ownstrm pplition lint s viw my somwt irnt onstrint systms n t two tur irris my not vn rinmnts o on notr, ut intrtwin. S Fiur. Furtrmor, viw oul ontin irnt rrnin sp lmnts tt r not prt o t nt sp n tror, not prt o t otr viws. Tis is prtiulrly t s wn ts rrnin sp lmnts r tully nrt urin t oprtions o istory-s prourl rprsnttion.. Contriution n Orniztion Our strtin point is t rntly vlop Frontir Vrtx Aloritm ( FA), DR-plnnr [9] [0], [], [], [5], [], wi uils upon nrly o rlir work on omtri onstrint solvin n ls wit nrl D n D vritionl onstrint systms. W iv tur inorportion loritm tt sits top t FA DR-plnnr n prmits it to inorport input sin ompositions n tur irris, wil prsrvin its iiny n otr sirl proprtis. T nw loritm s n implmnt in t opnsour D n D omtri onstrint solvr FRONTIER [5], [], [9]. In Stion, w iv t nssry kroun on omtri onstrint rps, r o rom nlysis, its limittions, DR-plns n tir ssntil n sirl proprtis. Stion ivs orml sttmnt o t prolm n irntits it rom notr, similr prolm. Stion ivs tos loritmi ssntils o t FA DR-plnnr tt r solutly nssry to iv our tur inorportion loritm wi is sri in Stion 5. A psuoo is provi in [9] n t oumnt o n oun in [5]. Bsi Bkroun Gomtri onstrint systms v n stui in t ontxt o vritionl onstrint solvin in CAD or nrly s For rnt rviws o t xtnsiv litrtur on omtri onstrint solvin mor lort sriptions n xmpls or t initions low, s,., [0, 8, 0, 8]. A omtri onstrint systm onsists o init st o primitiv omtri ojts su s points, lins, plns, onis t. n init st o omtri onstrints twn tm su s istn, nl, inin t. T onstrints n usully writtn s lri qutions n inqulitis wos vrils r t oorints o t prtiiptin omtri ojts. For xmpl, istn onstrint o twn two points (x, y ) n (x, y ) in D is writtn s (x x ) +(y y ) =. In tis s t istn is t prmtr ssoit wit t onstrint. Most o t onstrint solvrs so r l wit D onstrint systms. Wit t xption o work [7, 8,, ], [], [5,,,, 9,, 0, ], rlt to t FRONTIER omtri onstrint solvr [5], to t st o our knowl, work on stnlon D omtri onstrint solvrs is rltivly sprs [, ]. A solution or rliztion o omtri onstrint systm is t (st o) rl zro(s) o t orrsponin lri systm. In otr wors, t solution is lss o vli instntitions o (t position, orinttion n ny otr prmtrs o) t omtri lmnts su tt ll onstrints r stisi. Hr, it is unrstoo tt su solution is in prtiulr omtry, or xmpl t Eulin pln, t spr, or Eulin imnsionl sp. A onstrint systm n lssii s ovronstrin, wllonstrin, or unronstrin. Wll-onstrin systms v init, lit potntilly vry lr numr o rii solutions; i.., solutions tt nnot ininitsimlly lx to iv notr nry

4 7 intrnl onstrints S i 8 9 xtrnl onstrints mix prurl 5 Prourl History Bs 6 CSG BREP 0 Prmtri Constrints D Vritionl Constrints n Swps 0 P i Fiur : Constrint systm S i n unrlyin tur irry P i. Fturs 0,..., r in mix rprsnttion;,,,, r prtiiptin tur nls; 0,..., r turs t ir lvl o irry Fiur : Input sin omposition or tur irry n orrsponin irt yli rp on rit.

5 X A D F B X A B C Y Y E C D E F X A B C D E F Y Y Fiur : Aov: Constrint systm n onstrint rp sowin multipl viws (ott n s) wit irnt rrnin lmnts x n y; numrs rprsnt os. Blow: t two viws v intrtwin tur irris; ox ontins nt sp lmnts ommon to ot viws solution: t solution sp (moulo rii oy trnsormtions su s rottions n trnsltions) onsists o isolt points - it is zro-imnsionl. Unronstrin systms v ininitly mny solutions; tir solution sp is not zro-imnsionl. Ovronstrin systms o not v solution unlss ty r onsistntly ovronstrin. In tt s, ty oul m witin ovrll unronstrin systms. Systms tt r not unronstrin r ll rii systms.. Constrint Grps n Drs o From DR-plns, ormlly in in t nxt stion, provi t orml sis o our tur irry inorportion loritm. Gomtri onstrint rp rprsnttions o onstrint systm r typilly us to vlop DR-plns. Spiilly ts rps r us or omintoril nlysis o lri proprtis o t systm (su s wllonstrinnss, riiity t.,), tt ol nrilly, i.., or ll nri vlus or t onstrint prmtrs (or xmpl, or lmost ll istn vlus, in t s o istn onstrint systms). Pls s, or xmpl, [0] or mor tils rlt to t initions ivn low. A omtri onstrint rp G = (V, E, w) orrsponin to omtri onstrint systm is wit rp wit vrtx st (rprsntin omtri ojts) V n st (rprsntin onstrints) E; w(v) is t wit o vrtx v n w() is t wit o, orrsponin to t numr o rs o rom vill to n ojt rprsnt y v n numr o rs o rom (os) rmov y onstrint rprsnt y rsptivly. For xmpl, Fiurs 5, 6, 9 sow D n D onstrint systms n tir rsptiv o onstrint rps. Mor D onstrint systms wos rps v vrtis o wit (points) n s o wit, n oun in Fiurs 7 8. Not tt t onstrint rp oul yprrp, ypr involvin ny numr o vrtis. A surp A G tt stisis w(v) () A w() + D v A is ll ns, wr D is imnsion-pnnt onstnt, to sri low. Funtion (A) = A w() v A w(v) is ll nsity o rp A. T onstnt D is typilly ( ) + wr is t imnsion. T onstnt D pturs t rs o rom o rii oy in imnsions. For D ontxts n Eulin omtry, w xpt D = n or sptil ontxts D = 6, in nrl. I w xpt t rii oy to ix wit rspt to lol oorint systm, tn D = 0. A trivil surp is sinl vrtx (in D) n vrtx or (in D). 5

6 Fiur 5: D onstrint systm xmpl wit vril rius irls, points n istn onstrints - ll qul istns. T orrsponin rp s 7 vrtis, vrtis rprsntin vril rius irls ( wit rs o rom or t ntr position) n vrtis rprsntin points ( wit positionl rs o rom). T s r t istn onstrints sown in t onstrint systm. E s wit sin istn onstrint rmovs r o rom rom on o t two prtiiptin points. Fiur 6: Simpl D onstrint systm rwn on D nvs wit points n istns n DR-pln Fiur 7: D onstrint systm rwn on D nvs wit 5 point ojts (P - P6) n 6 ix lnt lin smnt ojts (twn point P i n P i+ ); istn onstrints (qul istns); 0 inin onstrints wr linsmnts r inint t o t points P -P6; n 5 qul nl onstrints twn jnt linsmnts. T orrsponin rp s vrtis o wit (points n ix lnt linsmnts v rs o rom ); s o wit (n inin onstrint twn points rmov t rs o rom o on o t points) n s o wit ( istn onstrint twn points rmov r o rom rom on o t points; n nl onstrint twn lin smnts rmovs r o rom rom on o t lins). Solution (rit) 6

7 Fiur 8: D onstrint systm wit nns typ [] nri onstrint pnn not ttl y simpl o ount: t istn twn P n P0 is inpnntly trmin y t lt n rit lmoon lustrs. It is owvr wll-ovronstrin n onsistntly onstrin or t ivn st o istns: orrsponin DR-pln s sinl root Nxt w iv purly omintoril proprtis rlt to nsity tt r us to tt nri lri proprtis. A ns, nontrivil rp wit nsity stritly rtr tn D is ll o-ovronstrin. A rp tt is ns n ll o wos surps (inluin itsl) v nsity t most D is ll o-wllonstrin. A rp G is ll o-wll-ovronstrin i it stisis t ollowin: G is ns, G s tlst on ovronstrin surp, n s t proprty tt on rplin ll ovronstrin surps y o-wllonstrin surps (in ny mnnr), G rmins ns. Intuitivly, tis inition is us to prvnt som ovronstrin surps wit i nsity rom skwin t lssiition o t ntir rp. In prtiulr, n xtrm xmpl oul rp tt s surps tt r svrly ovronstrin, ut wit no onstrints twn tm. By tis inition, su rp woul not wll-ovronstrin n woul orrtly lssii s unronstrin. A rp tt is wllonstrin or wll-ovronstrin is ll o-lustr. A nontrivil ns rp is miniml i it s no nontrvil ns propr surp. All miniml ns surps r o-lustrs ut t onvrs is not t s. A rp tt is not o-lustr is si to unronstrin. I ns rp is not miniml, it oul in t n unronstrin rp: s point out, t nsity o t rp oul t rsult o min surp o nsity rtr tn D. Nxt w isuss ow t rp torti proprtis r o rom (o) nlysis rlt to orrsponin proprtis o t orrsponin onstrint systm. In imnsions, orin to Lmn s torm [9], i ll omtri ojts r points n ll onstrints r istn onstrints twn ts points tn ny miniml ns lustr rprsnts nrilly rii systm. In nrl, owvr, wil nrilly rii systm lwys ivs lustr, t onvrs is not lwys t s. In t, tr r wllonstrin, ns lustrs wos orrsponin systms r not nrilly rii n r in t nrilly not rii u to t prsn o nri onstrint pnns. S Fiur 8 wit D points n istn onstrints, wi illustrts t so-ll nns prolm o [], wi nrlizs to t so-ll in prolm [6, 7]. To t, tr is no known, trtl, omintoril rtriztion o nri riiity o systms or or ir imnsions vn wn only points n istns r involv [6], [], ltou svrl onjturs xist. Tr r no 7

8 C0 C6 C7 i C C C6 C7 C C C C0=S(D(G)) C C C0 C C0 C7 C9 C C9 C6 C5 C C8 C C8 C Fiur 9: D onstrint rp G n DR-pln; ll vrtis v wit n s wit known nrl omintoril rtriztions o D riiity, wn otr onstrints sis istns (su s nls) r involv. For onstrint systms wit nl n inin onstrints, ut no istns su rtriztion is ivn in []. For D points n istns, t notion o moul-rii lustrs in [] (n xtnsion o o-rii lustrs in ov) ls wit ll spts o t nns n in prolms, i.., it orrtly rtrizs nri riiity in ll known ss. Currntly, no ountrxmpls r known - o moul-rii onstrint rps tt r not nrilly rii. Not. In tis ppr, w rstrit our ttntion to o-rii lustrs, n DR-plnnrs n tur inorportion loritms tt r s on tm. S isussion t t n o Stion 5.. Forml inition o DR-plns n Essntil Proprtis Hr w ormlly in n iv ssntil proprtis o DR-plns n DR-plnnrs. Tis stion is importnt sin w rquir our tur inorportion loritm to prsrv ts proprtis. Pls rr to [, ] or til isussion o ts proprtis. Formlly, o-dr-pln o onstrint rp G is irt yli rp () wos nos rprsnt o-lustrs in G, n s rprsnt ontinmnt. T lvs or sinks o t r ll t vrtis (primitiv o-lustrs) o G. T roots or sours r omplt st o t mximl o-lustrs o G. For wll or wll-ovronstrin rps, t DR-plns v sinl sour. Tr oul mny DR-plns or G. S Fiurs 9, 5, 6, 7, 8... Ovronstrints First, o-lustr C in t DR-pln soul ompni y trtl rprsnttion o omplt list o ruil ovronstrint sts irtly ssoit wit C. I.., sts o onstrints tt o not li ntirly witin ny il lustr o C n n rmov witout tin t o-riiity o C. T DR-plnnr soul itionlly mit n iint mto o rmovin ovronstrints y mkin t pproprit ns to t DR-pln... Optimlity T siz o lustr in DR-pln is its n-in or numr o its ilrn (it rprsnts t siz o t orrsponin susystm, on its ilrn r solv). Sin t lri-numri solvrs tk tim xponntil in t siz o t susystms ty solv, n t numr o solutions is lso typilly xponntil, minimizin t siz o DR-pln is ssntil to t ESM mto prsnt r. An optiml DR-pln is on tt minimizs t mximum n-in. It is sown in [], [], tt t prolm o inin t optiml DR-pln o vn D istn onstrint r p is NP-r, n pproximility rsults r sown only in spil ss. Nonpproximility rsults r not known. On msur us in liu o solut optimlity is s on t t tt most DR-plnnrs mk o ois urin omputtion (sy t orr in wi vrtis r onsir) n w n sk ow wll (los to optiml) t st omputtion pt o su DR-plnnr woul prorm (on t worst s input). W ll tis t st-oi pproximtion tor o t DR-plnnr. 8

9 A mor stistory msur o optimlity is s on t ollowin ltrntiv proprty. A trtl DR-pln or systmti nvition soul nsur tt lustr C soul ompni y smll st o its ilrn C i tt orm n optiml ovrin st o mximl lustrs proprly ontin in C. A ovrin st o lustrs is on wos union ontins ll omtri lmnts witin C. T siz o C is simply t siz o tis optiml ovrin st. T optimlity r rrs not to t siz o t ovrin st, ut to ny suitl omintoril msur o t lri omplxity o t tiv onstrint systm or solvin C, ivn t solutions o t il lustrs in t ovrin st. Tis ls to t notion o ompltnss o DR-plns, ivn low... Compltnss Any mto tt ooss n optiml ovrin st or lustr C rquirs s input nrliz omplt omposition o C into mximl propr sulustrs, ormlly in s ollows. T omposition o ny lustr C lls into on o typs. A Typ lustr C s xtly il lustrs, wi intrst on nontrivil surp, n tir union ovrs ll t omtri lmnts in C. A Typ lustr C s st o il lustrs C i wit t ollowin proprty. T union o C i s ovrs ll t omtri lmnts in C; ny pir o C i s intrst on t most trivil surp; n vry C i is propr mximl sulustr o C, i.., tr is no propr sulustr o C tt stritly ontins C i. Compltnss is lso n or ttin impliit onstrint pnns n or mor urt, moul-rii DR-plnnrs, s isussion t t n o Stion 5... Complxity Anotr si proprty o DR-pln is its wit i., numr o lustrs in t DR-pln to smll, prrly linr in t siz o G: tis rlts t omplxity o t plnnin pross n ts t omplxity o t solvin pross tt is s on t DR-pln. Clrly, tis proprty ompts wit ompltnss. Otr sirl proprtis o DR-plnnrs not mntion ov inlu systmti orrtion o unronstrin systms, n mnility to iint upts o omtri primitivs or onstrints. Forml Prolm Sttmnt W now sri t loritmi prolm o tur inorportion in DR-pln s n input-output spiition. Input: () A D or D omtri onstrint systm (s Stion ), S. () An prtil tur omposition o t onstrint systm ivn s irt yli rp () P otin rom on or mor tur, prt or sussmly irris or sin viws. A tur or sussmly is no o P wit its immit suturs rprsnt s its ilrn. S Fiurs 0,,. Dsir Output: A DR-pln or t input onstrint systm wos nos inlu tos turs in t input prtil tur omposition tt r o-lustrs. For turs tt r not o-lustrs, omplt st o mximl o-lustrs witin tm soul ppr s nos in t DR-pln. Otr output Rquirmnts: On t output DR-pln s n otin, upts to t input tur irry soul iv iintly witout vin to ro t ntir DR-pln. T sirl proprtis o DR-plns n DR-plnnrs ivn in Stion DR-plnnrs) must prsrv y t tur inorportion loritm n its output DR-pln. Som o ts proprtis rquir stritorwr r-inition or tur inorportin DR-plns, sin otn no DR-pln vn xists tt inorports ivn input tur omposition n prsrvs ts proprtis s su. For xmpl, it my not possil to prsrv optimlity sin rtin turs my simply rquir lustrs wit lr n-in; similrly it my not possil to prsrv omplxity n smll wit, sin t numr o ivn turs my or lr wit. Howvr, w woul lik to nsur tt our loritm 9

10 Fiur 0: Lt: input prtil ompositions or D onstrint systm sown in Fiur 7; roups r turs or sussmlis. Rit: irnt DR plns inorportin orrsponin input omposition. Fturs ppr s lustrs or, i unronstrin, tir omplt st o mximl lustrs. Fturs my (not) intrst on (non) trivil surps. Top: lt roup pprs s C n rit s C8. Mil: lt roup pprs s C n rit roup pprs s C. Bottom: lt roup pprs s C60 n rit roup pprs s C5. Fiur : D onstrint systm o points n istn onstrints, wit input turs or sussmlis G n G. Two DR-plns r sown, on tt inorports t turs n on tt os not 0

11 Fiur : Trinl-omposl D onstrint systm o points n istn onstrints tt is ountrxmpl to t Cur-Rossr proprty n n to t tur inorportion proprty o trinl omposition s DR-plnnrs. I trinl in t irl prt is irst pik, DR-plnnr nnot ontinu. Rit sows trinl omposition strtin rom irnt trinl. nrt DR-plns tt prsrv ts proprtis s mu s possil, ivn t rstrition tt ts DR-plns r or to inorport ivn input tur omposition.. A rlt, irnt prolm T ility o DR-plnnr to inorport ny input tur omposition into DR-pln o onstrint rp G implis tt t DR-plnnr n in t lustrs o t rp in ny orr tt is onsistnt wit ontinmnt. Tis is ll t Cur Rossr proprty o t DR-plnnr n s n invstit in [, ] n []. A orml inition is t ollowin. Assum tt DR-plnnr P onstruts its output DR-pln ( ) or onstrint rp G ottom up, i.., i lustr C ontins lustr D, n ot ppr in t output, tn D s n oun n insrt into t DR-pln or C. Expt or tis rstrition, t DR-plnnr is llow to in n insrt ny lustr o G into t output t ny st. T DR-plnnr P s t Cur-Rossr proprty i ny tt it outputs is vli DR-pln or t input rp G. T Cur Rossr proprty is wkr n os not imply t tur inorportion proprty, spilly wn t DR-plnnr in qustion os not prorm nrliz nlysis, ut works only on rps tt v rstrit typ o DR-pln. For xmpl, onsir DR-plnnr tt rquirs t rp G to omposl into lustrs tt v spii strutur or topoloil pttrn su s [, ], n only ins su lustrs. T DR-plnnr oul v Cur-Rossr proprty in tt it woul sussully in DR-pln rrlss o t orr in wi su lustrs r oun n pross. Howvr, y simply pikin tur wi is lustr tt in G tt os not v tt pttrn, t DR-plnnr woul not l to inorport tt tur in its DR-pln n n woul not possss t tur inorportion proprty. Not tt in t spii s o [, ], it n sown tt rp is wll-onstrin n trinl omposl i n only i ll lustrs in it r trinl-omposl, (ltou tis proo os not ppr in [, ]). So in tis s t Cur-Rossr proprty implis t tur inorportion proprty; owvr in tis s ot proprtis o not ol or ovronstrin rps (s Fiur ); n s point out ov, t prsn o onsistnt ovronstrints my ruil in orr to tivly mix tur-s n onstrint-s rprsnttions. Finlly, vn i DR-plnnr s t Cur-Rossr proprty n is in t sown to mnl to tur inorportion, t tul loritmi prolm still rmins: o iintly inorportin ivn

12 input tur omposition into DR-pln wil minimlly ltrin its otr sirl proprtis. Tis is t prolm tt w rss r. T Frontir Vrtx Aloritm (FA) DR-Plnnr Hr w skt t ssntil loritmi tils o t o-rii Frontir vrtx (FA) DR-plnnr wi r solutly nssry to sri our tur inorportion loritm. T o-rii FA DR-plnnr stisis t proprtis isuss in t Stion. T si i o tis DR-plnnr n its prormn ws prsnt in []; omplt orml sription lon wit nlysis o orrtnss n (ui) omplxity n proo o t ompltnss proprty r ivn in [], [], [8]. A psuoo is provi in [9] n t oumnt o n oun in [5]. A mto o omintorilly otinin n optiml, stl lri systm or solvin C, i.., or otinin optiml ovrin sts is ivn in [, ]. T mto or otinin ll possil sts o ruil onstrints or ovronstrin lustrs o o-rii Frontir vrtx DR-plns, ot or D n D, n moiyin t DR-pln on ty v n rmov, is prsnt in [5]. Otr sirl proprtis su s systmti orrtion o unronstrints, n mnility to iint upts o omtri primitivs or onstrints r ivn in [8, 9].. T FA DR-plnnr strutur n ky mtos T input is t onstrint rp G n t output is DR-pln o qurti wit, stisyin t ompltnss proprty, wit onstnt st-oi pproximtion tor. Rpt pik lustr C rom lustrquu CQ Distriutlustr(C) in lustr rp $CG$ i nw lustr C is oun (ontinin C) tn Complt (C G) [rursiv prour uils omplt su DR-pln or lustr rp rstrit to C, i.., input onstrint rp rstrit to C, strtin rom tos sulustrs o C tt r lry prsnt in DR-pln; moiis DR-pln, insrtin C n nw oun sulustrs o C into it.] insrt C t t n o CQ Comin CQ (itrtiv prour tt moiis ot DR-pln n CQ until no urtr ominin is possil) upt CG y (rontir vrtx) Simpliy(lustrs in CQ) rmov C rom CQ Until CQ is mpty i DR-pln s mor tn root, tn Complt(CG) W sri t mtos: DistriutClustr, Simpliy, n Comin to t xtnt nssry to sri n nlyz our tur inorportion loritm. DistriutClustr in turn rlis on mtos s on ntwork low or lotin o-rii lustrs: DistriutVrtx, DistriutE n PusOutsi. It is not nssry to sri Complt sin it is unt y t tur inorportion loritm. For ts, t rr is rr to [, ]. Brily: n Complt rursivly lls itsl s wll s Distriutlustr, Comin n Simpliy n rquirs r sin ompltnss ompts wit t smll wit proprty o DR-plns s mntion in Stion.

13 t t t s 0 0 s s 0 * Fiur : From Lt. Constrint rp G wit wit istriution. D is ssum to 0 (systm ix in oorint systm); A orrsponin low in iprtit G. Anotr possil low. Initil low ssinmnt tt rquirs ristriution n sr or n umntin pt.. Distriutin Es T ns surp isoltion loritm is us rptly to in o-rii lustrs. It ws irst ivn in [9, 0] s moii inrmntl ntwork mximum low loritm. W ssum stnr workin knowl o ntwork low. T ky routin is t istriution o n (s t DR-plnnr psuoo in [9], n opnsour o in [5]) in t onstrint rp G. For, w try to istriut t wit w() + D + to on or ot o its npoints s low witout xin tir wits, rrr to s istriutin t. n ll t DistriutE routin. Tis is st illustrt on orrsponin iprtit rp G : vrtis in on o its prts rprsnt s in G n vrtis in t son prt rprsnt vrtis in G; s in G rprsnt inin in G. As illustrt y Fiur, w my n to ristriut (in n umntin pt). I w r l to istriut ll s, tn t rp is not ns. I no ns surp xists, tn t low s loritm will trmint in O(n(m + n)) stps n nnoun tis t. I tr is ns surp, tn tr is n wos wit plus D + nnot istriut (s r istriut in som orr, or xmpl y onsirin vrtis in som orr n istriutin ll s onntin nw vrtx to ll t vrtis onsir so r). T sr or t umntin pt wil istriutin tis mrks t rquir ns rp. I t oun surp is not ovronstrin, tn it is in t miniml. I it is ovronstrin, [9, 0] iv n iint loritm to in miniml lustr insi it... Simpliyin Clustrs W now sri t mto Simpliy. Tis rontir vrtx simpliition ws ivn in [, ]. T oun lustr C intrts wit t rst o t onstrint rp trou its rontir vrtis; i.., t vrtis o t lustr tt r jnt to vrtis not in t lustr. T vrtis o C tt r not rontir, ll t intrnl vrtis, r ontrt into sinl or vrtx. Tis or is onnt to rontir vrtx v o t simplii lustr T(C) y n wos wit is t t sum o t wits o t oriinl s onntin intrnl vrtis to v. Hr, t wits o t rontir vrtis n o t s onntin tm rmin unn. T wit o t or vrtx is osn so tt t nsity o t simplii lustr is D, wr D is t omtry-pnnt onstnt. Tis is importnt or provin mny proprtis o t FA DR-pln: vn i C is ovronstrin, T(C) s ovrll wit is tt o wllonstrin rp, (unlss C is rottionlly symmtri - in wi s, it lks on o). Tnilly, T(C) my not wllonstrin in t pris sns: it my ontin n ovronstrin surp onsistin only o rontir vrtis n s, ut its ovrll o ount is tt o wllonstrin rp. Fiur illustrts tis itrtiv simpliition pross nin in t inl DR-pln o Fiur 9. T lows on t intrnl s n t or vrtx r inrit rom t ol lows on t intrnl s n intrnl vrtis. T unistriut wits on t intrnl s simply isppr. T unistriut wits on t rontir s r istriut (witin t lustr) s mu s possil. Howvr, unistriut wits on t rontir s (s twn rontir vrtis) my still rmin i t rontir portion o t lustr is svrly ovronstrin... Dtstruturs Four tstruturs r mintin. T lustr tstrutur or lustr C ontins t on t simpliition o t lustr (rontir vrtis, s t.), t oriinl rp vrtis n s orrsponin

14 7 i C C C C Fiur : From lt: FA s simpliition o rp ivin DR-pln in Fiur 9; lustrs r simplii in tir numr orr: C is simplii or C7 t. to C, n pointrs to t roots o t urrnt su DR-pln root t C (my or my not omplt su-dr-pln pnin on t st o t loritm). T DR-pln tstrutur just ontins pointrs to t lustrs tt r urrntly t top-lvl lustrs in t DR-pln. T low or lustr rp, CG ontins t urrnt simplii rp wr t oun lustrs v n simplii usin t rontir vrtx simpliition. It lso ontins ll t urrnt low inormtion on t s. lustr quu, CQ wi is t top-lvl lustrs o t DR-pln tt v not n onsir so r, in t orr tt ty wr oun. Initiliztion. W strt wit t oriinl rp wi srvs s t lustr or low rp initilly, wr t lustrs r sinlton vrtis, T DR-pln onsists o t l or sink nos wi r ll t vrtis. T lustr quu onsists o ll t vrtis in n ritrry orr... Distriutin Vrtis n Clustrs T mto DistriutVrtx istriuts ll s (lls DistriutE) onntin t urrnt vrtx to ll t vrtis onsir so r. Wn on o t s nnot istriut miniml ns lustr C is isolt, s sri in Stion... Now w sri t mto DistriutClustr. Assum ll t vrtis in t lustr quu v n istriut (itr ty wr inlu in ir lvl lustr in t DR-pln, or ty il to us t ormtion o lustr n ontinu to top lvl no o t DR-pln, ut v isppr rom t lustr quu). Assum urtr tt t DR-pln is not ntir, i.., its top lvl lustrs r not mximl. T nxt lvl o lustrs r oun y istriutin t (omplt) lustrs urntly in t lustr quu. Tis is on y illin up t ols or t vill rs o rom o lustr C in istriut y D units o low. Tn t PusOutSi mto sussivly onsirs inint on t lustr wit npoint outsi t lustr. It istriuts ny unistriut wit on ts s + xtr wit unit on o ts s. It is sown in [, ] tt i C is ontin insi lrr lustr, tn tlst on su lustr will oun y tis mto on ll t lustrs urrntly in t lustr quu v n istriut...5 Cominin lustrs Nxt, w mpsiz t prts o t loritm tt nsur ruil proprty o t output DR-pln tt t tur inorportion loritm must l wit, nmly smll wit, wil mintinin omplt omposition o lustrs (two typs) in in Stion. T FA DR-plnnr ontrols t wit o t FA DR-pln to nsur FA ivs qurti oun on DR-pln wit y mintinin t ollowin invrint o t lustrquu wnvr DistriutClustr is ll, vry pir o lustrs in t lustr

15 quu n lustr rp intrst on t most trivil surp (i.., surp wi on rsolvin inin onstrints itr rprsnts to sinl point in D or vril or ix lnt lin smnt in D). FA os tis y rptly prormin t Comin oprtion tim nw lustr is isolt. T oprtion is to itrtivly omin N D wit ny lustrs D, D,... s on nontrivil ovrlp. In tis s, N D D, N D D D t. ntr t DR-pln s stirs, or in, ut only t sinl lustr N D D D..... ntrs t lustr rp n lustr quu tr rmovin D, D, D Inorportin n input tur omposition into n FA DRpln W vlop nw loritm tt solvs t loritmi prolm o Stion. I.., it prmits t FA DR-plnnr ([, ], skt in Stion ) to inorport into its output DR-pln n input tur omposition or onptul, sin omposition. 5. Mor til rquirmnts on t mto A rursiv mto ll DistriutGroup rivs t nw FA DR-plnnr It is ll on t root no o t input tur omposition, wi, y onvntion, rprsnts t ntir rp. Lt G, G,.., G m t ilrn o som prnt tur G in t input omposition. Now tt t FA DR-plnnr s strutur s n sri, w n iv mor til st o rquirmnts on t tur inorportion loritm s on t orml prolm rquirmnts o Stion.. First o ll, w n to nsur tt tur G i itsl pprs in t output DR-pln provi it is vli lustr. I it is not lustr, tn omplt mximl omposition o it is otin. Tis implis tt sprt DR-pln or G i ns to otin n ll o ts DR-plns soul ppr witin t DR-pln or G.. Sonly, wil w nnot mintin t sm wit W s n FA DR-pln tt is not rquir to inorport ny turs (or t sm onstrint rp), w woul lik to v wit siniintly smllr tn O(WF), wr F is t numr o turs in t input tur irry. Inst o spiyin t sir omplxity, w simply rquir t st possil. I.., w rquir tt onsistny twn t DR-plns o t irnt G i s ns to stlis, in s ty v sr ojts. In prtiulr, in t D onstrint systm in Fiur 5, ssum G s DR-pln s n otin n s n intrmit lustr. Wn t DR-pln or G is otin, t sm lustr soul ppr.. Tirly, or iiny, w woul lik t nw DR-plnnr - wil workin on som G i - to us s mu o t low, lustr n DR-pln inormtion tt it lry otin wil workin on som rlir G k, so tt t ntir omplxity is proportionl to t wit or numr o lustrs in t inl DR-pln. Tus t ky issus r: ow r t low or lustr rp (s sription o FA in Stion ) n t output DR-pln iintly mintin n moii s G i is work on, so tt t ov rquirmnts r stisi. Finlly, not tt sin t input omposition my, n not tr, som ilrn my v mor tn on prnt. Howvr, DistriutGroup must prorm only on on ny ivn roup G; Wn notr prnt o G lls DistriutGroup(G) t ltr point in tim, t stor rsults or t low or lustr rp n DR-pln or G soul rturn. 5. Distriutin Groups or Fturs W isuss istint ss tt n to lt wit irntly t t ruil stps o t loritm. Cs : t ilrn o G r mutully pirwis isjoint, i., o not ovrlp otr 5

16 Fiur 5: Consistny twn t DR-plns o roups o rp in Fiur Cs : G i s ovrlp wit G k ( k i ) onsists ntirly o rontir vrtis o t top lvl lustrs in t mr DR-pln o t G k s. Cs : or t lst on o t G k s( k i ), G i s ovrlp wit G k inlus vrtis (o t oriinl rp) tt mp to t or o on o t lustrs o G k. T mto DistriutGroup(G). onsists o stps. First, or il roup G i o G, it prorms t ollowin stps. Not. For lrity o xposition, w prr to not iv psuoo or DistriutGroup(G), ut rtr xplin sussivly o t stps or ll tr o t ss. Wn w rr to t ol DR-plnnr, w mn t psuoo o Stion ; n t nw rrs to t ol FA DR-plnnr umnt y t DistriutGroup rivr. A til psuoo o t ntir FA DR-plnnr inluin t tur inorportion loritm n oun in [9]. Stp : T ol FA DR-plnnr o Stion is ll on G i n strts nw DR-pln or G i (wi will vntully t mr wit t DR-plns o t otr ilrn o G tr Stp low). Tn t nw DR-plnnr uss irnt options or t low or lustr rps or t tr ss. Cs : Us t urrnt low or lustr rp y rzin ll t s n vrtis outsi o G i ; Cs : Us t urrnt low or lustr rp, moiy t low on rully slt s n vrtis outsi o G i, mrkin tm n rzin tm; I.. or wit on npoint in G i n notr npoint outsi G i, i tr is ny low on tis towrs G i, tn rmov it n inst to t unistriut low pity on tis. Mrk tis s vin unistriut low. Cs : Crt nw lol opy o low rp or G i lon (wi will vntully us to upt t urrnt low rp in Stp low). Stp : T DR-plnnr ontinus wit rursiv ll to DistriutGroup(G i ) urin wi it nsurs tt DistriutE is run on t unistriut low on ll mrk s witin G i. An is unmrk only i istriution is sussul on ll unistriut units on. Stp : T DR-plnnr mrs t DR-Pln o G i wit t DR-Plns o G trou G i. Tis inlus mrin opis o lustrs tt oul v n inpnntly oun y t DistriutGroup mto on irnt roups. It itionlly inlus puttin lustrs totr to orm lrr lustrs, s on mount o ovrlp; umntin urrnt low rp (mr low rps so r) usin t lol low or lustr rp or G i. Tis lttr prt inlus not only ominin lustrs ut lso moiyin lows. Mor spiilly, t ollowin mrin oprtions r prorm. First, i lustr in t DR-plns or G, G,..., G i pprs in in t DR-pln or G i, t two opis r link to prvnt rplition o ort urin t solvin st. Wil t istint roups o t input omposition tt r prsnt in t su-dr-pln o lustr opy will v to solv, t lustr itsl n ny lustr in its su-dr-pln tt is not sust o roup in t input omposition, will only v to solv on. Not tt mrin t lustr opis (y tkin t union o tir prnts n union o tir ilrn) will violt t so-ll lustr minimlity proprty o oo DR-pln (mntion in Stion ), sin propr sust o t ilrn woul lry orm t lustr. 6

17 G G G G G G I II G G G G G G G III IV Fiur 6: Cs : Cil Clustrs o not ovrlp Nxt, t DR-plnnr looks t t top lvl lustrs o t mr DR-pln or G, G,..., G i n t top lvl lustrs o t DR-pln or G i. I ny pir o ts sy C n D intrst nontrivilly, on mor tn points in D n points in D, tn rt prnt lustr C D o C n D, in t DR-pln; (tis is t sm s t Comin oprtion on lustrs prorm y t si FA - Stion ), n mkin t orrsponin simpliition in t low or lustr rp, sri low. Css n : Bus t lol low rp is inrit rom t low rp or G, no itionl moiition is n; Cs : rmovs ll low rom t non-lustr s tt r in t ovrlpp prt twn G i n tt prt o G tt s n omplt so r i.: G,...,G i n mrks tir ntir wit s unistriut. Ts s will ristriut wn t lustrs tt ontin tm r istriut. Stp : On t DR-plns o ll t G i s v n omin, t DR-plnnr pros s sri in Stion on t rsultin low or lustr rp o G, prormin DistriutClustr on t lustrs in tm, potntilly isoltin n simpliyin nw lustrs tt ontin t G i s, moiyin t lustr quu n t DR-pln, until t DR-pln or G is omplt. W now sri ow t ov loritm works on xmpls tt rprsnt t ss. 5.. Exmpl For t D xmpl o points n istn onstrints in Cs in Fiur 6, Prt I sows t low rp n t lustr quu (s Stion ) tr t DR-pln or G s n onstrut. Wn t ol DR-plnnr strts to istriut G, it rts nw lustr quu or G n inrits t low rp in Stp. Atr t ol DR-plnnr in inis wit G, in Stp, t DR-pln o G n t low or lustr rp r sown in Prt II. Tn t nw DR-plnnr tris to omin tm in Stp n t rsults r sown in Prt III o t iur. T inl DR-pln o G wi is otin tr Stp is sown in Prt IV. For t D xmpl o points n istn onstrints in Cs in Fiur 7, Prt I sows t low rp n lustr quu tr t DR-pln or G s n omplt. Wn t DR-plnnr strts to istriut G, it rts nw lustr quu or G n inrits t low rp in Stp. It lso rmovs t lows on t n n mrks tm. Atr t ol DR-plnnr is inis wit G in Stp, t DR-pln o G n t low rp r sown in Prt II. Sin G n G ovrlp on points, t nw DR-plnnr omins tm in Stp n t rsults r sown in Prt III. T inl DR-pln o G wi is otin tr Stp is sown in Prt IV. 7

18 G G G G G G G G I II G G G G G G G G G III IV Fiur 7: Cs : input roups ovrlp on rontir vrtis G G G G G G G G I II G G G G G G G G G III IV Fiur 8: Cs : t ovrlpp prt inlus non-rontir vrtis 8

19 For t D xmpl o points n istn onstrints in Cs in Fiur 8, Prt I sows t low rp n lustr quu tr t DR-pln or G s n onstrut. Wn t ol DR-plnnr strts to istriut G, it rts nw lustr quu or G n t low rp in Stp. Atr t ol DR-plnnr is inis wit G in Stp, t DR-pln o G n t low rp r sown in Prt II. Sin G n G ovrlp on points, t DR-plnnr omins tm in Stp n t rsults r sown in Prt III. T inl DR-pln o G wi is otin tr Stp is sown in Prt IV. 5. Proo o Corrtnss n Complxity W sow ow t rquirmnts o Stion 5. r mt. From Stps n, w know tt or omposition G i, t loritm rts nw DRpln. Tis nsurs, y t proprtis o t ol FA DR-plnnr ivn in Stion, tt tur G i pprs in t output DR-pln provi it is vli lustr, n otrwis omplt omposition into mximl lustrs is otin. In Stp, t DR-plnnr ks t pir o t top lvl lustrs o DR-pln or G i n tos o t omin DR-pln or G, G,..., G i n omin tm i possil. Bus tis pross is xut or G i tr its DR-pln is stlis, t loritm omins t DR-plns or ll t G i s to iv t DR-pln or G. Tus Rquirmnt is mt. In Stp, t nw loritm links two opis o t sm lustr in irnt G i s. Tus, tr is only on opy o tis lustr in t DR-pln o G. So, t onsistny twn t DR-plns is nsur, stisyin Rquirmnt. Finlly, t low inormtion in t ss is itr us s su, opi n rstor, or is lk up s unistriut units wi will r-istriut in Stp y t ol DR-plnnr. Tis urnts tt low inormtion rmins urt trouout, ivn orrtnss o ol DR-plnnr. Also, noti tt or ny ivn, t numr o tims it is r-istriut is no mor tn t numr o roups or turs in t tur irry tt sr tt. Tis nsurs tt ny itionl tim spnt (yon tt o t oriinl DR-plnnr) is proportionl to t numr o turs in t input tur irry; try nsurin Rquirmnt. 5. Prsrvin proprtis o t ol FA DR-plnnr Inorportion o turs into t DR-pln lvs ntirly unn mny sirl proprtis o t t output DR-pln mntion in Stion, simply us t ol DR-plnnr is ll y t Distriut- Groups rivr t st to tully onstrut t DR-pln. Ts inlu proprtis su s ttion n rtriztion o ovr n unr onstrinnss [, 5], ompltnss [, ], systmti orrtion o unronstrinnss y ivin so-ll ompltion onstrints mnility to iint upts (ition or ltion or moiition) o omtri primitivs or onstrints [8, 9], or nvition o t solution sp [0]. Nxt, w rily isuss som sirl proprtis o t FA DR-pln tt r t y t tur inorportion loritm s wll s proprtis tt r only rlvnt in t prsn o turs. 5.. Complxity n Wit Lt n is t numr o vrtis o t input onstrint rp n k is t numr o turs in t input tur omposition. Usin t rumnt ivn in t proo o orrtnss n omplxity, n t omplxity o t ol DR-plnnr, t ollowin ol. I ll t turs r itr isjoint or ontin on notr (Cs ). nw loritm s tim omplxity is O(n ), wit O(n ) (t omplxity o t ol DRplnnr). I t turs intrst on trivil surps or ontin on notr (Cs ), t omplxity is O(n +k), wit O(n +k). Finlly i t turs oul intrst on nontrivil surps, t st oun on omplxity is O(n k), wit O(n k) (w omit inr, ut siniintly mor umrsom omplxity xprssion in trms o sizs o t intrstions t.) Ts r t st omplxitis on n xpt. T irst tor is t omplxity o t unrlyin ol FA DR-plnnr n in typil ss, t son tor is insiniint. 9

20 5.. Optimlity Conrnin optimlity, t FA DR-plnnr s st oi pproximtion tor is unt y t nw umnttion. T proo is t sm s or t ol DR-plnnr. [] Amon ll DR-plns tt inorport t ivn input tur omposition, t FA DR-plnnr umnt y t tur inorportion loritm n in on wos mximum n-in lustr s n-in tt is tmost onstnt tor lrr tn t optimum. Also, s mntion rlir, t tur inorportion os not t ompltnss proprty, so t loritms o [] n [] n still us to in n optiml ovrin st. 5.. Corrtion o Ovronstrints Anotr proprty o FA DR-plns tt is supriilly t y t prsn o turs is t systmti orrtion o ovronstrints, i.., t mto prsnt in [5]. Clrly tur inorportion os not t t ility to tt ovronstrints n isolt o omplt st o ovronstrints tt n rmov witout mkin t ntir rp unronstrin. Howvr, orrtion o ovronstrints typilly rsults in rmovin som lustrs in t DR-pln, sin ty om unronstrin, ltou t ntir rp rmins wll-onstrin. In t prsn o turs, it is rsonl to rquir tt no tur tt ws prviously lustr is m unronstrin y t orrtion, i.., t st o so-ll ruil ovronstrints is smllr. Howvr, t ovronstrint orrtion mto o [5] xpliitly provis list o ruil ovronstrints irtly ssoit wit lustr in t DR-pln. S Stion or initions. Hn, t rquir moiition is stritorwr: t nw st o ruil ovronstrints tt prsrv turs is t union o ll t ruil sts o ovronstrints irtly ssoit wit lustr tur in t DR-pln, totr wit t union o ll t ruil sts o ovronstrints or lustrs tt r not snnts o ny lustr tur in t DR-pln. 5.. Uptin t Ftur irry Finlly, proprty list unr t output rquirmnts o t tur inorportion prolm in Stion is t ility to upt t input tur omposition n orrsponinly iintly upt t DR-pln. Rmovl o tur is stritorwr. I t tur is lustr it simply ntils t rmovl o t orrsponin no C rom t DR-pln, n ll o its snnts tt r inssntil ilrn o tir otr prnts wo r not snnts o C (s Stion or inition o ssntil lustrs). I t tur is not lustr, tn ll o its mximl propr lustrs r prsnt s nos in t DR-pln n ts r trt lik C ov. T DR-plnnr os not n to involv in tis simpl it o t DR-pln. Aition o tur is mor involv. Tr r two ss. In t s wr t tur is not ontin witin n xistin lustr o t DR-pln (it oul ontin in t sinl root i t rp is wllonstrin), tn t ition o t tur is stritorwr sin it will ntr t uppr most lvl o t DR-pln. It is simply trt y t Distriutroups mto s tou it is (lst) il o t root o t input tur omposition. I t tur ontins nw omtri lmnts n onstrints, ts r pross usin t upt mto or FA DR-plns, ivn in [8, 9]. In s t nw tur F is ontin witin on or mor o t xistin lustrs C i in t DR-pln, it is irst ssum to lustr n insrt in t DR-pln s il o t C i n s prnt o ll t mximl lustrs D i tt r ontin in F n r prsnt in t su-dr-pln root t ny o t C i s. Sin F lis insi n lry pross lustr, no low inormtion is vill. A lustr or low rp o F is rt y usin t rontir vrtx simpliitions o ts mximl lustrs. Ts rontir vrtis r onnt usin s rom t oriinl rp. A lustr quu wit ts lustrs is rt n ts lustrs r trt s t top lvl o t DR-pln or F onstrut so r. I.., Stps n o t DistriutGroup mto on F r xut, tkin t su-dr-pln root t F s t input tur omposition n ssumin tt DistriutGroup s lry n ll on t ilrn o F in tis tur omposition. Durin Stp, sin non o t s in t low or lustr rp onstrut or F v n istriut, Pusoutsi n otr mtos nnot ssum tt t s in t lustr rp or F v n istriut, DistriutE is run in on ts s. 0

21 5..5 Implmnttion A til psuoo tt inlus t nw tur inorportion loritm n oun in [9]. Doumnt opnsour o n ownlo t [5] (us post-dmr-00 vrsions or D) To us t tur inorportion option tr opnin t min sktr winow, n tr pullin up (or rwin) skt: prss trl ky n lt lik t ojts wi soul in t sm tur. You will s t olor o ll slt ojts is n. Tip: you n us lt mous utton to rw rtnl to slt ojts quikly, tn us trl + lt lik to moiy slt st. ( trl + lt lik on slt ojt woul unslt it.) Clik Dsin mnu, tn lik Mk nw tr to rt tur irry (inpnnt tur irris or t sm onstrint rp provi y irnt usrs or multipl viws, r omin to orm sinl omposit tur irry, intrnlly). Tn lik Mk nw roup to rt t tur. For turs, you oul simply slt t primitiv ojts in tm n lik Mk nw roup. You n k t turs y likin t roup t in rit-ottom o t winow. Not. Rll rom Stion tt t o-rii FA DR plnnr onsir r os not l wit impliit onstrint pnns. Howvr, t mor nrl, moul-rii FA DR-plnnr [] ls wit ll known typs o onstrint pnns su s nns n ins. Wil inorportion o tur irry into t t moul-rii FA DR-plnnr [] s n implmnt in FRONTIER [, 5], its sription n nlysis r yon our urrnt sop. W woul lik to not tt t ttion o moul-riiity ruilly rlis on t ompltnss o t unrlyin o-rii DR-plnnr, wi is unn y t tur inorportion loritm. Mor siniintly, t notion o moulrii lustr inlus so-ll pnnt lustrs tt r not sl-ontin, ut ty n to rsolv tr otrs, imposin solvin priority orr. DR-plnnrs tt n l wit su lustrs v n in inorportin tos turs - s in prourl istory s rprsnttions - wos vry inition is s on prviously in turs. Rrns [] S. Ait-Aoui, R. Jou, n D. Milui. Rution o onstrint systms. In Compurpis, ps 8 9, 99. [] V. All n S. Ann. Ftur-s molin ppros or intrt mnuturin: stt-o-trt survy n utur rsr irtions. Intrntionl Journl or Computr Intrt Mnuturin, 8: 0, 995. [] W. Boum, I. Fuos, C. Homnn, J. Ci, n R. Pi. A omtri onstrint solvr. Computr Ai Dsin, 7:87 50, 995. [] B. Brurlin. Construtin tr-imnsionl omtri ojt in y onstrints. In ACM SIG- GRAPH. Cpl Hill, 986. [5] G. Bruntti n B. Golo. A tur s ppro towrs n intrt prout mol inluin onptul sin inormtion. Computr Ai Dsin, : , 000. [6] Hnry Crpo. Struturl riiity. Struturl Topoloy, :6 5, 979. [7] Hnry Crpo. T ttrrl-otrl truss. Struturl Topoloy, 7:5 6, 98. [8] K.J. Krkr, M. Domn, n W.F. Bronsvoort. Mintinin multipl viws in tur molin. In ACM/SIGGRAPH Symposium on Soli Molin Fountions n CAD/CAM Applitions, ps 0. ACM prss, 997. [9] U. Dorin n B. Brurlin. A lrtiv molin systm. In P. Brunt, itor, CAD Systms Dvlopmnt -Tools n Mtos, p To ppr. SprinrVrl, 999. [0] I. Fuos. Gomtri Constrint Solvin. PD tsis, Puru Univrsity, Dpt o Computr Sin, 995.

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