Computing the 3D Voronoi Diagram Robustly: An Easy Explanation

Transcription

1 Computing the 3D Voronoi Digrm Roustly: An Esy Explntion Hugo Leoux Delft University of Tehnology (OTB setion GIS Tehnology) Jffln 9, 2628BX Delft, the Netherlns Astrt Mny lgorithms exist for omputing the 3D Voronoi igrm, ut in most ses they ssume tht the input is in generl position. Beuse of the mny egeneries tht rise in 3D geometri omputing, their implementtion is still prolemti in prtie. In this pper, I esrie simple 3D Voronoi igrm (n Deluny tetrherliztion) lgorithm, n I explin, y giving s mny etils n insights s possile, how to ensure tht it outputs orret struture, regrless of the sptil istriution of the points in the input. 1. Introution Mny omputer sientists n mthemtiins onsier the Voronoi igrm (VD) s eing the most funmentl sptil struture euse it is very simple, n yet is so powerful tht it helps in solving mny theoretil prolems, n is lso useful in gret mny pplition omins [2, 32]. Algorithms tht n ompute effiiently n roustly the struture re therefore of the utmost importne. Severl optiml lgorithms se on ifferent prigms, n lso implementtions of these lgorithms, hve een propose over the yers. The mjority of these lgorithms o not ompute iretly the VD, ut losely-relte struture, the Deluny tringultion. As expline in Setion 2, the two strutures re tully ul, whih mens tht the knowlege of one implies the knowlege of the other one (i.e. if one hs only one struture, he n lwys extrt the other one). As first pointe out y Boots [5], it is esier, from n lgorithmi n t struture point of view, to mnge tringles over ritrry polygons (they hve onstnt numer of verties n jent ells). When the VD is neee, it n simply e extrte from its ul. This hs the itionl vntge of speeing up lgorithms euse when the VD is use iretly intermeite Voronoi verties tht will not neessrily exist in the finl igrm nee to e ompute n store. The uthors of most ppers in the omputtionl geometry literture ssume tht they re working in perfet worl, i.e. they stte tht their lgorithms re vli if n only if the input is in generl position (whih n hve ifferent menings epening on the geometri prolem t hn). When eveloping n lgorithm n nlysing its omplexity, this ssumption usully mkes sense euse one wnts to voi eing overwhelme with tehnil etils (tht re usully seen s implementtion etils), n lso wnts to simplify the omplexity nlysis (to e le to ompre with other lgorithms). Moreover, s stte in [8, pge 9], egenerte ses n lmost lwys e hnle without inresing the symptoti omplexity of the lgorithm. The result is tht the hnling of egenerte ses is usully left to the progrmmer. However, moifying n lgorithm to mke it roust for ny inputs n e in some ses n intrite, timeonsuming n error-prone tsk euse egeneries hve the effet of inresing onsierly the numer of speil ses tht must e hnle. For non-geometri lgorithms suh s sorting lgorithm, it n e esy sine only the se of two keys eing equl must e hnle, ut geometri lgorithms n hve ozens, or even hunres of speil ses [18]. In some ses, moifying the originl lgorithm so tht it is roust ginst ll input n e even more time-onsuming n teious thn the esign of the originl solution in generl position. A simple GISrelte exmple is tht of Dougls [12] who esries his erly ttempts to implement routine to etermine where two line segments in plne interset. Wht seeme like simple prolem t first, turne out to e n implementtion nightmre euse of the prolems use y vertil lines, tolerne, segments tht re lose to prllel, et. (his routine finlly h 36 ifferent ses). He sttes: All of these inonsistenies eventully rg the progrmmer own from his high

2 level mth (i.e. lger), through omputer lnguge (i.e. FORTRAN), into the relm of the mhine methos tully use to perform rithmeti opertions n their restritions. Amittely, this prolem n now e solve rther esily if vetor representtions re use inste of mthemtil equtions [33], ut I elieve it gives goo exmple of the iffiulties of eling with speil ses. This pper is onerne with the implementtion of n lgorithm to onstrut the 3D Deluny tetrherliztion (DT), n thus the 3D Voronoi igrm. Geometri omputing in 3D spe is known to e plgue with speil ses, n the roust implementtion of the 3D DT is iffiult prolem in prtie [14, 38]. In the following, I esrie, y giving s mny etils n insights s possile, 3D DT lgorithm n I explin how to ensure tht it outputs orret solution, regrless of the sptil istriution of the points in the input. Note tht ertin numer of implementtions re known (it is for instne implemente in CGAL 1 ), so tht wht is presente in this pper is not entirely new. However, there is to my knowlege no omprehensive esription tht tkes into ount ll the possile egeneries, n explins how to hnle them. The etils n insights given in the following re se on my own experiene while implementing suh n lgorithm, n lso on the few etils tht re ville in sientifi ppers n ooks. It is my hope tht this pper will help others implement the 3D VD/DT, n tht they will not spen ountless hours fixing their oe to mke it roust. The lgorithm, whih is esrie in Setions 4 n 5, is flip-se inrementl lgorithm, s first propose y Joe [21]. Notie tht it is not the fstest solution in prtie, ut oneptul simpliity n roustness were fvoure over theoretil results n solutions tht woul e intrite to implement. Still, s emonstrte in Setion 7, it tkes in prtie resonle time for the prolem tht nees to e solve, n it is lso reltively esy to implement. The importnt prolem of the roustness of the rithmeti use y omputer is lso isusse in Setion Dulity Between the VD n the DT Let S e set of n points in 3-imensionl Eulien spe R 3. The Voronoi ell of point p S, efine V p, is the set of points x R 3 tht re loser to p thn to ny other point in S. The union of the Voronoi ells of ll generting points p S form the Voronoi igrm of S, efine VD(S). In three imensions, V p is onvex polyheron. 1 The Computtionl Geometry Algorithm Lirry. p α () () () () Figure 1. Dulity in 3D etween the elements of the VD n the DT. The Deluny tetrherliztion is ul to the VD, n is efine y the prtitioning of the spe into tetrher where the verties of the tringles re the points in S (generting eh Voronoi ell) tht stisfy the empty irumsphere test ( sphere is empty is no points re in its interior, ut points n lie iretly on the sphere). The ulity etween the VD n the DT in R 3 is simple, eh element of struture orrespons to one n only one element in the ul: eh polyheron eomes point n eh line eomes fe, n vievers. For exmple, s shown in Figure 1: Deluny vertex p eomes Voronoi ell (Figure 1()), Deluny ege α eomes Voronoi fe (Figure 1()), Deluny tringulr fe κ eomes Voronoi ege (Figure 1()), n Deluny tetrheron τ eomes Voronoi vertex (Figure 1()). A Voronoi vertex is lote t the entre of the sphere irumsrie to its ul tetrheron, n two verties in S hve Deluny ege onneting them if n only if their two respetive ul Voronoi ells re jent Degeneries For the VD n the DT, set S of points is in generl position when the istriution of points oes not rete ny miguity in the strutures. For the VD/DT in R, the egeneries, or speil ses, our when + 1 points lie on the sme hyperplne n/or when + 2 points lie on the sme ll. For exmple, in two imensions, when four or more points in S re oirulr there is n miguity in the efinition of DT(S). As shown in Figure 2, the qurilterl n e tringulte with two ifferent igonls, n n ritrry hoie must e me sine oth respet the Deluny riterion (points shoul not e on the interior of irumirle, ut more thn three n lie iretly on the irumirle). This implies tht in the presene of four or more oirulr points, DT(S) is not unique. Notie tht even in the presene of oirulr points, VD(S) is still unique, ut it hs ifferent properties. For exmple, in Figure 2, the Voronoi vertex in the mile hs egree 4 κ τ

3 o 1 DT(S) Figure 2. The DT (lk lines) for four oirulr points in two imensions is not unique, ut the VD (she lines) is. o 2 o 3 Figure 3. The set S of points is ontine y ig tringle forme y the verties o 1, o 2 n o 3. (when S is in generl position, every vertex in VD(S) hs egree 3). When three or more points re olliner, DT(S) n VD(S) re unique, ut prolems with the omputtion of the strutures n rise, s expline in Setion 5. All the previous oservtions generlise strightforwrly to three n higher imensions. 3. Construting DT Minly three prigms of omputtionl geometry n e use for omputing Deluny tringultion in two n three imensions: ivie-n-onquer, sweep plne, n inrementl insertion. Eh one of these prigms yiels n optiml lgorithm in two imensions, i.e. DT of n points is ompute in O(n log n). Exmples of these re the ivie-nonquer n the inrementl insertion lgorithms of Guis n Stolfi[17], n the sweep line lgorithm tht onstruts iretly the VD [15]. In three imensions, things re it more omplite. An lgorithm, lle DeWll n se on the ivie-n-onquer prigm, hs een evelope for onstruting the DT in ny imensions [7]. Although the worst-time omplexity of this lgorithm is O(n 3 ) in three imensions, the uthors ffirm tht the spee of their implementtion is omprle to the implementtion of known inrementl lgorithms, n is suqurti. Shewhuk [36] proposes sweep lgorithms for the onstrution of the onstrine Deluny tringultion in R, n these n e use for the norml DT. As is the se with DeWll, his lgorithm is suoptiml for the three-imensionl se. In R 3, only inrementl insertion lgorithms hve omplexity tht is worst-se optiml, i.e. O(n 2 ) sine the omplexity of the DT in R 3 is qurti. With these lgorithms, eh point is inserte one t time in vli DT n the tetrherliztion is upte, with respet to the Deluny riterion, etween eh insertion. Oserve tht the insertion of single point p in DT only moifies lolly the DT, i.e. only the tetrher whose irumsphere ontins p nee to e elete n reple y new one respeting the Deluny riterion (see Figure 8 for two-imensionl exmple). In shrp ontrst to this, ivie-n-onquer n plne sweep lgorithms uil DT in one opertion (this is th opertion), n if nother point nees to e inserte fter this, the whole onstrution opertion must e one gin from srth. Inrementl insertion lgorithms re therefore mntory for uiling ynmi sptil moel, whih is useful in mny pplitions. Consier T, the DT(S) of set S of n points in R 3. The insertion of single point p, thus getting T p = T {p}, n e one with two inrementl insertion lgorithms: the Bowyer-Wtson lgorithm [6, 39], or one se on flipping in tringultion. The ie ehin the former lgorithm is reltively simple: ll the tetrher onfliting with p, i.e. whose irumsphere ontins p, re elete from T, n the hole thus rete is fille y reting eges joining p to eh vertex of the hole. It n e rgue tht this lgorithm is more error-prone euse hole in the tetrherliztion is rete. It is lso intrite from n lgorithmi n t struture point of view s the geometri n topologil reltionships of the tetrherliztion re temporrily estroye. The rest of the pper fouses on the lterntive metho: flipse lgorithm. 4. Bsi DT Opertions 4.1. Initilistion: Big Tetrheron The lgorithm in Setion 5 ssumes tht the set S of points is entirely ontine in ig tetrheron (τ ig ) severl times lrger thn the rnge of S; the onvex hull of S, enote onv(s), therefore eomes τ ig. Figure 3 illustrtes two-imensionl exmple. The onstru-

4 x y z 1 Orient(,,, p) = x y z 1 x y z 1 p x p y p z 1 x y z 2 x + 2 y + 2 z 1 x y z 2 x + 2 y + 2 z 1 InSphere(,,,, p) = x y z 2 x + 2 y + 2 z 1 x y z 2 x + 2 y + 2 z 1 p x p y p z p 2 x + p 2 y + p 2 z 1 e flip14 flip23 flip41 flip32 () () Figure 5. The 3D istellr flips. e Figure 4. The two preites neee. tion of DT(S) is for exmple lwys initilise y first onstruting τ ig, n then the points in S re inserte. Doing this hs mny vntges, n is eing use y severl implementtions [30, 4, 29]. First, when single point p nees to e inserte in DT(S), this gurntees tht p is lwys insie n existing tetrheron. We o not hve to el expliitly with verties e outsie the onvex hull, s in [21]. Seon, we o not hve to el with the (nsty) se of eleting vertex tht ouns onv(s), if suh n opertion is neee. Thir, sine tringulr fe is lwys gurntee to e shre y two tetrher, point lotion lgorithms never fll off the onvex hull. The min isvntge is tht more tetrher thn neee re onstrute. For exmple in Figure 3 only the she tringles woul e prt of DT(S). The extr tetrher n nevertheless e esily mrke s they re the only ones ontining t lest one of the four points forming τ ig Preites Construting DT n mnipulting it essentilly require two si geometri tests (lle preites): Orient etermines if point p is over, uner or lies on plne efine y three points, n ; n InSphere etermines if point p is insie, outsie or lies on sphere efine y four points,, n. As shown in Figure 4, oth tests n e reue to the omputtion of the eterminnt of mtrix [17]. We n stte tht pplying n ientil trnsltion to every point will not hnge the result of the eterminnt, n, for this reson, fter trnslting ll the points y p the two preites re respetively implemente s the eterminnts of 3x3 n 4x4 mtries. I will not prove here the orretness of those preites (see [17] for the two-imensionl se) ut simply stte tht Orient returns positive vlue when the point p is ove the plne efine y, n ; negtive vlue if p is uner the plne; n extly 0 if p is iretly on the plne. Orient is onsistent with the left-hn rule: when the orering of, n follows the iretion of rottion of the urle fingers of the left hn, then the thum points towrs the positive sie (the ove sie of the plne). In other wors, if the three points efining plne re viewe lokwise from viewpoint, then this viewpoint efines the positive sie the plne. The preite InSphere follows the sme ie: positive vlue is returne if p is insie the sphere; negtive if p is outsie; n extly 0 if p is iretly on the sphere. Oserve tht to otin these results, the points,, n in InSphere must e orere suh tht Orient (,,, ) returns positive vlue Three-imensionl Bistellr Flips A istellr flip is lol (topologil) opertion tht moifies the onfigurtion of some jent tetrher [25, 13]. Consier the set S = {,,,, e} of points in generl position in R 3 n its onvex hull onv(s). There exist two possile onfigurtions, s shown in Figure 5: 1. the five points of S lie on the ounry of onv(s); see Figure 5(). Aoring to Lwson [25], there re extly two wys to tetrherlize suh polyheron: either with two or three tetrher. In the first se, the two tetrher shre tringulr fe, n in the ltter se the three tetrher ll hve ommon ege e. 2. one point e of S oes not lie on the ounry of onv(s), thus onv(s) forms tetrheron; see Figure 5(). The only wy to tetrherlize S is with four tetrher ll inient to e. Bse on these two onfigurtions, four types of flips in R 3 n e esrie: flip23, flip32, flip14 n flip41 (the numers refer to the numer of tetrher efore n fter the flip). When S is in the first onfigurtion, two types of flips re possile: flip23 is the opertion tht trnsforms one tetrherliztion of two tetrher into nother one with three tetrher; n flip32 is the inverse opertion. If S is tetrherlize with two tetrher n the tringulr fe is not lolly Deluny, then flip23 will rete three tetrher whose fes re lolly Deluny. A flip14 refers

5 () flip12 flip13 f flip44 () Figure 6. The egenerte flips () flip12 n flip13, n () flip44. to the opertion of inserting vertex insie tetrheron, n splitting it into four tetrher; n flip41 is the inverse opertion tht eletes vertex. Bistellr flips o not lwys pply to jent tetrher [20]. For exmple, in Figure 5(), flip23 is possile on the two jent tetrher n e if n only if the line e psses through the tringulr fe (whih lso mens tht the union of n e is onvex polyheron). If not, then flip32 is possile if n only if there exists in the tetrherliztion thir tetrheron jent to oth n e Degenerte ses e To el with egenerte ses, other flips nee to e efine. Shewhuk efines n uses egenerte flips for the onstrution n the mnipultion of onstrine Deluny tringultions in R [37]. A flip is si to e egenerte if it is non-egenerte flip in lower imension. It is neee for speil ses suh s when new point is inserte iretly onto simplex. For instne, Figure 6 shows the flip12 tht splits tetrheron into two tetrher when new point is inserte iretly onto the ege, n the flip13 tht splits into three tetrher when new point is inserte iretly onto the fe. The flip12 is equivlent to flip in one imension, n the flip13 to the one neee to insert new point in tringle in two imensions. These flips re neee when eling with onstrine DT, ut in our se they re not expliitly neee, s the opertion they perform n e simulte, s shown in the next setions, with the flip23 n the flip32. The only egenerte flip neee for onstruting DT is the following. Consier the set S = {,,,, e, f} of points onfigure s shown in Figure 6, with points,, n e forming plne. If S is tetrherlize with four tetrher ll inient to one ege this onfigurtion is lle the onfig44 then flip44 trnsforms one tetrherliztion into nother one lso hving four tetrher. Note tht the four tetrher re in onfig44 efore n fter the flip44. A flip44 is tully omintion in one step of flip23 (tht retes flt tetrheron e) followe immeitely y flip32 tht eletes the flt tetrheron; flt tetrheron is tetrheron spnne y four oplnr verties (its volume is zero). The flip44 is oneptully equivlent to the well-known flip22 in two imensions Point Lotion Given the DT T of set S of n points n query point p, the point lotion prolem onsists of etermining insie whih tetrheron of T lies p. This is neessry opertion for onstruting inrementlly DT, n the lgorithm esrie in Setion 5 uses it. Mny point lotion lgorithms for the DT re optiml [13, 9], ut they use itionl storge, they require preproessing for reting the itionl t struture, n they re lso often very omplite to implement. As Müke et l. [31] note, optiml lgorithms o not neessrily men etter results in prtie euse of the mount of preproessing involve, the extr storge neee, n lso euse the optiml lgorithms o not lwys onsier the ynmi se, where points in the DT oul e elete. For mny prolems like this one, prtitioners hve een known to fvour su-optiml lgorithms tht re esier to implement. Müke et l. [31] n Devillers et l. [10] nlyse theoretilly su-optiml lgorithms, for the DT in three imensions, tht yiel fst prtil performnes. These lgorithms re esirle in the se of ynmi struture euse they o not use ny itionl storge or preproessing. The jeny reltionships etween the simplies in DT re use to perform the point lotion. Bse on experimentl results, oth onlue tht the wlking (refer to s Wlk in the following) strtegy is the fstest solution. It ws esrie in the erliest ppers out the onstrution of the DT in two imensions [16]: in DT in R, strting from - simplex σ, we move to one of the neighours of σ (σ hs ( + 1)-neighours; we hoose one neighour suh tht the query point p n σ re on eh sie of the ( 1)-simplex shre y σ n its neighour) until there is no suh neighour, then the simplex ontining p is σ. The lgorithm, illustrte in Figure 7, is not etile here euse of spe onstrints, ut n e esily implemente fter reing [31] n [10]. Also,

6 strting tringle p Figure 7. The Wlk lgorithm for DT in two imensions. The query point is p. mking Wlk roust is trivil, s it is not ffete y egenerte ses. 5. An Inrementl Flip-se Algorithm The lterntive to reting hole is to use istellr flips to moify the onfigurtion of tetrher in the viinity of p. It shoul first e notie tht flips re opertions vli in ny imensions, n not only in three imensions [25]. They permit us to keep omplete tetrherliztion uring the insertion proess, n hene lgorithms re reltively simple to implement n lso numerilly more roust. Although flip-se lgorithm requires somewht more work thn n lgorithm where hole is rete Liu n Snoeying [29] stte tht on verge times more tetrher re rete n tht flippility tests lso slow own the whole proess its implementtion is simplifie n less error-prone sine the mintenne of jeny reltionships in DT is enpsulte in the flip opertions Two Dimensions The first flip-se lgorithm ws esigne to onstrut the DT in two imensions. It ws evelope y Lwson [24] who prove tht, strting from n ritrry tringultion of set S of points in the plne, flipping eges (i.e. repling the igonl of qurilterl s in Figure 2) n trnsform this tringultion into ny other tringultion of S, inluing the Deluny tringultion. The totl running time of this lgorithm is O(n 2 ), sine there is O(n) tringles tht must e teste ginst eh other. An inrementl insertion lgorithm se on ege flipping n improve this to O(n log n) [17]. Wht follows re the min steps for the insertion of single point p in DT, n the proess is shown in Figure 8. First, the tringle τ ontining p is ientifie n then split into three new tringles y joining p to Figure 8. Step-y-step insertion, with flips, of single point in DT in two imensions. every vertex of τ. Seon, eh new tringle is teste oring to the Deluny riterion ginst its opposite neighour (with respet to p); if it is not Deluny tringle then the ege shre y the two tringles is flippe n the two new tringles will lso hve to e teste lter. This proess stops when every tringle hving p s one of its verties respets the Deluny riterion Three Dimensions While the onept of flipping generlises to three imensions, Lwson s lgorithm [24] oes not, s Joe [20] proves. Inee, strting from n ritrry tetrherliztion, it is possile tht uring the proess of flipping, non-lolly Deluny fet e impossile to flip (this se hppens when the union of two tetrher is onve). Joe [21] nevertheless lter irumvente this prolem y proving tht, given DT(S) n point p in R 3, there lwys exists t lest one sequene of istellr flips to onstrut DT(S {p}). In this se, there will e non-deluny fets impossile to flip, ut he prove tht there will lwys e flip possile somewhere else suh tht the lgorithm progresses. This n form the sis of n inrementl insertion lgorithm for the onstrution of the Deluny tetrherliztion, tht is strightforwr generlistion of the two-imensionl se. Let S e set of points in R 3 n let T e DT(S), Figure 9 shows the lgorithm, lle InsertOnePoint, neee to restore the Delunyness in T when single point p is inserte. The lgorithm is pte from [21], n is oneptully the sme s [13]. As is the se with the two-imensionl lgorithm, the point p is first inserte in T with flip (flip14 in the se here), n the new tetrher rete must e teste to mke sure they re Deluny. The sequene of flips neee is ontrolle y stk ontining ll the tetrher tht hve not een teste

7 Input: A DT(S) T in R 3, n new point p to insert Output: T p = T {p} 1: τ Wlk {to otin tetr ontining p} 2: insert p in τ with flip14 3: push 4 new tetrher on stk 4: while stk is non-empty o 5: τ = {p,,, } pop from stk 6: τ = {,,, } get jent tetrheron of τ hving s fet 7: if is insie irumsphere of τ then 8: Flip(τ, τ ) 9: en if 10: en while Figure 9. Algorithm InsertOnePoint(T, p). yet. The stk strts with the four resulting tetrher of the flip14, n eh time flip is performe, the new tetrher rete re e to the stk. The lgorithms stops when ll the tetrher inient to p re Deluny, whih lso mens tht the stk is empty. The Flip metho in InsertOnePoint nees to e refine euse, unlike in two imensions, ifferent geometri ses yiel ifferent flips. Assume tht p ws inserte in T, n tht ertin numer of flips hve een performe ut tht T p = T {p} is not Deluny yet. All the tetrher inient to p, whih form the str 2 of p, must e teste to ensure tht they re Deluny, n notie tht the ones tht hve not een teste yet re in the stk. Let τ = {p,,, } e the next tetrheron poppe from the stk. To ensure τ is Deluny, we nee only to test it with the tetrheron τ = {,,, } outsie str(p) n inient to the fet ; we re tully testing if is lolly Deluny. If the irumsphere of τ ontins, ifferent options for the flip require re possile, oring to the geometry of τ n τ. Oserve tht if we look from p, we n see one, two or even three fets of τ (epite in Figure 10). When S is in generl position, two ses re possile (oth ses refer to Figure 11): Cse #1: only one fe of τ is visile, n therefore the union of τ n τ is onvex polyheron. In this se, flip23 is performe. Cse #2: two fes of τ re visile, n therefore the union of τ n τ is non-onvex. If there exists tetrheron τ = p in T p suh tht the ege is shre y τ, τ n τ, then flip32 n e performe. If there exists no suh tetrheron, 2 The str of vertex v in -imensionl tringultion, enote str(v), onsists of ll the simplies tht ontin v; it forms str-shpe polytope. 1 fet 2 fets 3 fets Figure 10. In 3D, three types of tetrher re possile, when viewe from fixe viewpoint (the reer in this se). p p se #1 se #2 se #3 se #4 Figure 11. Four ses re possile for the Flip lgorithm. For se #3, verties p,, n re oplnr. For se #4, the tetrheron p is flt. then no flip is performe. The non-lolly Deluny fet will e retifie y nother flip performe on jent tetrher, s [21] proves. When three fes of τ re visile, no tion is tken n the next tetrheron in the stk is proesse. The implementtion of the test tht etermines whih flip shoul e pplie for τ n τ oes not test if their union is onvex or onve. Wht is use inste is test tht etermines if the ege joining the two pexes of τ n τ rosses the interior of their ommon fet. If the ege oes, it mens the union is onvex, n if it oes not the union is onve Time Complexity When n inrementl insertion lgorithm se on InsertOnePoint is use to ompute DT(S) of set S of n points in R 3, the lgorithm tkes O(n 2 ), whih is worst-se optiml [13]. However, in prtie, the lgorithm will most likely e fster. It ws prove tht if the n points re uniformly istriute in unit ue, then the expete running time goes own to O(n log n), provie tht the history of the flips is use for point lotion [13]. Furthermore, the work neee to insert single point p in T is proportionl to s, the numer of tetrher onfliting with p [13]. Inee, notie tht every flip in InsertOnePoint is tully eleting one n p p

8 Input: Two jent tetrher τ n τ Output: A flip is performe, or not 1: if se #1 then 2: flip23(τ, τ ) 3: push tetrher p, p n p on stk 4: else if se #2 AND T p hs tetrheron p then 5: flip32(τ, τ, p) 6: push p n p on stk 7: else if se #3 AND τ n τ re in onfig44 with τ n τ then 8: flip44(τ, τ, τ, τ ) 9: push on stk the 4 tetrher rete 10: else if se #4 then 11: flip23(τ, τ ) 12: push tetrher p, p n p on stk 13: en if Figure 12. Algorithm Flip(τ, τ ). Note tht the 4 ses refer to Figure 11. only one non-deluny tetrheron, n repling it with some new ones inient to p. The first flip14 removes only one tetrheron n reples it y four new ones; flip23 eletes onfliting tetrheron τ n reples τ (the tetrheron inient to p in str(p)) n τ y three tetrher ll inient to p; n flip32 lso removes one onfliting tetrheron (the one outsie str(p)) n retes two new tetrher inient to p. Also, one tetrher is elete fter flip, it is never re-introue in T p. Therefore, if s tetrher in T re onfliting with p, then extly s flips re neee to otin T p Degenerte Cses Joe [21] not only prove tht flip-se inrementl insertion works in three imensions, he prove it even when S hs egeneries. He etile the ifferent onfigurtions possile when flip is to e performe. I present in the following the mny egeneries tht n rise n I esrie how to solve them. These solutions re minly tken from [21, 35, 37]. When S ontins egenerte ses, the intersetion tests etween the two pexes of τ n τ n return other results (oth ses refer to Figure 11): Cse #3: the line segment p intersets iretly one ege of (ssume it is here). Thus, verties p,, n re oplnr. Oserve tht flip23, tht woul rete flt tetrheron p, is possile if n only if τ n τ re in onfig44 (see Figure 6) with two other tetrher τ n τ, suh tht the ege is inient to τ, τ τ n τ. Sine the four tetrher re in onfig44, flip44 n e performe. If not, no flip is performe. Cse #4: the vertex p is iretly on one ege of the fe (ssume it is here). Thus, p lies in the sme plne s, ut lso s. The tetrheron p is oviously flt. The only reson for suh se is euse p ws inserte iretly on one ege of T in the first ple (the ege ), n the first flip14 to split the tetrheron ontining p rete two flt tetrher. Beuse p ws inserte on the ege, ll the tetrher in T inient to must e split into two tetrher. This oul e one with the egenerte flip12, ut it n lso e one simply with sequene of flip23 n flip32. When the se #4 hppens, it suffies to perform flip23 on τ n τ. Oviously, nother flt tetrheron p will e rete, ut this one will elete y nother flip. Oserve tht lthough flt tetrher re not llowe in DT (the irumsphere of flt tetrheron is unefine n ontins ll the points in set), they re llowe uring the proess of upting tetrherliztion, s long s the omintoril struture stys oherent. The only egenerte ses remining to hnle re: (1) there exists vertex t the ext lotion where p ws suppose to e inserte; (2) p lies iretly on the irumsphere of tetrheron in T ; (3) p ws inserte iretly on fe of tetrheron in T. The first se n e hnle trivilly: simply test the istne from p to eh vertex of the tetrheron returne y Wlk ginst tolerne. If one istne is smller thn the tolerne, o not insert p t ll. The seon se is lso esy to hnle: o not perform flip sine this is n unneessry opertion tht will slow own the lgorithm. Sine p is iretly on the irumsphere n not insie, no opertion will still result in orret DT. The lst se oes not require ny speil tretment: the flip14 tht inserts p in tetrheron will rete one flt tetrheron, n this tetrheron will e elete when teste ginst the tetrheron outsie str(p) shring the sme fe Extrting the VD from the DT Let T e the DT of set S of points in R 3. The simplies of the ul D of T n e ompute s follows (ll the exmples refer to Figure 1: Vertex: single Voronoi vertex is esily extrte:it is

9 lote t the entre of the sphere pssing through the four verties of its ul tetrheron τ. Ege: Voronoi ege, whih is ul to tringulr fe κ, is forme y the two Voronoi verties ul to the two tetrher shring κ. Fe: Voronoi fe, whih is ul to Deluny ege α, is forme y ll the verties tht re ul to the Deluny tetrher inient to α. The ie is simply to turn roun Deluny ege n extrt ll the Voronoi verties. These re gurntee to e oplnr, n the fe is gurntee to e onvex. Polyheron: the onstrution of one Voronoi ell V p ul to vertex p is similr: it is forme y ll the Voronoi verties ul to the tetrher inient to p. Sine Voronoi ell is onvex y efinition, it is possile to ollet ll the Voronoi verties n then ompute the onvex hull (e.g. see [3] for n lgorithm); the retrievl of ll the tetrher inient to p n e one y performing BFS-like lgorithm on the grph ul to the tetrher. A simpler metho onsists of first ientifying ll the eges inient to p, n then extrting the ul fe of eh ege. Given T, we must oviously visit ll its 3-simplies to e le to extrt D. This mens tht omputing D from T hs omplexity of Θ(n) when S ontins n points. 6. Roustness Consier n mittely simple lgorithm tht s the vlue 0.1 one hunre times, n then returns true if the totl is 10.0, n flse otherwise. It shoul oviously lwys returns true, ut if it is implemente on omputer with floting-point rithmeti, the os tht it will return true re rther low (if not nil). This is euse floting-point rithmeti offers only n pproximtion to rel numers, whih re lwys roune to the losest possile vlue in the omputer. When one hooses to implement n lgorithm with floting-point rithmeti, it is importnt he unerstns the onsequenes of his hoie. Floting-point rithmeti to represent rel numers is uiquitous euse it hs mny vntges: it is ville lmost on every pltform, n, more importntly, it hs een highly optimise so tht rithmeti opertions re performe very fst. I egn implementing the lgorithm esrie in this pper uner the nïve ssumption tht flotingpoint rithmeti ws goo enough n tht I woul e le to unerstn the speil ses, n fix them y ing some more oe. My metho ws the most wiely use to fix numeril non-roustness: the tolerne. In other wors, if two vlues re very lose to eh other, then they re equl. Tolernes n proly yiel stisftory results for simple prolems (e.g. the intersetion of two line segments), ut for more omplex ones like the onstrution of the DT, where the omintoril struture oul e invlite y smll movements of the verties, it is risky. How shoul one efine the optiml tolerne? I oserve tht tweking the tolerne to fix given prolem ws usully esy, the prolem eing tht it ws lso reting nother prolem somewhere else. Even with the tolerne well-efine (or so I thought), uring the evelopment of lgorithms I eme frustrte euse my progrm woul sometimes rsh, or even output something tht ws not vli tetrherliztion. Ws it the rithmeti? My lgorithm? Or mistke I me while oing it? I wonere ll the time. The evelopment of the lgorithm ws hinere y the ft tht the soure of the prolem ws never known. The only solution ws to use ext rithmeti: my first question ws therefore nswere tht left two unnswere ut tht ws etter thn nothing! The mjor ostle to using ext rithmeti is the spee of omputtion: it is very slow [40]. Unlike floting-point rithmeti, n rithmeti opertion n not e ssume to e performe in onstnt time. The omplexity of n opertion epens on the numers n of its use to store numer, n the multiplition of two numers n hve for instne omplexity of O(n 2 ). Krsik et l. [23] reporte tht the nïve implementtion of ivie-n-onquer lgorithm to onstrut the two-imensionl DT ws slowe own rmtilly y ftor of , lthough they showe tht y refully seleting when to use ext rithmeti, they oul reue the ftor to roun 5 for points uniformly istriute. As mentione in Setion 4.2, the inrementl insertion lgorithm mkes its only importnt eisions se on the result of two geometri preites. For this reson, if one wnts to uil roust lgorithm, only these two preites nee to e implemente with ext rithmeti. Oserve tht we re not intereste in the ext vlues returne y Orient n InSphere, ut rther y the sign of the result (lthough we must e le to etet when the vlue of the eterminnt is extly 0). Floting-point rithmeti will most likely ompute orretly the sign of the eterminnt when the points involve in preites re lerly in generl position, ut when four points re nerly oplnr, there is hne tht Orient returns tht they

10 re oplnr. Similrly, five ospheril points n e mistkenly onsiere not ospheril y InSphere euse the result of the eterminnt is not extly 0, ut perhps something like 1, The prolem of roustness of n lgorithm is therefore tightly linke to the prolems of egeneries, s it is speil ses tht will rete prolems when omputing preite. An esy solution for the roustness prolem is ue to Shewhuk [34, 35]. He evelope n implemente his oe is pulily ville on the Internet 3 fst geometri preites tht re ptive, whih mens tht their spee is inversely proportionl to the egree of unertinty of the result. His metho works like filter tht will tivte ext rithmeti only when it is neee, ut sine the ext result of the preite is not sought, s soon s the sign of the eterminnt n e orretly ompute, the proess stops. He shows tht using his ptive preites for uiling DT in two imensions slows own y out 8% the totl running time for rnomly generte points, n y out 30% when those points form tilte gri. In three imensions, his preites slows own y out 35% the totl running time for rnomly generte points, n y ftor of 11 when those re generte on the surfe of sphere. As he sttes: these preites ost little more thn orinry nonroust preites, ut never srifie orretness for spee [35]. 7. Prtil Performnes The lgorithm InsertOnePoint, s esrie in Setion 5, ws implemente with the Borln Delphi environment, the ojet-oriente version of the lnguge Psl. I report in this setion experiments tht were me on Pentium 4 (2.8 GHz), with 1 GB of RAM n running the Winows XP Professionl operting system. The prtil performnes of Insert- OnePoint re nlyse with ifferent tsets, n ompre with wht is rguly the e fto stnr in omputtionl geometry implementtions, CGAL [4]. The soure oe of CGAL ws written in C++, n n e ompile uner ifferent pltforms, inluing Winows. It shoul e notie tht CGAL is not the fstest implementtion for the omputtion of DT in R 3, s there re reports of implementtions tht outperform it, e.g. [1] n [28]. The im of this setion is not to show tht InsertOnePoint is the fstest euse it is not ut to emonstrte tht it is omprle to other solutions ville n n relistilly e use for rel-worl pplitions. 3 quke/roust.html For the experiments esrie in the following, the running time shown is the time only for the opertion itself, i.e. the time to re the input file n output the results re not tken into ount. Unless expliitly mentione, the ptive preites of Shewhuk [34] re use for oth Orient n InSphere for ll the opertions (whih inlues for instne Wlk), ut stnr floting-point rithmeti is use for ll the other opertions. The CGAL oe (relese 3.1, Deemer 2004) ws ompile uner Winows with MinGW ( port to Winows for the g ompiler), n ws me omprle to my implementtion. In other wors, the preites Orient n InSphere lso use roust rithmeti ut ll other opertions re me with floting-point rithmeti 4. Also, ll the heks in CGAL hve een isle; these re funtions verifying tht the omintoril struture of the DT stys oherent fter n opertion (isling them spe up the insertion n the eletion y roun 20%) Dtsets Use InsertOnePoint ws teste with four ifferent tsets, whih represent vriety of sptil istriutions tht one is likely to fin, n where egeneries re present: 50k: set of points rnomly istriute in unit ue; sphere: set of points rnomly istriute on the surfe of sphere; ues: set of 15x15x15 points regulrly istriute in the x y z iretions, forming ue with points; oen: rel oenogrphi tset of the Bering Se 5. Figure 13 shows the tset, where the points re istriute long wter olumns. It ontins mny of suh wter olumns, n the points re regulrly istriute long eh olumn (t regulr intervls of epth), lthough this is not lwys the se s some t points were remove euse of instrument errors. Also, notie tht the wter olumns o not ll go to the sme epth. Tle 1 ontins the etils of the four tsets. The numer of tetrher is the totl numer inluing the ones existing euse of the ig tetrheron (see Setion 4.1), n k is the egree of vertex in the 4 The CGAL kernel Ext preites inext onstrutionskernel ws use. 5 Tken from the Oenogrphi In-Situ Dt Aess: http: //

11 mine (roust) mine (flot) CGAL 50k sphere ues ues r oen oen r Tle 2. Running times (in seons) for the onstrution of the DT of the tsets. Figure 13. Perspetive view of the oen tset, forme y mny wter olumns. # pts # tet k min k mx k vg 50k sphere ues oen Tle 1. Detils onerning the tsets use for the experiments. DT. Oserve tht in the se of the oen tset, the nisotropi istriution mens tht one vertex hs egree of 123, ut tht the numer of tetrher is still liner. It shoul e notie tht the oen n ues tsets hve strong sptil oherene: in the originl oen file, the wter olumns re liste one fter the other, n in eh wter olumn the points re liste from the surfe to the ottom of the se; n the ue file ws onstrute using three imrite loops (so the tset is onstrute line y line ). For this reson, the orer of the points in the two tsets were rnomly shuffle to otin two new tsets: oen r n ues r Results To onstrut DT, CGAL lso uses n inrementl insertion lgorithm, ut inste of flipping, the Bowyer- Wtson lgorithm is use. For the point lotion strtegy, CGAL implements the Deluny hierrhy [9]: for set S of points, the first DT(S) is onstrute, n then other levels re rete y smpling some points in S n reting the other DTs. The tetrher shring the verties etween two levels re linke together. The point lotion involves wlking (with Wlk) to the tetrheron t the top level ontining the trget point, n then going own one level n ontinuing this wy until the tetrheron in DT(S) ontining the trget point is foun; this is theoretilly fster thn the lgorithm Wlk esrie erlier. The running times of my implementtion versus CGAL for the six tsets re reporte in Tle 2. The results otine re rther surprising s my implementtion is fster for five of the tsets, n y ftor of lmost 14 for the ues r tset; the only tset where my implementtion is slower is for 50k, ut only y 10%. Notie lso tht shuffling the ues n the oen tsets improve the onstrution of the DT of my implementtion y ftor of respetively 4 n 3; while with CGAL the ftors were respetively 1.6 n lmost no hnge. The resons of suh importnt ifferenes etween my implementtion n CGAL re not totlly ler to me euse in mny reports CGAL performs rther well, n n ompre esily with the other implementtions ville. For instne, Boissonnt et l. [4] report onstruting the DT of points rnomly istriute in ue in 12.1s, on omputer reltively slower thn the one use for my experiments ( Pentium 3, 500 MHz with 512 MB of RAM). By omprison, for points rnomly istriute, CGAL tkes 19.7s on the omputer I use. The mjor ifferenes were tht they use CGAL uner Linux, n lso use integers to store the oorintes of the points. The use of integers n spee up n implementtion (omputtion with integers is fster), ut unfortuntely n not e use in ll situtions. When eling with rel-worl t, the oorintes of the points re usully onverte to integers y multiplying them y onstnt; this n le to prolems in ynmi ontext if the next points to e e re unknown, s there might e ig ifferene in the orer of mgnitue of the preision of point, whih woul invli the onstnt use. Integers re lso limite to 4 ytes, whih oul e prolemti for some tsets. Two resons oul explin the somewht poor re-

12 sults of CGAL I otine uring my experiments. Firstly, euse CGAL performs very poorly for tsets ontining egeneries (even when they re shuffle), the roust rithmeti use y CGAL is proly the ottlenek. The rithmeti use in my implementtion is ptive, while CGAL uses ompletely ifferent sheme [22], whih might not e s fst, espeilly uner Winows. Seonly, it seems tht the ompiler use, the ompiler options, n the pltform itself, n hve n influene on the spee of the oe. Inee, Liu n Snoeyink [28] report hving ifferenes in CGAL s running time of ftor of s muh s 2.5 when it ws ompile on Linux with ifferent kernels, n tht similr ifferenes were foun etween versions of CGAL. The spee of my implementtion, with roust n with stnr floting-point rithmeti, ws lso ompre to verify the lims me in Shewhuk [34]. The running times for eh tset re given in Tle 2 ( sh mens tht the progrm rshe). As expete, the DT of the tsets ontining mny egenerte ses oul not e onstrute, lthough, surprisingly, the ues n ues r tsets oul e onstrute (this y no mens implies tht ll regulrly spe tsets oul hve een onstrute, sine uring other experiments mny regulrly-spe tsets rshe). The onstrution of the DT for the 50k tset is roun 40% slower when roust rithmeti is use, n the ues n ues r tsets re slower y mgnitue of respetively 6 n 4.6. These numers orroorte the results in [34]. More etils onerning these experiments n e foun in [26]. 8. Disussion Although mny spets of the omputtion of 3D VD/DT hve een isusse, severl others were left out, mostly euse of spe onstrints, ut lso euse they re full reserh topis on their own. One of them is the t struture tht n e use to store the DT (or the VD). A very simple one, the tetrheron-se t struture, ws use to implement the lgorithm n perform the experiments. It onsiers the tetrheron s eing its tom n stores eh tetrheron with four pointers to its verties n four pointers to its jent tetrher. Most implementtions use tht struture, euse it is speeffiient n fst. CGAL notly uses it ut hs e to eh vertex pointer to one of its inient tetrher, to spee up the extrtion of the Voronoi ells [4]. It is lso possile to store the tetrher impliitly y onsiering t struture where the tom is tringulr fe hving three pointers to its verties n six pointers to its jent fes [35]. Furthermore, it is possile to store simultneously, oth the DT n the VD [11, 27]. Another spet is the onstrution of the DT for very lrge tsets, n we re tlking out mny millions of points here. The lgorithm s esrie is not optimise in ny wys for tsets for whih the t strutures o not fit in min memory. Severl solutions to this prolem hve een propose in the lst few yers. The most promising methos moify the orer of insertion of the points, so tht the swpping etween the memory n the isk is minimise, see for instne [1, 29, 19]. The goo news is tht the use of these methos requires only slight moifitions to the lgorithm esrie in this pper no rewriting of new lgorithm or t struture is neessry. Finlly, it shoul e si tht if one oes not wnt to implement the 3D VD/DT lgorithm himself, he n lwys use one of the freely ville implementtions or lirries, suh s CGAL! Referenes [1] N. Ament, S. Choi, n G. Rote. Inrementl onstrutions on BRIO. In Proeeings 19th Annul Symposium on Computtionl Geometry, pges , Sn Diego, USA, ACM Press. [2] F. Aurenhmmer. Voronoi igrms: A survey of funmentl geometri t struture. ACM Computing Surveys, 23(3): , [3] C. B. Brer, D. P. Dokin, n H. T. Huhnp. The quikhull lgorithm for onvex hulls. ACM Trnstions on Mthemtil Softwre, 22(4): , [4] J.-D. Boissonnt, O. Devillers, S. Pion, M. Teillu, n M. Yvine. Tringultions in CGAL. Computtionl Geometry Theory n Applitions, 22:5 19, [5] B. Boots. Deluny tringles, n lterntive pproh to point pttern nlysis. In Proeeings, Assoition of Amerin Geogrphers, 6, pges 26 29, [6] A. Bowyer. Computing Dirihlet tesselltions. Computer Journl, 24(2): , [7] P. Cignoni, C. Montni, n R. Sopigno. DeWll: A fst ivie & onquer Deluny tringultion lgorithm in E. Computer-Aie Design, 30(5): , [8] M. e Berg, M. vn Krevel, M. Overmrs, n O. Shwrzkopf. Computtionl geometry: Algorithms n pplitions. Springer-Verlg, Berlin, seon eition, [9] O. Devillers. The Deluny hierrhy. Interntionl Journl of Fountions of Computer Siene, 13(2): , [10] O. Devillers, S. Pion, n M. Teillu. Wlking in tringultion. Interntionl Journl of Fountions of Computer Siene, 13(2): , 2002.

13 [11] D. P. Dokin n M. J. Lszlo. Primitives for the mnipultion of three-imensionl suivisions. Algorithmi, 4:3 32, [12] D. Dougls. It mkes me so CROSS. Unpulishe mnusript from the Hrvr Lortory for Computer Grphis n Sptil Anlysis. Reprinte in Peuquet DJ, Mrle, DF, eitors, Introutory Reings in Geogrphi Informtion System, pges Tylor & Frnis, Lonon, [13] H. Eelsrunner n N. R. Shh. Inrementl topologil flipping works for regulr tringultions. Algorithmi, 15: , [14] D. A. Fiel. Implementing Wtson s lgorithm in three imensions. In Proeeings 2n Annul Symposium on Computtionl Geometry, volume , Yorktown Heights, New York, USA, ACM Press. [15] S. Fortune. A sweepline lgorithm for Voronoi igrms. Algorithmi, 2: , [16] C. M. Gol, T. D. Chrters, n J. Rmsen. Automte ontour mpping using tringulr element t strutures n n interpolnt over eh tringulr omin. In Proeeings Siggrph 77, volume 11(2), pges , [17] L. J. Guis n J. Stolfi. Primitives for the mnipultion of generl suivisions n the omputtion of Voronoi igrms. ACM Trnstions on Grphis, 4:74 123, [18] C. M. Hoffmnn. The prolems of ury n roustness in geometri omputtion. Computer IEEE Computer Soiety Press, 22:31 42, [19] M. Isenurg, Y. Liu, J. R. Shewhuk, n J. Snoeyink. Streming omputtion of Deluny tringultions. ACM Trnstions on Grphis, 25(3): , [20] B. Joe. Three-imensionl tringultions from lol trnsformtions. SIAM Journl on Sientifi n Sttistil Computing, 10(4): , [21] B. Joe. Constrution of three-imensionl Deluny tringultions using lol trnsformtions. Computer Aie Geometri Design, 8: , [22] V. Krmheti, C. Li, I. Pehthnski, n C. K. Yp. A CORE lirry for roust numeri n geometri omputtion. In Proeeings 15th Annul Symposium on Computtionl Geometry, volume , Mimi, Flori, USA, ACM Press. [23] M. Krsik, D. Lieer, n L. R. Nkmn. Effiient Deluny tringultion using rtionl rithmeti. ACM Trnstions on Grphis, 10(1):71 91, [24] C. L. Lwson. Softwre for C 1 surfe interpoltion. In J. R. Rie, eitor, Mthemtil softwre III, pges Aemi Press, [25] C. L. Lwson. Properties of n-imensionl tringultions. Computer Aie Geometri Design, 3: , [26] H. Leoux. Moelling three-imensionl fiels in geosiene with the Voronoi igrm n its ul. PhD thesis, Shool of Computing, University of Glmorgn, Pontypri, Wles, UK, [27] H. Leoux n C. M. Gol. Simultneous storge of priml n ul three-imensionl suivisions. Computers, Environment n Urn Systems, In press. [28] Y. Liu n J. Snoeyink. A omprison of five implementtions of 3D Deluny tesselltion. In J. E. Goomn, J. Ph, n E. Welzl, eitors, Comintoril n Computtionl Geometry, volume 52, pges Cmrige University Press, [29] Y. Liu n J. Snoeyink. tess3: A progrm to ompute 3D Deluny tesselltions for well-istriute points. In Proeeings 2n Interntionl Symposium on Voronoi Digrms in Siene n Engineering, pges , Seoul, Kore, [30] E. P. Müke. A roust implementtion for threeimensionl Deluny tringultions. Interntionl Journl of Computtionl Geometry n Applitions, 8(2): , [31] E. P. Müke, I. Sis, n B. Zhu. Fst rnomize point lotion without preproessing in two- n three-imensionl Deluny tringultions. Computtionl Geometry Theory n Applitions, 12:63 83, [32] A. Oke, B. Boots, K. Sugihr, n S. N. Chiu. Sptil tesselltions: Conepts n pplitions of Voronoi igrms. John Wiley n Sons, seon eition, [33] A. Slfel. It oesn t mke me nerly s CROSS: Some vntges of the point-vetor representtion of line segments in utomte rtogrphy. Interntionl Journl of Geogrphil Informtion Systems, 1(4): , [34] J. R. Shewhuk. Aptive preision floting point rithmeti n fst roust geometri preites. Disrete & Computtionl Geometry, 18: , [35] J. R. Shewhuk. Deluny Refinement Mesh Genertion. PhD thesis, Shool of Computer Siene, Crnegie Mellon University, Pittsurg, USA, [36] J. R. Shewhuk. Sweep lgorithms for onstruting higher-imensionl onstrine Deluny tringultions. In Proeeings 16th Annul Symposium on Computtionl Geometry, pges , Hong Kong, ACM Press. [37] J. R. Shewhuk. Upting n onstruting onstrine Deluny n onstrine regulr tringultions y flips. In Proeeings 19th Annul Symposium on Computtionl Geometry, pges , Sn Diego, USA, ACM Press. [38] K. Sugihr n I. Hiroshi. Why is the 3D Deluny tringultion iffiult to onstrut? Informtion Proessing Letters, 54: , [39] D. F. Wtson. Computing the n-imensionl Deluny tesselltion with pplition to Voronoi polytopes. Computer Journl, 24(2): , [40] C. K. Yp n T. Dué. The ext omputtion prigm. In D.-Z. Du n F. K. Hwng, eitors, Computing in Eulien Geometry, pges Worl Sientifi Press, seon eition, 1995.