Generalized swap operation for tetrahedrizations

Transcription

1 Gnrliz swp oprtion for ttrhriztions B. Lhnr 1, B. Hmnn 2, G. Umluf 3 1 Dprtmnt of Computr Sin, Univrsity of Kisrslutrn, Grmny lhnr@s.uni-kl. 2 Institut for Dt Anlysis n Visuliztion (IDAV), Dprtmnt of Computr Sin, Univrsity of Cliforni, Dvis, USA hmnn@s.uvis.u 3 Dprtmnt of Computr Sin, HTWG Constn, Grmny umluf@htwg-konstnz. Astrt. Msh optimiztion of 2D n 3D tringultions is us in multipl pplitions xtnsivly. For xmpl, msh optimiztion is ruil in th ontxt of ptivly isrtizing gomtry, typilly rprsnting th gomtril ounry onitions of numril simultion, or ptivly isrtizing th ntir sp ovr whih vrious pnnt vrils of numril simultion must pproximt. Togthr with oprtions ppli to th vrtis th so-ll g or f swp oprtions r th uiling lok of ll optimiztion pprohs. To sp up th optimiztion or to voi lol minim of th funtion msuring ovrll msh qulity ths swps r omin to gnrliz swp oprtions with lss lol impt on th tringultion. Dspit th ft tht ths swp oprtions hng only th onntivity of tringultion, it pns on th gomtry of th tringultion whthr th gnrliz swp will gnrt inonsistntly orint or gnrt simplis. Bus ths r unsirl for numril rsons, this ppr is onrn with gomtri ritri tht gurnt th gnrliz swps for 3D tringultion to yil only vli, non-gnrt tringultions. Ky wors: 3D tringultion; gomtri onitions; swp oprtions. 1 Introution Tringultions of points in 2D sp for msh of tringls or points in 3D sp for msh of ttrhrr ruilly importnt for numrous pplitions nountr in sintifi n nginring pplition, inluing numril simultion, shp pproximtion, or visuliztion. In sttr t pproximtion [8, 15, 20] 2D tringultions r us to fin piwis linr ors pproximtion of ns t st, ssigning hight vlu for vry vrtx. This thniqu n lso us for img omprssion [4, 5, 18, 21] n vio omprssion [17, 19]. For rvrs nginring [1, 6, 9, 12], th 2-mnifol surf to ronstrut is pproximt y 3D tringultion tht ontins no ttrhr. For mhnil nginring n physil simultions [14, 24], 3D tringultions r us s mshs for finit lmnt mthos.

2 2 B. Lhnr, B. Hmnn, G. Umluf For ll of ths pplitions th tringultion ns to optimiz with rspt to n pplition-pnnt ost funtion msuring msh qulity s on multitu of propr msh qulity vrils, inluing, for xmpl, point istriution, pproximtion rror [7,18], tringl shp [10], ihrl ngls [14], t. Th optimiztion pross is usully s on simpl, lol hngs in th tringultions suh s rpositioning of vrtis [15], insrtions n rmovl of vrtis [7, 11] n g n f swps [22]. Whil th first of ths oprtions hng gomtry n onntivity of th tringultion th swps hng only th onntivity of tringultion. To sp up th optimiztion or to voi lol minim uring msh optimiztion multipl g n f swps r omin to gnrliz swp oprtions tht hng th onntivity of mor thn thr ttrhr of th tringultion [13, 17, 23, 25], s Stion 2. Howvr, it pns on th gomtry of th tringultion if gnrliz swp will gnrt flipp or gnrt simplis. W prsnt in this ppr gomtri ritri tht gurnt tht gnrliz swp oprtion in 3D tringultion will gnrt only vli, non-gnrt tringultions. 2 Rlt Work In gnrl, swp oprtion rpls -imnsionl simplis of tringultion ( 1) y othr simplis. It usully ffts only lol r of th tringultion, n hngs th onntivity of th tringultion without hnging th numr or position of th vrtis. Lwson [16] ws ws mong th first sintists stuying n pulishing swp oprtions systmtilly. H show tht + 2 points in imnsions, whih o not ll li in hypr-pln, hv ithr on uniqu tringultion T or two possil tringultions T 1 n T 2. Whih s hppns pns on th vrtx positions, s Figur 1 for th 2D s. In th lttr s, T 1 n T 2 iffr only in onntivity n th trnsformtion from T 1 to T 2 is ll swp oprtion s 1 2 (T 1 ) = T 2. Th opposit trnsformtion is s 2 1 (T 2 ) = T 1. Bus s 1 2 s 2 1 = s 2 1 s 1 2 = i, s 1 2 n s 2 1 r invrs oprtions. s 1 2 s 2 1 Fig. 1. Tringultions of four points in th 2D s.

3 Gnrliz swp oprtion for ttrhriztions 3 If T 1 is sust of lrgr tringultion T, th swp oprtion n ppli y rpling only th simplis of T 1 with thos of T 2, n lving ll simplis of T unhng, i.., T = (T \ T 1 ) T 2. Not tht th sust T 1 hs to tringultion, i.. it hs to fill th onvx hull of its vrtis, n must onvx. Aitionlly to ths si swps, on n onstrut gnrliz swp oprtions tht rpl st of simplis C of th tringultion y iffrnt st of simplis C. Thus, C n C r not rquir to ovr th onvx hull of thir vrtis. Sin th gnrliz swps r usully mor powrful, thy n l to goo tringultion with lss swp oprtions, ut r oftn lss ffiint. On wy to onstrut gnrliz swp oprtion is to omin squn of si swp oprtions to so-ll ompos swp oprtion. For th 2D s Yu t l. [25] us omintion of two g swp oprtions. If simpl g swp os not ru th ost funtion, thy swp th g n on of its jnt gs. Thus, th fft fs o not n to form onvx polygon for th ompos swp oprtions. Using th ompos swp oprtions n improv th optimiztion rsults signifintly. Conrning th 3D s, th st of swp oprtions is lrgr n mor vri thn in th 2D s. Agin, w n tgoriz thm into si swp oprtions n ompos swp oprtions. Aoring to Lwson [16], thr r fiv iffrnt sttings of fiv points,,, n in 3D sp, only two of whih hv two iffrnt tringultions n thrfor provi swp oprtions, s Figur 2. If thr points r ollinr, or four points,,, r oplnr with onv(,, ), or onv(,,, ) thr is only on possil tringultion, s Figurs 2 I., III., n V. If xtly four points r oplnr n form onvx quriltrl q thr r two possil tringultions with flipp igonls of q, s Figur 2 II. Bus th tringultion onsists of two lls for n ftr th swp, th swp is ll 2-2 swp. For th most gnrl s in whih ll fiv points r ornrs of onv(,,,, ) thr r lso two possil tringultions, s Figur 2 IV. Bus this swp rpls thr lls y two n vi vrs, it is ll 3-2 swp or 2-3 swp, rsptivly. Whn ppli to sust of tringultion T, th 2-2 swp is only possil if th two fs {,, } n {,, } r orr fs of T. If thy r intrior fs, th inint two lls lso hv to swpp, s Figur 3. This ls to th 4-4 swp, whih rpls four lls with four othr lls. In 3D lso omintion of si swp oprtions n mor powrful. Jo [13] systmtilly nlyz th possil sttings. Evry f of tringultion is ssign to nin iffrnt tgoris, sriing thir lol stting n thir sttus of ing trnsforml y si swp oprtion. H proposs st of ompos swp oprtions to trnsform fs tht r initilly not trnsforml, y first swpping jnt fs. For vry ompos swp oprtion, h lists th lls tht r rmov n rt. From this list, h provis ritri in [13] to omput th hng of ost funtion rsulting from h of th oprtions, if is th minimum of th osts of th iniviul lls. Anothr lss of ompos swp oprtions is th lss fin y th gnrliztions of th 3-2 n 2-3 swps, s [3, 24].

4 4 B. Lhnr, B. Hmnn, G. Umluf 2 2 swp 2 2 swp I. II. II. III. 3 2 swp 2 3 swp IV. IV. V. Fig. 2. Th iffrnt sttings of fiv points in th 3D s. Gnrliz 3-2 swp (G 32 ) A gnrliz 3-2 swp (G 32 ) n ppli to n g = {, } with n 3 inint lls C = { 1,..., n }, with i = {,, v i, v i+1 } n v n+1 v 1, s Figur 4 (lft). Th loop (v 1,..., v n ) is split into st of n 2 onnt fs F = {f 1,..., f n 2 }. Not tht th hoi of F is not uniqu. G 32 rpls th g with th fs F, whr th n lls C r rpl y th 2(n 2) lls C = {,1,,1,...,,n 2,,n 2 } with,i = f i {} n,i = f i {}. Gnrliz 2-3 swp (G 23 ) W sy f f = {v 1, v 2, v 3 } is snwih twn vrtis n, if th two lls inint to f r 1 = {, v 1, v 2, v 3 } n 2 = {, v 1, v 2, v 3 }. A gnrliz 2-3 swp (G 23 ) is ppli to st F = {f 1,..., f n 2 } of fs, whih r snwih twn two points n, s Figur 4 (right). A nw g = {, } is insrt into th tringultion, n th orr gs of F r onnt to th nw g to form th nw lls. Lt C = {,1,,1,...,,n 2,,n 2 } th st of lls inint to th fs,i = f i {} n,i = f i {} of F, n (v 1,..., v n ) th loop of vrtis fin y th orr gs of F. G 23 rpls th fs of F y th g = σ {,}, n th 2(n 2) lls of C r rpl y th n lls C = { 1,..., n }, with i = {,, v i, v i+1 }, n v n+1 v 1.

5 Gnrliz swp oprtion for ttrhriztions swp 4-4 swp f f Fig. 3. Th 4-4 swp is us if th fs of th 2-2 swp r no orr fs. G 23 is th invrs of G 32. Sin th hoi of fs is not uniqu in ithr irtion, pplying th on swp oprtion ftr th othr ls to th strt tringultion only if for oth swps th sm fs r hosn. Also not tht th 2-3 swp is spil s of G 23, th 3-2 swp of G 32, n th 4-4 swp spil s of G 23 n G 32. Th xution of G 32 n G 23 n rsult in invli tringultions. In Stions 4 n 5 w isuss nssry n suffiint gomtri onitions to nsur th vliity of th rsulting tringultion. Shwhuk [23] nots tht ths swps n rpl y sris of 2-3 n 3-2 swps, whr th intrmit tringultions r topologilly orrt, ut my ontin gnrt or invrt lls. In Stion 6 w show tht thr is lwys squn of 2-3, 3-2, n 4-4 swps to rpl G 23 or G 32 swp without gnrt or invrt lls. 3 Nottion In orr to fin th gnrliz swp oprtion in trms of onntivity hngs n ssoit gomtri onitions, w first just our nottion proprly. A 3D tringultion T = (V, E, F, C) (ttrhriztion) onsists of st of vrtis V, gs E V 2, fs F V 3 (tringls), n lls C V 4 (ttrhr). Thus, n g is pir of vrtis, f tripl of vrtis, n ll qurupl of vrtis. All ths ntitis r orr suh tht T is n orint simpliil 3-omplx, whr th gs of jnt fs n th fs of jnt lls r orr rvrsly. In this s, w ll T vli tringultion. W will us st oprtions to fin nw fs n lls, i.., for v 1 V, = (v 2, v 3 ) E, f = (v 2, v 3, v 4 ) F n C w fin n {v 1 } = (v 1, v 2, v 3 ) F, f {v 1 } = (v 1, v 2, v 3, v 4 ) C f (v 2, v 3 ) is su-tupl of f, f (v 2, v 3, v 4 ) is su-tupl of.

6 6 B. Lhnr, B. Hmnn, G. Umluf v 5 v 4 v 5 G 32 v 4 v 1 v G 1 23 f 1 f 2 f 3 v 2 v 3 v 2 v 3 Fig. 4. Th gnrliz 3-2 n 2-3 swps. Whil V, E, F, n C sri only th onntivity of th tringultion, gomtri rliztion of T is fin y ssoiting point v R 3 to vry vrtx v V. Th gomtri rliztions of n g V 2, f f V 3, or ll V 4 r thn fin s th onvx hull of th gomtri rliztions of thir vrtis, n r lso not in olf lttrs, f, n, rsptivly. Furthrmor, for st M of gs, fs, or lls, w not y M th union of th gomtri rliztions of th lmnts of M. Throughout this ppr, gomtri rliztions of lmnts of tringultion r not y olf lttrs. W sy vli tringultion T is onsistntly orint, whn th gomtri rliztions of ll lls hv th sm gomtri orinttion. Th orinttion of ll inus notion of orinttion on ll of its ontin k-su-simplis for k = 1, 2. A k-su-simplx is ll positivly orint if it is positivly orint in th k-imnsionl hyprpln ouning th nlosing (k + 1)-su-simplx with outwr pointing norml. This mns in prtiulr, tht ll fs of ll r positivly orint with rspt to th hlf-pln ouning th ll with norml pointing to th outsi of th ll. If th vrtis of ll r not ffinly inpnnt, it is ll gnrt, n if ll or ny of its k-su-simplis r not positivly orint, w ll it inonsistntly orint. Borr fs r fs of tringultion T tht r inint to only on ll in T, ll othr fs r ll innr fs. Anlogously, orr gs r inint to only on innr f, ll othr gs r ll innr gs Th orr of tringultion T is th st of ll its orr fs. If T is vli, onsistntly orint tringultion, th gomtri rliztion of its orr is 2-mnifol.

7 Gnrliz swp oprtion for ttrhriztions 7 Th ounry S of sust S of mnifol M r th points in S for whih vry ε-ll in M ontins points in M \ S. Not tht th trm orr is n ttriut of th onntivity of tringultion, whrs ounry is proprty of its gomtri rliztion. W n to provi som finitions onrning sphril projtions, whih w will us to stlish gomtri onitions for llowl swp oprtions. Dfinition 1. Th sphril projtion of point p R 3 onto th sphr S q with ntr q R 3 n rius r is fin s Π q (p) = q + r(p q)/ p q 2, p q. A projtion of st of points P R 3 \ {q} is th st of th projt points, Π q (P ) = {Π q (p) p P }. Som proprtis of th sphril projtion (without proof) r: If P is lin, Π q (P ) is ithr two ntipol points (for q P ), or hlf grt irl (for q / P ) of S q. If P is pln, Π q (P ) is ithr grt irl (for q P ), or n opn hlf sphr (for q / P ) of S q. If P = onv(p 1, p 2, p 3 ) is tringl n th pln fin y P os not ontin q, Π q (P ) is sphril tringl, oun y th projtion of th gs Π q (onv(p 1, p 2 )), Π q (onv(p 2, p 3 )), Π q (onv(p 3, p 1 )), whih r sgmnts of grt irls of S q. 4 Gomtri Conitions for G 32 For th gomtri onitions to stisfi for G 32 -swp s fin in Stion 2 w hv n g = (, ) with n inint lls tht is swpp. Th tringultion for n ftr th G 32 -swp is not y T n T. Conition 1 Th tringultion T = (V, E, F, C) is vli, n ll lls of T hv positiv orinttion. Conition 2 Th g is n innr g of T, i.., vry f f inint to is inint to xtly two lls f,1 f,2. Not tht th lst onition implis tht is not on th orr of T. Furthrmor, ths onitions inu n orr of th fs inint to. Lmm 1. All fs ontining n orr to form yli squn G = (g 1,..., g n ), i.., th inx i = 1,..., n of g i is unrstoo moulo n. Furthrmor, th ihrl ngls θ i twn g i n g i+1 (in th irtion to ) r in th intrvl (0, π), n sum to 2π.

8 8 B. Lhnr, B. Hmnn, G. Umluf Proof. Du to Conition 2, f g = (,, v) inint to is inint to two lls g,1, g,2. Both hv two fs inint to, on of th two is g, th othr ons r g 1 n g 2, rsptivly. Th sussor of g is th f g k of ll g,k on th positiv si of g (in th irtion to ), k = 1, 2. Th prssor of g is th othr f. Du to Conition 2 this rltion trmins yli sussor-grph without rnhs. Th ihrl ngl θ i twn f g i n its sussor g i+1 is th ihrl ngl t of th ll tht ontins oth fs. Thrfor, 0 < θ < π, us othrwis th ll woul invrt or gnrt, ontriting Conition 1. Sin th squn of fs is yli, it surrouns. It n only yl xtly on roun, us othrwis lls twn th fs woul intrst in thir intrior, whih ontrits Conition 1. Th sum of th ihrl ngls twn th fs is thrfor 2π. W not th ll twn g i n g i+1 s i, n th thir vrtx of g i s v i. Thus, Lmm 1 inus lso yli orr on th lls C = ( 1,..., n ) n vrtis V = (v 1,..., v n ) roun. Bus G 32 rpls th lls of C y othr lls, w ll C th fft rgion, n th orr fs of it r givn y C := {(, v 2, v 1 ), (, v 1, v 2 ),..., (, v 1, v n ), (, v n, v 1 )}, i.., C = f C f. Th lin through n is not y l = { + λ( ) λ R}. (1) Lmm 2. Thr is los loop of gs B = { 1,..., n } tht wins roun l xtly on. Proof. This follows from Lmm 1, whr i is th g of i opposit to. For th sphr S roun ontin in th onvx hull of ll lls ontining w st t I = Π () n not y t O th ntipol point of t I. Lt B = Π (B) th sphril projtion of B onto S. Sin B is los loop on S, it splits S into two prts SI n SO, whih r hrtriz y t I S I n t O S O, s Figur 5. Anlogously, S, t I, t O, B, SI, n S O r fin. Dfinition 2. A prtition of B is st F = {f 1,..., f m } of fs f i / F, whr 1. ll vrtis of f i long to gs of B, i.., f i V, 2. ll gs of f i r ithr gs of B or innr gs I, n 3. () vry g of B is inint to xtly on f of F, () vry g of I is inint to xtly two fs of F. Lmm 3. Evry prtition F of B hs n 3 innr gs n m = n 2 fs. Proof. As onsqun of Lmm 1 prtitioning B is quivlnt to tringultion of simpl polygon B in pln prpniulr to l without introuing nw vrtis. This polygon is th orthogonl projtion of B long irtion l. Sin vry simpl polygon with n vrtis n tringult with n 2 tringls (s [2]), i.., n 3 innr gs, th lim follows.

9 Gnrliz swp oprtion for ttrhriztions 9 t O S O B v 1 v 5 t I B v 4 S I v 2 v 3 Fig. 5. Trms us in sphril projtion with B in lu n B in grn. Th prtition F of B fins th lls tht r rt y th G 32 -swp. Evry f of th prtition is onnt to n to form two nw lls. Th st of nw lls is C = {,1,,1,...,,m,,m } with,j = f j {} n,j = f j {} for f j F. Not tht for n > 3 th prtitions n lso th G 32 -swp is not uniqu. It n hppn tht C ontins inonsistntly orint or gnrt lls. Thrfor, th G 32 -swp woul rsult in n invli tringultion n must not ppli. Whthr this is hppns pns on n B n lso on th hoi of F. W ll F vli prtition if ll lls in C r vli. Dpning on th gomtry, thr r thr iffrnt ss. For vry s w prsnt n xmpl for n = 4, so tht two iffrnt prtitions xist: F 1 = {(v 1, v 2, v 3 ), (v 1, v 3, v 4 )} n F 2 = {(v 1, v 2, v 4 ), (v 2, v 3, v 4 )}. For vry xmpl, = (0, 0, 1) n = (0, 0, 1). Furthrmor, th x n y oorints of v 1 to v 4 r ( 0.3, 0.3), (0.7, 1.3), (1.7, 0.3), n (0.7, 0.7), rsptivly. Evry prtition is vli For vry prtition F, ll lls in C r vli. For our xmpl, w hoos th z oorints to z 1 = z 2 = z 3 = z 4 = 0. Both prtitions F 1 n F 2 r vli in this s. Not, tht vry prtition is vli s long s th fft rgion C is onvx. whih is only th s if (s in this xmpl) ll v i r oplnr. But lso for non-onvx fft rgion ll prtitions n vli. Som prtitions r invli For som prtitions, thr r lls in C tht r invrt or gnrt. But othr prtitions r vli. For onrt xmpl, st th z oorints to z 1 = z 2 = z 3 = 0.8 n z 4 = 0.8. Hr,

10 10 B. Lhnr, B. Hmnn, G. Umluf F 1 is n invli prtition, us th ll (, v 1, v 3, v 4 ) is invrt, whil prtition F 2 is vli. All prtitions r invli It n lso hppn tht no vli prtition xists t ll. In this s, G 32 nnot ppli to. An xmpl for this s is z 1 = z 3 = 0.8 n z 2 = z 4 = 0.8. Hr, F 1 is invli us of th invrt ll (, v 2, v 4, v 3 ), F 2 is invli us of th invrt ll (, v 1, v 3, v 4 ). Ths xmpls show tht w n nothr onition tht nsurs tht F is vli prtition. Unr th ssumption tht Conitions 1 n 2 r stisfi, w foun four quivlnt formultions 3.2., 3.1., 3.3., n 3.4. for th missing onition. Will prov thir quivln ltr in Thorm 2. Bfor w sri th missing onition in til w n to fin th supporting pln pl(t) of tringl t s th ffin hull of its vrtis. Conition All lls,j n,j hv positiv orinttion Evry f i hs on its positiv si, n on its ngtiv si Th sphril projtion of th fs f i onto S is ontin in SI B, n th intrior of th innr gs is projt into SI (for S nlogously), Π p (f i ) S p I Bp, for ll i = 1,..., n, (2) for p {, }. Π p ( ) S p I, for ll I, (3) 3.4. Th intrior of th innr gs is sust of th intrior of th fft rgion, n th supporting plns of ll fs f i intrsts th lin l in th intrior of, C \ C, for ll I, (4) pl(f i ) l. (5) Thorm 1. If Conitions 1, 2, n 3 r mt, th tringultion T = (V, E, F, C ) with C = (C \ C) C (n E n F oringly) is vli. Proof. Du to Conitions 1 n 3.1., ll lls of C hv positiv orinttion. To prov tht thr r no hols in C, w hk for orr fs of th lls of C : Th fs i {p} for p {, } r orr fs of oth C n C. Th fs f j r inint to,j n,j, i.., f j is not on th orr of C. For th fs f = {p}, I, p {, }, th g is inint to two f s f j n f k, i.., f is inint to p,j n p,k. So, f is not on th orr of C. Thus, thr r no nw orr fs, i.., thr r no hols in C. Lmm 4. Conition 3.1. n Conition 3.2. r quivlnt. Proof. By finition, is on th positiv si of f i if n only if th ll,i hs positiv orinttion. Furthrmor, is on th ngtiv si of f i if n only if th ll,i hs positiv orinttion.

11 Gnrliz swp oprtion for ttrhriztions 11 Lmm 5. Conitions 3.1. n 3.2. imply Conition 3.3. Proof. To prov (2) w first show tht Π (F) is onnt rgion on S tht is oun y B. Thn w show tht t I Π (F). Du to Conition 3.1. is not in F, sin this woul us gnrt lls, n Π (F) is onnt rgion on S. For f i, f j F with ommon g I, th sphril tringls Π (f i ) n Π (f j ) shr th sphril g Π (). Du to Conition 3.1. th oth lls,i n,j hv positiv orinttion, so thy r on opposit sis of th pln P through n. Thrfor, Π (f i ) n Π (f j ) r lso on opposit sis of Π (), s Figur 6. This implis tht th intrior of ll innr gs of I is not projt to th ounry of Π (F). Th sm hols tru for ll intrior points of F. Thus, th ounry of Π (F) onsists of projtions of th orr gs of B. Consquntly, th intrior of Π (F) is not intrst y B, so Π (F) is ithr ompltly in SI B, or in SO B. f i f j P Fig. 6. Th projtions of f i n f j r on opposit sis of th projtion of th g. Sin B wins roun l on, th l lin intrsts F in t lst on f f i. Lt p = l f i. Bus is on th positiv si of f i (Conition 3.2.), Π (p) = t I. Thrfor, Π (F) S I B. Anlogously, on n show Π (F) S I B. Espilly, th intrior of Π ( ) os not intrst B, whih implis (3). Lmm 6. Conition 3.3. implis Conition 3.4. Proof. Lt I n innr g of F, n p n intrior point of. Du to Conition 3.3., p = Π (p) SI. W split S I into sphril tringls y ing gs from Π (v i ) to t I. At lst on of ths tringls ontins p. Lt this tringl t = (t I, Π (v l ), Π (v l+1 )), s Figur 7. Th ounry B (grn) is prtition into sphril tringls (r lins), is (v 1, v 4 )

12 12 B. Lhnr, B. Hmnn, G. Umluf (lu lin) n p. In this s, Π (p) is within th sphril tringl t = (t I, Π (v 4 ), Π (v 5 )). Th st of points tht r projt into t is fin s th intrstion of th hlf sps fin y th plns spnn y n on of th gs of t, i.., g 1 = (,, v l ), g 2 = (, v l+1, ), n g 3 = (, v l, v l+1 ), whih ontins th fourth point {,, v l, v l+1 } \ g i. Th point p nnot on th ngtiv si of g 1, g 2 or g 3, s this woul mn tht its img is not in t. Also, it nnot in th pln fin y n g 3, s this woul mn tht it is projt to B. With th sm rgumnt for Π, w otin th fs g 4 = (, v l, ), g 5 = (, v l+1, ), n g 6 = (, v l+1, v l ). Rmoving th runnt fs g 4 g 1 n g 5 g 2, w n onlu tht p is not on th ngtiv si of g 1 n g 2, n it is on th positiv si of g 3 n g 6. Ths four fs fin th ll i. Thus, p C, n p / C, proving (4). W still hv to prov (5). Assum thr xists f f in F with {q} = pl(f) l n, without loss of gnrlity, λ 0. This f hs t lst on intrior g I n w hos n ritrry point p. Now, p is projt to p whih lis outsi of SI. This ontrits (3) n, thus, provs (5). v 5 v k+1 v 4 p q q v k t I g k v 1 v 3 v k 1 v 2 Fig. 7. SI is ivi into sphril tringls (r lins), on of whih ontins Π (p). Fig. 8. Th intrstion of th xtnsion of sgmnt v k to q with l is twn n. Lmm 7. If Conitions 1 n 2 r stisfi, Conition 3.4. implis Conition 3.2. Proof. For n = 3 w hv I = n F = {f 1 }. Sin B irls roun l, thr must n intrstion of l n f 1. Du to Conition 3.4., this is twn n

13 Gnrliz swp oprtion for ttrhriztions 13, n us of th orr of th vrtis of f 1 s inu y Lmm 1, is on th positiv n on th ngtiv si of f 1, n Conition 3.2. is stisfi. W now onsir n > 3. Th prtition F ontins n 2 fs, th orr B hs n gs (s Lmm 3). If vry f of F h t most on g of B, thr woul t lst two gs in B lft. Sin no f of F n hv thr gs of B (othrwis B woul hv su-yl of thr gs), t lst two fs of F must hv two gs of B. Lt ˆF F th st of fs with two gs in B. Th lin l intrsts ithr on f of F in its intrior, or it intrsts n innr g of I n thrfor two fs of F on thir orr. In th s tht l intrsts n innr g, n th jnt fs of F r th only two fs in ˆF, thr n no othr fs in F, u to th following: if two fs with h two gs in B n oth shring ommon innr g, thir gs in B lry fin yl. Sin B os not ontin ny su-yls, thr n no furthr gs in B. In this s w hv n = 4. Sin th intrstion of l with f 1 n f 2 is twn n (Conition 3.4.), n us of th orr of th vrtis of f 1 n f 2, is on th positiv si of f 1 n f 2, n is on th ngtiv si. Thus, in this s Conition 3.2. is stisfi. For th rmining s thr is t lst on f f in ˆF tht hs no intrstion with l, us othrwis Conitions 1 n 2 wr violt. Lt f = (v k 1, v k, v k+1 ). Bus f os not intrst l, θ k 1 + θ k < π. Thus, th innr g = (v k 1, v k+1 ) nnot ross ny othr ll sis k 1 n k. Du to Conition 3.4., k 1 k. With g k = (,, v k ), th intrstion g k = {q}, with q in g k. Whn xtning th lin sgmnt from v k to q, it intrsts th sgmnt in its intrior in point q, us of (5) (s Figur 8). Sin v k n q r points in f, th lin through v k n q is lso in th pln of f, n so is q. From ths onsirtions n th vrtx orr of f, it follows tht is on th positiv n is on th ngtiv si of f. Thus, f fulfills Conition Now w rmov f from F, i.., F oms F \{f}, B oms (B\{ k 1, k )}) {}, n I oms I \ {}. This nw g yl B still stisfis Conitions 1 n 2, ut hs on g lss. This prour n rpt until n = 3, or n = 4 n l intrsts oth fs in F. Thorm 2. If Conitions 1 n 2 r stisfi, Conitions 3.2., 3.1., 3.3., n 3.4. r quivlnt. Proof. This follows irtly from Lmmt 4, 5, 6, n 7. 5 Gomtri Conitions for G 23 W us th sm nottion s in Stion 4, i.., F = {f 1,..., f m } is st of fs snwih twn n, suh tht F is onnt 2-mnifol. Th g st B = { 1,..., n } r th orr gs of F. Th orr of gs in B is inu y th orr of ounry gs in F. Th tringultion for n ftr th G 23 -swp is not y T n T. W fin th orinttion of f i so tht is on th positiv si of f i. Th lls

14 14 B. Lhnr, B. Hmnn, G. Umluf inint to ths fs r C = {,1,,1,...,,m,,m } with p,i = f i {p} for i = 1,..., m n p {, }. Th nw g in T is = (, ). Nxt w fin th onitions for whih G 23 will rsult in vli tringultion. Conition 4 Th tringultion T = (V, E, F, C ) is vli, n ll lls of T hv positiv orinttion. Conition 5 Th gs of B form xtly on simpl yl (v 1,..., v n ). This onition nsurs tht th fs in F r onnt vi gs, tht thr is only on onnt omponnt of fs, n tht th fs form oun 2-mnifol without hols. Exmpls of sts F tht violting Conition 5 r shown in Figurs 9(), 9(), n 9(). Conition 6 All vrtis inint to f in F r on th orr B. Conition 6 th sn of intrior vrtis in F tht r not prt of B. Thos intrior vrtis woul rmov y G 23, ut swp my only moify th onntivity, ut not, rmov, or mov vrtis. Figur 9() shows n xmpl of st F tht violts Conition 6 u to n intrior vrtx. f 1 f 2 f 3 f 11 f 10 f 1 f 2 f 3 f4 f 5 f 6 f 1 f 2 f 3 f 4 f 1 f 2 f 3 f 4 f 9 f 8 f 7 f 4 f5 () Violting Conition 5. Con- () Violting ition 5. () Violting Conition 5. Con- () Violting ition 6. Fig. 9. Exmpls of sts F violting Conitions 5 or 6. Lmm 8. If Conition 6 is stisfi, th numr of vrtis in B is n = m+2. Proof. If Conition 6 is stisfi, F is prtition of B. Consiring Lmm 3 w n onlu m = n 2. Thrfor, n = m + 2. Th G 23 -swp will now rpl th lls C y th lls C = { 1,..., n } with i = (,, v i, v i+1 ) n fs g i = (,, v i ), whr th inx i is unrstoo moulo n. Conition 7 On of th quivlnt following onitions hols:

15 Gnrliz swp oprtion for ttrhriztions All lls i hv positiv orinttion Th ihrl ngl θ i twn th fs g i n g i+1 (in ountrlokwis irtion, sn from in irtion ) is in (0, π). Lmm 9. Conition 7.1. n Conition 7.2. r quivlnt. Proof. i hs positiv orinttion if n only if i is onsistntly orint or nongnrt. This is tru if n only if th innr ihrl ngl θ i is in (0, π). Thorm 3. If Conitions 4, 5, 6, n 7 r mt, th tringultion T = (V, E, F, C) with C = (C \ C ) C (n E n F oringly) is vli. Proof. Du to Conitions 4 n 7.1., ll lls of C hv positiv orinttion. To prov tht thr r no hols in C, w hk for orr fs of th lls of C: Th fs i {p} for p {, } r orr fs of oth C n C. Th fs g i r inint to i 1 n i, i.., g i is not on th orr of C. Thus, thr r no nw orr fs, i.., thr r no hols in C. 6 Rpling Gnrliz Swps y Sris of Bsi Swps In [23] Shwhuk show tht th multi-f rmovl (quivlnt to G 23 ) n g rmovl (quivlnt to G 32 ) n rpl y sris of si 2-3 n 3-2 swps. Th intrmit tringultions r topologilly orrt, ut my ontin inonsistntly orint or gnrt ttrhr. W will show tht thr lwys xists sris of si 2-3, 3-2, n 4-4 swps to mimi th fft of G 23 - n G 32 -swp, whr ll intrmit tringultions r vli. This rsult shows tht th G 23 - n G 32 -swps o not itionl potntil tht is not lry possil with 2-3, 3-2 n 4-4 swps. An optimiztion prour lik simult nnling shoul thortilly l to fin nr-optiml solution lso without utilizing G 23 n G 32. In prti, th onvrgn rt n inrs y implmnting G 23 n G Rpling G 32 Lt n innr g of tringultion T, B th st of orr gs, n F vli prtition of B, so tht th Conitions 1, 2, n 3 for G 32 r stisfi. Thorm 4. Th sm fft s th G 32 swp oprtion of n prtition F n otin y sris of ithr n 3 si 2-3 swps follow y 3-2 swp, or n 4 si 2-3 swps follow y 4-4 swp, for n 4.

16 16 B. Lhnr, B. Hmnn, G. Umluf Proof. W us th sm rgumnts s in th proof of Lmm 7. For n = 3, F onsists of xtly on f f 1, n th vrtis of f 1 irl roun xtly on. Thrfor, th onitions r stisfi to pply 3-2 swp to, so w n sustitut G 32 y singl 3-2 swp. For n = 4, n th singl innr g = (v i, v i+2 ) with i {1, 2} intrsts with, th quriltrl (v i,, v i+2, ) is plnr n onvx, fulfilling th onitions of 4-4 swp. This 4-4 swp rpls y n th four lls of C with th four lls of C. Thus, th G 32 swp n rpl y singl 4-4 swp. If n = 4 n n o not intrst, or if n > 4, thr is t lst on f in F with two gs in B tht os not intrst. Lt this f f j = (v i 1, v i, v i+1 ). As in th proof for Lmm 7, th g = (v i 1, v i+1 ) intrsts th f g i in its intrior, so th lls i 1 n i fulfill th onition for 2-3 swp. This swp rmovs i 1 n i from th tringultion, s,j n,j n tmporry nw ll = (,, v i 1, v i+1 ). Th rmining lls (C \ { i 1, i }) {} togthr with th ru prtition F \ {f j } n th ru orr (B \ {(v i 1, v i ), (v i, v i+1 )}) {} fulfill Conitions 1 3. So, G 32 n ppli to th ru stting. By inution, th ru stting n pross with ithr (n 1) swps, follow y 3-2 swp, or with (n 1) swps, follow y 4-4 swp. Aing th 2-3 swp to rmov f j, th lim follows. 6.2 Rpling G 23 Sin th G 23 oprtion is th invrs of th G 32 oprtion for th sm prtition F, G 23 n rpl y sris of si swps. Thorm 5. Th sm fft s G 23 oprtion of prtition F snwih twn n n otin y sris of ithr singl 2-3 swp, follow y n swps, or singl 4-4 swp, follow y n swps. Proof. Whil G 23 rpls th lls C y lls C, G 32 os th invrs. G 32 n sustitut y sris of si swp oprtions G 32 = s 1 s 2 s m, with m ing ithr n 2 (s m ing 3-2 swp) or n 3 (s m ing 4-4 swp), s in Thorm 4. For th sm hoi of F, w hv G 23 = G 32 1 = (s 1 s m ) 1 = s 1 m s 1 1. Th invrs of 3-2 swp is 2-3 swp n vi vrs, n th invrs of 4-4 swp is orrsponing 4-4 swp. W strt in G 23 with s 1 m, whih is ithr 2-3 swp or 4-4 swp. Thn w pro with ithr n 3 or n swps. 7 Conlusions W hv prsnt iffrnt gomtri onitions for gnrliz swp oprtions 3D tringultion. Ths onitions r prov to quivlnt, suh tht on

17 Gnrliz swp oprtion for ttrhriztions 17 n us tht prtiulr onition in prti tht is most pproprit givn th spifi ns of n implmnttion. In msh optimiztion pplition ths swp oprtions r us to sp up th optimiztion pross n to ttnut gtting stuk in lol minim. Furthrmor, w hv shown tht th gnrliz swp oprtions n rliz y simpl 3-2, 2-3, n 4-4 swps, whih simplifis th implmnttion signifintly. This omposition of th gnrliz swp gurnts t th sm tim, tht ll intrmit tringultions r onsistntly orint n o not ontin gnrt lls, using numril prolms in rtin pplitions. Bs on ths onitions, our futur rsrh plns r fous on pplitions of 3D msh optimiztions,.g., in vio omprssions or io-mil n io-mhnil simultions. Aknowlgmnts This work ws prtly fun y th Grmn Rsrh Fountion (DFG) within th Intrntionl Rsrh Trining Group 1131, Visuliztion of Lrg n Unstrutur Dt Sts t th Univrsity of Kisrslutrn. This work ws lso support y th Ntionl Sin Fountion unr ontrt ACI (CAREER Awr) n lrg Informtion Thnology Rsrh (ITR) grnt. Rfrns 1. N. Amnt, S. Choi, n R. K. Kolluri. Th powr rust, unions of lls, n th mil xis trnsform. Computtionl Gomtry, 19(2 3): , M. Brg, M. vn Krvl, M. Ovrmry, n O. Shwrzkopf. Computtionl Gomtry. Springr, 2n ition, H. L. Cougny n M. S. Shphr. Prlll rfinmnt n orsning of ttrhrl mshs. Int. J. Numr. Mth. Engng., 46: , L. Dmrt, N. Dyn, n A. Isk. Img omprssion y linr splins ovr ptiv tringultions. Signl Pross., 86(7): , L. Dmrt, N. Dyn, A. Isk, n M. Flotr. Aptiv thinning for trrin molling n img omprssion. In N. A. Dogson, M. S. Flotr, n M. A. Sin, itors, Avns in Multirsolution for Gomtri Molling, pgs Springr, Klus Dnkr, Burkhr Lhnr, n Gorg Umluf. Onlin tringultion of lsrsn t. In R. Grimll, itor, Proings of th 17th Intrntionl Mshing Rountl 2008, pgs , N. Dyn, D. Lvin, n S. Ripp. Dt pnnt tringultions for piwis linr intrpoltions. IMA Journl of Numril Anlysis, 10(1): , Jn M. Grln n P. Hkrt. Fst polygonl pproximtion of trrins n hight fils. Thnil rport, CS Dprtmnt, Crngi Mllon Univrsity, M. Grln n P. S. Hkrt. Surf simplifition using quri rror mtris. In SIGGRAPH 97, pgs , M. Grln n Y. Zhou. Quri-s simplifition in ny imnsion. ACM Trns. Grph., 24(2): , H. Hopp. Progrssiv mshs. In SIGGRAPH 96, pgs , 1996.

18 18 B. Lhnr, B. Hmnn, G. Umluf 12. H. Hopp, T. DRos, T. Duhmp, J. MDonl, n W. Stützl. Surf ronstrution from unorgniz points. Computr Grphis, 26(2):71 78, B. Jo. Constrution of thr-imnsionl improv-qulity tringultions using lol trnsformtions. SIAM J. Si. Comp., 16(6): , B. M. Klingnr n J. R. Shwhuk. Aggrssiv ttrhrl msh improvmnt. In 16th Int. Mshing Rountl, pgs 3 23, O. Krylos n B. Hmnn. On simult nnling n th onstrution of linr splin pproximtions for sttr t. IEEE TVCG, 7(1):17 31, C. L. Lwson. Proprtis of n-imnsionl tringultions. CAGD, 3: , B. Lhnr. Mshing Thniqus for Img/Vio Comprssion n Surf Ronstrution. PhD thsis, TU Kisrslutrn, Grmny, B. Lhnr, G. Umluf, n B. Hmnn. Img omprssion using t-pnnt tringultions. In G. Bis, itor, ISVC 2007, pgs , B. Lhnr, G. Umluf, n B. Hmnn. Vio omprssion using t-pnnt tringultions. In Computr Grphis n Visuliztion 08, pgs , H. Prini. An improv rfinmnt n imtion mtho for ptiv trrin surf pproximtion. In Proings of WSCG, pgs , V. Ptrovi n F. Küstr. Optimiz onstrution of linr pproximtions to img t. In Pro. 11th Pifi Conf. on Comp. Grphis n Appl., pgs , L. L. Shumkr. Computing optiml tringultions using simult nnling. CAGD, 10(3-4): , J. R. Shwhuk. Two isrt optimiztion lgorithms for th topologol improvmnt of ttrhrl mshs, Unpulish mnusript, jrs/pprs/g.pf. 24. J. R. Shwhuk. Wht is goo linr lmnt? Intrpoltion, onitioning, n qulity msurs. In 11th Int. Mshing Rountl, pgs , X. Yu, B. S. Mors, n T. W. Srrg. Img ronstrution using tpnnt tringultion. IEEE Comput. Grph. Appl., 21(3):62 68, 2001.