A certified conflict locator for the incremental maintenance of the Delaunay graph of semi-algebraic sets

A ce i ied con lic loca o o he inc emen al main enance o he Delaunay g aph o semi-algeb aic se s F an¸cois An on, Depa men o Compu e Science, Uni e si y o B i ish Columbia, 201-2366 Main Mall, Vancou e , B.C., Canada, V6T 1Z4 Abs ac Mos o he cu es and su aces encoun e ed in geome ic modelling a e de ined as he se o solu ions o a sys em o algeb aic equa ions o inequali ies (semi-algeb aic se s). The Vo onoi diag am o a se o si es is a decomposi ion o he space in o p oximal egions (one o each si e). Vo onoi diag ams ha e been used o answe p oximi y que ies. The dual g aph o he Vo onoi diag am is called he Delaunay g aph. Only app oxima ions by conics can gua an ee a p ope con inui y o he i s o de de i a i e a con ac poin s, which is necessa y o gua an eeing he exac ness o he Delaunay g aph. The cen al idea o his pape is ha a (one ime) symbolic p ep ocessing may accele a e he ce i ied nume ical e alua ion o he Delaunay g aph con lic loca o . The symbolic p ep ocessing is he compu a ion o he implici equa ion o he gene alised o se o conics. The ce i ied compu a ion o he Delaunay g aph con lic loca o elies on heo ems on he uniqueness o a oo in gi en in e als (Kan o o ich, Moo e-K awczyk). Fo conics, he compu a ions ge much as e by conside ing only he implici equa ions o he gene alised o se s. Key wo ds: Delaunay g aph, semi-algeb aic se s, conics, con lic loca o 1. In oduc ion Mos o he cu es and su aces encoun e ed in geome ic modelling a e de ined as he se o com- mon ze oes o a se o polynomials (algeb aic a i- e ies) o subse s o algeb aic a ie ies de ined by one o mo e algeb aic inequali ies (semi-algeb aic se s). Many p oblems om di e en ields in ol e p oximi y que ies like inding he nea es neigh- bou , inding all he neighbou s, o quan i ying he neighbou liness o wo objec s. The e ac ion planning [´ OY85] p oblem ( ha add esses he op- imal ajec o y o a obo a ound obs acles) in obo ics and spa ial analysis and in luence zones in Geog aphic In o ma ion Sys ems [VC90] a e s ongly linked o ques ions o p oximi y among eal-wo ld objec s in eal-wo ld en i onmen s. The Vo onoi diag am [Vo 08] (see Fig. 1) o a se o si es is a decomposi ion o he space in o p ox- Email add ess: an [email protected] echnique. (F anc¸ois An on). imal egions (one o each si e). The p oximal e- gion (o Vo onoi zone) o a si e is he locus o poin s close o ha si e han o any o he one. Vo onoi diag ams allow one o answe p oximi y que ies a - e a que y poin has been loca ed in he Vo onoi zone i belongs o. The Vo onoi diag am de ines a neighbou hood ela ionship among si es: wo si es a e neighbou s i , and only i , hei Vo onoi egions a e adjacen . The g aph o his neighbou hood e- la ionship is called he Delaunay g aph. The De- launay g aph o si es in he plane sa is ies he ol- lowing emp y ci cle c i e ion (see Fig. 2): no si e in- e sec s he in e io o he ci cles ouching ( angen o wi hou in e sec ing he in e io o ) he si es ha a e he e ices o any iangle o he Delau- nay g aph (see Fig. 3). The e ha e been a emp s [OBS92] o compu e Vo onoi diag ams o cu es by app oxima ing cu es by line segmen s o ci cu- la a cs, bu he exac ness o he Delaunay g aph is no gua an eed [RF99a]. Indeed, he Vo onoi di- ag am is e y sensi i e o he o de o con inui y a con ac poin s (see [RF99a]). Only app oxima- 20 h EWCG Se ille, Spain (2004) 20 h Eu opean Wo kshop on Compu a ional Geome y Fig. 1. The Vo onoi diag am (ligh ) o a ci cle, an ellipse and a hype bola (da k) Fig. 2. The 3 emp y ci cles o he si es o Fig. 1 HYPERBOLA CIRCLE ELLIPSE Fig. 3. The Delaunay g aph o he si es o Fig. 1 ions by conics can gua an ee a p ope con inui y o he i s o de de i a i e a con ac poin s, which is necessa y o gua an eeing he exac ness o he Delaunay g aph [RF99a]. O he app oxima ion al- go i hms ha e used a New on-Raphson scheme o compu e and classi y Vo onoi e ices o cu es wi h a a ional pa ame e isa ion [RF99a,RF99b]. These do no di ec ly add ess he exac ness o he Delaunay g aph. 2. P elimina ies We will ecall now he o mal de ini ions o he Vo onoi diag am and o he Delaunay g aph. Fo his pu pose, we need o ecall some basic de ini- ions. De ini ion 1 (Me ic) Le Mbe an a bi a y se . A me ic on Mis a mapping d:M×M→R+ such ha o any elemen s a,b, and co M, he ollowing condi ions a e ul illed: d(a, b) = 0 ⇔ a=b,d(a, b) = d(b, a), and d(a, c)≤d(a, b) + d(b, c).(M, d)is hen called a me ic space, and d(a, b)is he dis ance be ween aand b. Le M=RN, and δdeno e he Euclidean dis- ance be ween poin s. Le S={s1, ..., sm}⊂ M, m ≥2 be a se o mdi e en subse s o M, which we call si es. The dis ance be ween a poin xand a si e si⊂Mis de ined as d(x, si) = in y∈si{δ(x, y)}. De ini ion 2 (In luence zone) Fo si, sj∈S, si(= sj, he in luence zone D(si, sj)o siwi h espec o sjis: D(si, sj) = {x∈M|d(x, si)< d (x, sj)}. De ini ion 3 (Vo onoi egion) The Vo onoi e- gion V(si,S)o si∈Swi h espec o he se Sis: V(si,S) = !sj∈S,sj"=siD(si, sj). De ini ion 4 (Vo onoi diag am) The Vo onoi di- ag am o Sis he union V(S) = "si∈S∂V(si,S) o all egion bounda ies. De ini ion 5 (Delaunay g aph) The Delaunay g aph DG (S)o Sis he dual g aph o V(S) de ined as ollows: – he se o e ices o DG (S)is S, – o each N−1−dimensional ace o V(S) ha belongs o he common bounda y o V(si,S)and o V(sj,S)wi h si, sj∈Sand si(=sj, he e is an edge o DG (S)be ween siand sjand ecip- ocally, and – o each e ex o V(S) ha belongs o he com- mon bounda y o V(si1,S),. . . ,V #siN+2 ,S$, wi h ∀k∈{1, ..., N + 2}, sik∈Sall dis inc , he e exis s a comple e g aph KN+2 be ween he sik, k ∈{1, ..., N + 2}, and ecip ocally. (see example on Fig. 3). Le us in oduce he gene alised o se and he gene alised Vo onoi e ex. We place ou sel es in he a ine space K2whe e K=C o he sake o in oducing hose no ions in an easie way. While he R−gene alised o se o νis he locus o he cen es o ci cles o adius R ha a e angen o ν, he ue R−o se o νis he locus o he cen es o ci cles o adius R ha a e angen o νand do no con ain any poin o νin i s in e io (see Fig. 4). De ini ion 6 (gene alised Vo onoi e ex) A gen- e alised Vo onoi e ex o h ee semi-algeb aic se s S1,S2, and S3is a poin o in e sec ion o he R−gene alised o se s o S1,S2, and S3(see Exam- Ma ch 25-26, 2004 Se ille (Spain) Fig. 4. The s ophoid and i s ue (le ) and gene alised ( igh ) o se s Fig. 5. A gene alised Vo onoi e ex (do ) o h ee conics ( hick lines) ple on Fig. 5). 3. The Delaunay g aph con lic loca o o semi-algeb aic se s Le X1, ..., XN+2, be semi-algeb aic se s [BR90,BCR98]. A semi-algeb aic se Xiis de- ined as: !si j=1 " i,j k=1 #x∈RN| i,j,k ⋆i,j,k 0$, whe e i,j,k is a polynomial wi h eal coe icien s in he a iables xi1, ..., xiNand ⋆i,j,k is ei he <o =, o i= 1,2,3,4, j= 1, ..., siand k= 1, ..., i,j. The Delaunay g aph con lic loca o de e mines which ones o he maximal dimensional ace s o he Delaunay g aph o N+ 1 semi-algeb aic se s X1, ..., XN+1 would be changed by he addi ion o he semi-algeb aic se XN+2. Le us assume wi hou loose o gene ali y ha each " i,j k=1 #x∈RN| i,j,k ⋆i,j,k 0$ o each Xiis de ined by a leas one non- i ial algeb aic equa- ion (i.e. di e en om he ze o polynomial). I ou s a ing assump ion is no alid in he case we ea , we can make i alid by adding he equa- ions co esponding o i,j,k = 0 o each (i, j, k) such ha jis he index o a componen ha is no de ined as in he assump ion and iis he index o he semi-algeb aic se o which he componen be- longs. Le us deno e Vias he in e sec ion o all he V( i,j,k) such ha ⋆i,j,k is = o each i= 1,2,3,4. Le Nibe he no mal space o Via he poin xi= (xi1, ..., xiN). Each i,j,k de ining Viinduces N−1 polynomials ni,j,k,l wi h l= 1, ..., N −1 ha a e he equa ions de ining he no mal o V( i,j,k) a xi. A poin q= (y1, ..., yN) belongs o Nii i s co- o dina es sa is y all he equa ions o he no mal spaces o V( i,j,k) a xisuch ha ⋆i,j,k is =. Fo a gi en q= (y1, ..., yN), le Mibe he he se o poin s mi= (zi1, ..., ziN)∈Xisuch ha q belongs o he no mal space o Via he poin mi. In he gene al case, each se Miis a ini e se o poin s. Howe e , i Vicon ains a po ion o hype - sphe e P HS (q, ρ) cen e ed on q, hen Micon ains ha po ion o hype sphe e. To ge in all cases a ini e se o poin s mio Vi, we use Si=Miwhen Miis ini e, and Si"P HS (q, ρ) = {wi} o an a bi a y poin wio P HS (q, ρ) when Vicon ains a po ion o hype sphe e P HS (q, ρ) cen e ed on q. We a e now able o w i e he sys em o alge- b aic equa ions and inequali ies ha de ine he ou come o he Delaunay g aph con lic loca o . Le us conside he map π:K3N→KNde ined by π(xi, q, mi) = q. The poin qis a he dis ance om he poin xii , and only i , he dis ance be ween qand xiis . This is exp essed algeb aically by he equa ion di(q, xi) = (y1−xi1)2+...+(yN−xiN)2− 2= 0. The gene alised -o se Oi o Xiis he image by πo he poin s o K3Nde ined by he ollowing sys em o equa ions and inequali ies:                            ∃j∈[1, si],∀k∈[1, i,j],                      i,j,k (xi)⋆i,j,k 0 i ⋆i,j,k is “ = ”,          di(xi, q) = 0 ∀l= 1, .., N −1, ni,j,k,l (xi, q) = 0 xi1(xi)&= 0 o . . . o xiN(xi)&= 0 The ue -o se o Xiis ob ained as he di e - ence o he gene alised -o se Oi o Xiand he union o each one o he images by πo he semi- algeb aic se s de ined by he ollowing sys em o equa ions and inequali ies o each poin mio Si:                            ∃j∈[1, si],∀k∈[1, i,j],                      i,j,k (mi)⋆i,j,k 0 i ⋆i,j,k is “ = ”,          (mi) = 0 ∀l= 1, .., N −1, ni,j,k,l (mi, q) = 0 d(mi, q)<0 I is ob ious ha a ue Vo onoi e ex o 20 h Eu opean Wo kshop on Compu a ional Geome y Running ime abo e sys ems gene alised o se s Gene al Sol e 6 min 38 s 12 min 37 s G adien Sol e 2 h 56 min 10 s 2 min 26 s Hessian Sol e 20 h 17 min 42 s 3 min 42 s Table 1 Some unning ime esul s o ellipses X1, ..., XN+1 is a poin o in e sec ion o he ue −o se s o X1, ..., XN+1 espec i ely. A ue Vo onoi e ex o X1, ..., XN+1 is a he dis ance R om XN+2, o al e na i ely, a ue Vo onoi e - ex o X1, ..., XN+1 belongs o he ue R−o se o XN+2. Conside he N+ 2−dimensional poin s whose i s Ncoo dina es a e he coo dina es o a ue Vo onoi e ex o X1, ..., XN+1, and he e- maining wo a e he dis ances be ween ha ue Vo onoi e ex and X1, ..., XN+1, and Rbe ween ha ue Vo onoi e ex and XN+2. The Delau- nay g aph con lic loca o should epo all he ue Vo onoi e ices such as he co esponding N+ 2−dimensional poin s sa is y R− < 0. We ha e e alua ed he Delaunay g aph con lic loca o wi hou sol ing any in e media y sys em by using an in e al analysis based lib a y (ALIAS [Me 00]) o sol ing ze o-dimensional sys ems o equa ions and inequali ies. The ce i ied compu a- ion o he Delaunay g aph con lic loca o elies on heo ems on he uniqueness o a oo in gi en in- e als (Kan o o ich and Moo e-K awczyk). This compu a ion uses a bisec ion p ocess on one o all he a iables using ei he only he equa ions o he sys em, o using he Jacobian o he sys em (Moo e-K awczyk es o inding “exac ly” he so- lu ions), o using he Jacobian and he Hessian o he sys em (wi h Kan o o ich, Moo e-K awczyk es s). We i s used ALIAS on he abo e sys em o algeb aic equa ions and inequali ies ha spec- i y he Delaunay g aph con lic loca o o semi- algeb aic se s. Then o conics, we used ALIAS on he sys em simpli ied by eplacing he equa ions i, niand dio he conics, no mals and dis ances be- ween he poin s on he conics and he ue Vo onoi e ex by he implici equa ions o he gene alised o se s o he conics (see [An 04]). This induces much as e compu a ions (see Table 1). 4. Conclusions We ha e p esen ed wha we belie e is he i s ce i ied con lic loca o o he inc emen al main- enance o he Delaunay g aph o semi-algeb aic se s. Fu he esea ch will y o imp o e he un- ning ime o he compu a ions. Re e ences [An 04] F an¸cois An on. Vo onoi diag ams o semi- algeb aic se s. PhD hesis, Uni e si y o B i ish Columbia, Janua y 2004. [BCR98] Jacek Bochnak, Michel Cos e, and Ma ie- F an¸coise Roy. Real algeb aic geome y. Sp inge - Ve lag, Be lin, 1998. T ansla ed om he 1987 F ench o iginal, Re ised by he au ho s. [BR90] Ricca do Benede i and Jean-Jacques Risle . Real algeb aic and semi-algeb aic se s. He mann, Pa is, 1990. [Me 00] Jean-Pie e Me le . Alias: an in e al analysis based lib a y o sol ing and analyzing sys em o equa ions. In SEA, Toulouse, F ance, 14–16 June 2000. [OBS92] A suyuki Okabe, Ba y Boo s, and K¯okichi Sugiha a. Spa ial essella ions: concep s and applica ions o Vo ono¨ı diag ams. John Wiley & Sons L d., Chiches e , 1992. Wi h a o ewo d by D. G. Kendall. [´ OY85] Colm ´ O’D´unlaing and Chee-K. Yap. A “ e ac ion” me hod o planning he mo ion o a disc. J. Algo i hms, 6(1):104–111, 1985. [RF99a] Rajesh Ramamu hy and Rida T. Fa ouki. Vo onoi diag am and medial axis algo i hm o plana domains wi h cu ed bounda ies. I. Theo e ical ounda ions. J. Compu . Appl. Ma h., 102(1):119–141, 1999. Special issue: compu a ional me hods in compu e g aphics. [RF99b] Rajesh Ramamu hy and Rida T. Fa ouki. Vo onoi diag am and medial axis algo i hm o plana domains wi h cu ed bounda ies. II. De ailed algo i hm desc ip ion. J. Compu . Appl. Ma h., 102(2):253–277, 1999. [VC90] Ch is ine Voi on-Canicio. Analyse spa iale e analyse d’images pa la mo phologie ma h´ema ique. RECLUS, 1990. [Vo 08] Geo gi¨ı Feodose ich Vo ono¨ı. Nou elles applica ions des pa am`e es con inus `a la h´eo ie des o mes quad a iques. deuxi`eme m´emoi e. eche ches su les pa all´elo`ed es p imi i s. p emi`e e pa ie. pa i ion uni o me de l’espace analy ique `a ndimensions `a l’aide des ansla ions d’un mˆeme poly`ed e con exe. Jou nal ¨u die eine und angewand e Ma hema ik, 134:198–287, 1908.