Precise Voronoi Cell Exracion of Free-form Raional Planar Closed Curves Iddo Hanniel, Ramanahan Muhuganapahy, Gershon Elber Deparmen of Compuer Science Technion, Israel Insiue of Technology Haifa 32000, Israel e-mail: {iddoh, raman, gershon}@cs.echnion.ac.il Myung-Soo Kim School of Compuer Science and Engineering Seoul Naional Universiy Seoul 151-742, Korea e-mail: mskim@cse.snu.ac.kr Absrac We presen an algorihm for generaing he Voronoi cells for a se of raional C 1 -coninuous planar closed curves, which is precise up o machine precision. Iniially, bisecors for pairs of curves, (C(),C i (r)), are generaed symbolically and represened as implici forms in he r-parameer space. Then, he bisecors are properly rimmed afer being spli ino monoone pieces. The rimming procedure uses he orienaion of he original curves as well as heir curvaure fields, resuling in a se of rimmed-bisecor segmens represened as implici curves in a parameer space. A lower-envelope algorihm is hen used in he parameer space of he curve whose Voronoi cell is sough. The lower envelope represens he exac boundary of he Voronoi cell. Keywords: Voronoi cells, Skeleon, Free-form boundaries, Raional curves, Medial Axis Transform. 1 Inroducion Voronoi diagrams are one of he mos exensively sudied objecs in compuaional geomery. Algorihms for generaing Voronoi diagrams for raional curves ypically preprocess he inpu curved boundaries ino linear and circular segmens. This preprocessing generaes a Voronoi diagram ha is approximae boh in opology and geomery. Given a number of disjoin planar regions bounded by free-form curve segmens,,..., C n (r n ), heir Voronoi diagram [Aurenhammer 1991] is defined as a se of poins ha are equidisan bu minimal from wo differen regions. The erm minimal ensures ha for a poin on he boundary curve, he corresponding poin on he Voronoi diagram is he minimum in disance. This definiion excludes self-voronoi edges [Al and Schwarzkopf 1995]. The Voronoi cell of a curve is he se of all poins closer o han o C j (r j ), j > 0. The Voronoi diagram is hen he union of he Voronoi cells of all he free-form curves. In his paper, he erm Voronoi cell normally refers o he boundary of he Voronoi cell. A relaed eniy o he Voronoi diagram, he Medial Axis Transform (MAT) or Skeleon, was inroduced by Blum [Blum 1967; Blum 1973] o describe biological shapes. The MAT can be viewed as he locus of he cener and radius of a maximal ball as i rolls inside an objec. Since heir inroducion, boh Corresponding auhor. Voronoi diagrams and MATs have been used in a wide variey of applicaions ha primarily involve reasoning abou geomery or shapes. Skeleons have been used in paern and image analysis [Baja and Thiel 1994; Monanari 1969], finie elemen mesh generaion [Armsrong 1994; Gursoy and Parikalakis 1992], and pah planning [O Rourke 1993], o menion only a few. Algorihms for generaing Voronoi cells/diagrams have predominanly used linear or circular arc inpus [Kim e al. 1995; Srinivasan and Nackman 1987; Yap 1987]. Moreover, algorihms for generaing Voronoi diagrams for raional eniies ypically preprocess he inpu curved boundaries ino linear and circular segmens [Held 1998], due o he difficuly in processing raional curves direcly. This preprocessing no only leads o he generaion of arifacs and branches no presen in he original Voronoi diagram, which requires a pos-processing sage o remove hem, bu also i produces an approximaed Voronoi diagram [Ramamurhy and Farouki 1999; Ramanahan and Gurumoorhy 2003]. A Voronoi diagram of C 1 planar curves consiss of porions of bisecor curves for some pairs of curves. In paricular, he Voronoi cell of a curve will involve porions of bisecors beween a pair of curves in he se, one of which will be. However, generaing he bisecor for a pair of planar curves is rivial only if hey are simple, such as sraigh lines or circular arcs. When he curve is a raional free-form curve, he bisecor is raional only for a few special cases a poin and a raional curve in he plane [Farouki and Johnsone 1994] and wo raional space curves for which he bisecor is a raional surface [Elber and Kim 1998b]. However, for he case of coplanar curves (polynomial or raional), he bisecor has been shown o be, in general, algebraic bu no raional [Farouki and Johnsone 1994]. This has resuled in he need for numerical racing of he bisecor curves [Farouki and Ramamurhy 1998], which is compuaionally expensive. Alernaively, Elber and Kim [Elber and Kim 1998a] showed ha he bisecor for a pair of raional curves can be implicily represened symbolically in he parameric space. The bisecor so generaed can be represened in an implici form ha can be used for furher processing. Moreover, he bisecors can be accuraely represened up o machine precision. Voronoi diagram generaion algorihms ha do no preprocess curved boundaries are relaively harder o find. Lavender e al. [Lavender e al. 1992] suggesed a subdivision echnique (based on inerval arihmeic) o consruc he global srucure of he Voronoi diagram. This mehod is general in he sense ha i can handle arbirary se-heoreic objecs in any dimensions. Chou [Chou 1995] suggesed an algorihm ha is based on he ree srucure, whereas Al and Schwarzkopf [Al and Schwarzkopf 1995] presened a randomized incremenal algorihm. Neverheless, boh algorihms [Al and Schwarzkopf 1995; Chou 1995] appear o have only heoreical implicaions. Ramamurhy and Farouki [Ramamurhy and Farouki 1999], and Ramanahan and Gurumoorhy [Ramanahan and Gurumoorhy 2003] have used numerical racing mehods. [Ramamurhy and Farouki 1999] firs generaes he bisecors and hen rims hem, whereas [Ramanahan and Gurumoorhy 2003] generaes he rimmed segmens of he medial axis direcly.
In his paper, he problem of generaing a Voronoi cell of a planar free-form closed curve is addressed. The proposed algorihm direcly uses he free-form raional curves ha are no preprocessed ino line segmens or circular arcs. I is shown here ha he symbolically generaed bisecor ha is represened as an implici form in he parameric space beween a pair of planar raional curves can be processed o exrac he Voronoi cell of a closed curve. A lower envelope algorihm [Sharir and Agarwal 1995], a wellknown divide-and-conquer echnique, has been used for generaing he Voronoi cell [Kao 1999] (o be precise, [Kao 1999] generaes he medial axis). Though [Kao 1999] has described an algorihm ha uses lower envelope for generaing medial axis ransform of general curved objecs, he approximaes he inpu curves using biarc splines in his implemenaion. I is shown here ha he lower envelope algorihm can be used for processing raional curves direcly. The oupu of he proposed approach is he boundary of a Voronoi cell. This boundary is no raional in general and hence is represened as a piecewise implici form in he parameeric spaces of he original raional curves. This inroduced represenaion is precise o wihin machine precision, and while no exac, is probably amendable o exac arihmeic compuaions, a opic beyond his work. All he algorihms and examples presened in his paper were implemened and creaed wih he aid of ools available in he IRIT [Elber 2002] solid modeling sysem, developed a he Technion, Israel. The res of his paper is srucured as follows: Secion 2 oulines he basic idea of his paper. The bivariae implici represenaion of he bisecor is described in Secion 3. Spliing he funcion ino monoone pieces is hen described in Secion 4, followed by he descripion of several bisecor consrains in Secion 5. Secion 6 presens he algorihm for Voronoi cell exracion for a closed curve using a lower envelope algorihm. Resuls from our implemenaion and a discussion on he algorihm appear in Secion 7. Finally, Secion 8 concludes he paper. 2 Basic Idea Le,,...,C n (r n ) be he (n + 1) C 1 -coninuous raional parameric planar closed curves. Wihou loss of generaliy, we may assume ha he Voronoi cell is exraced for. The algorihm sars wih he symbolic compuaion of a bivariae polynomial funcion, F 3 (,r i ), following [Elber and Kim 1998a], beween wo planar curves and C i (r i ) so ha he zero-se of F 3 (,r i ), F 3 (,r i ) = 0, which is an implici algebraic curve in he r i -plane, corresponds o he unrimmed bisecor curve beween and C i (r i ). The formulaion of F 3 (,r i ), which is described in Secion 3, is based on a symbolic subsiuion of polynomial/raional funcions of and r i ino a simple polynomial expression in he r i - plane. Figure 1(a) shows he bisecor beween wo planar raional curves. The zero-se of is F 3 (,r i ) is shown in Figure 1(b). In Figure 1(a), he curve-curve bisecor ends before i passes hrough he narrow porion of he wo inpu curves. This early erminaion is due o he fac ha he curve-curve bisecor measures he disances o he wo inpu curves in he orhogonal direcions o he curves angens. Moreover, near hese narrow regions, he displayed curve-curve bisecor should connec o a poin-curve bisecor and hen o a poin-poin bisecor which are no considered, as we assume C 1 geomery for now. A opology analysis algorihm [Keyser e al. 2000], described in Secion 4, is hen used o decompose he bivariae funcion ino r i - monoone pieces. This decomposiion faciliaes he effecive manipulaion of he bivariae funcion when processed furher. Each C i (r i ) (a) The bisecor (in gray) beween wo raional C 1 planar curves. r i (b) The zero-se in he domain of F 3 (,r i ), defines he bisecor. Figure 1: The bisecor and he zero-se beween wo open C 1 raional curves C 1 () and C i (r i ). monoone piece is subsequenly subjeced o several consrain checks described in Secion 5. The consrains are applied based on he orienaion of he wo raional curves as well as heir curvaure fields. The orienaion consrain deermines he bisecor s side wih respec o he curves. The objec side is assumed o lie on he righ-hand side when he curve is raversed in he increasing direcion of he regular parameerizaion. Applicaion of he orienaion consrain purges away porions of he unrimmed bisecor ha do no belong o he desired side. This consrain can also be flipped o obain bisecor porions ha lie on he oher side of he curve. Following his, he resuling bisecor porions are subjeced o a curvaure consrain. This consrain deermines wheher he radius of curvaure of he inpu raional curve a a foopoin of he bisecor is greaer han he radius of curvaure of he disk deermined by he disance from he foopoin o he bisecor. Noe ha he applicaion of his consrain purges away some (bu no all) poins on he bisecor ha are no minimal in disance o he corresponding boundary curves. The above process is repeaed for all pairs of curves (,C i (r i )). One feaure of he algorihm described in his paper is he applicaion of he lower envelope algorihm [Sharir and Agarwal 1995] o generae he Voronoi cell of wih respec o C i (r i ), i > 0. The lower envelope algorihm here akes advanage of he correspondence beween he bivariae funcion and he disance funcion ha measures he disance from he foopoin o he corresponding poin on he bisecor. The lower envelope algorihm for he generaion of he Voronoi cell a is described in Secion 6. 3 Bivariae Funcion In he coming secions, r will replace r i for he purpose of clariy of represenaion. Le = (x 0 (),y 0 ()) and = (x 1 (r),y 1 (r)) be wo planar C 1 -coninuous regular raional curves. Though he funcions F 1 (,r) or F 2 (,r) in [Elber and Kim 1998a] can be used, he bivariae funcion F 3 (,r) in [Elber and Kim 1998a] has been seleced for he generaion of he unrimmed bisecor because i does no generae redundan branches, making i ideal for use in furher processing. For compleeness, he following formulaion is aken, hough no fully, from Elber and Kim [Elber and Kim 1998a].
[Elber and Kim 1998a] showed ha a bisecor poin P mus saisfy: P C0 (),C 0 () = 0, (1) P C1 (r),c 1 (r) = 0, (2) P (C 0()+ 2, = 0. (3) When he wo angens C 0 () and C 1 (r) are neiher parallel nor opposie, he poin P = (x,y) on he inersecion of he wo normal lines of he wo curves has a unique symbolic soluion for he following marix equaion; from Equaions (1) and (2): [ C 0 () C 1 (r) ][ x y ] [ C0 (),C = 0 () ]. C1 (r),c 1 (r) (4) (a) The unrimmed bisecor (in gray) of wo C 1 closed planar curves (in black). The bisecors exend o on boh sides. F 3 (,r) Using Cramer s rule, we can generae a planar bivariae raional surface: P(,r) = (x(,r), y(,r)), which is embedded in he xyplane: x 0 ()x 0 ()+y 0()y 0 () y 0 () x 1 (r)x 1 x(,r) = (r)+y 1(r)y 1 (r) y 1 (r) x 0 () y 0 (), x 1 (r) y 1 (r) x 0 () x 1 y(,r) = (r) x 0()x 0 ()+y 0()y 0 () x 1(r)x 1 (r)+y 1(r)y 1 (r) x 0 () y 0 (). (5) x 1 (r) y 1 (r) r Subsiuing he expressions for x(,r) and y(,r) ino Equaion (3), we ge 0 = Fˆ 3 (,r) = P(,r) (C 0()+ 2,. Equaion (6) is hen muliplied by he denominaor of P(,r) o yield F 3 (,r), 0 = F 3 (,r) = (x 0 ()y 1 (r) x 1 (r)y 0 ()) ˆ F 3 (,r). (7) Once Equaion (7) is solved, poins on he bisecor can be compued from Equaions (5). The bisecor curve has a considerably lower degree when represened in a parameer space (see [Elber and Kim 1998a] for furher deails), compared o he convenional represenaion of he bisecor curve in he xy-plane [Hoffmann and Vermeer 1991]. Figure 2(a) shows he unrimmed bisecor of wo closed curves and Figure 2(b) shows he surface (,r,f 3 (,r)) generaed by he bivariae funcion and is corresponding zero-se. I should be noed ha F 3 (,r) = 0 represens he bisecor wih an accuracy ha is bounded only by he machine precision. 4 Spliing ino Monoone Pieces In order o manipulae and effecively use (i.e., compue lower envelopes) he zero-se of he bivariae funcion, F 3 (,r) = 0, i is necessary o break i ino monoone pieces, in he r-space. Given he polynomial F 3 (,r) in he r-domain, he (6) (b) The bivariae funcion F 3 (,r) and is zero-se. Figure 2: The unrimmed bisecor (a) and he zero-se of F 3 (,r) (b) beween wo C 1 closed planar curves and. problem is o decompose he zeros of he funcion ino monoone pieces wihin he domain, while obaining he conneciviy beween he monoone pieces. This process resuls in branches of he funcion delimied by appropriae r-domain poins, such ha when one raverses he branch from one endpoin o he oher, he branch is increasing/decreasing in boh and r. The opology analysis algorihm presened in [Keyser e al. 2000] has been implemened o spli F 3 (,r) = 0 ino monoone branches in boh and r. The opology analysis algorihm in [Keyser e al. 2000] uses he urning poins (local maxima and minima) and edge poins as he inpu. The urning poins are isolaed by finding he common simulaneous soluions beween F 3 (,r) = 0 and eiher F 3 (,r) = 0 or F 3 (,r) = 0. The mulivariae solver of [Elber and Kim 2001] r is used o find hese local exrema. Edge poins are idenified by compuing all he inersecions of F 3 (,r) = 0 wih he boundaries of he domain of ineres. Recursive subdivision in he r-space is used o find he conneciviy beween all urning and edge poins. Each subdivision involves aking a verical or horizonal line and finding all inersecions of he curve wih ha line. The edge poins are furher classified depending on which border hey hi. The algorihm proceeds by analyzing he paern of urning poins and edges poins in he region, and eiher finding he connecions
Figure 3: The monoone pieces of he zero-se of F 3 (,r) shown in Figure 1(b). beween hem, or subdividing he region. For furher deails, please refer o [Keyser e al. 2000]. Figure 3 shows he monoone pieces (bounded by a box) obained by he applicaion of he opology analysis algorihm for he zerose of F 3 (,r) shown in Figure 1(b). The algorihm splis he given inpu funcion a he midpoin whenever wo or more urning poins are deeced. Hence, he wo monoone regions ha are visible in he middle region of Figure 3. This, however, does no affec our algorihm in any way. 5 Applying Consrains In his secion, wo consrains are described o rim he individual bisecors represened as he zero-se of he bivariae funcion F 3 (,r). The consrains are based on he orienaion of he inpu curves (Secion 5.1) and heir curvaure fields (Secion 5.2). 5.1 Orienaion Consrain The orienaion consrain uses he direcion of he parameerizaion of he inpu raional curves. The righ hand side is assumed o belong o he inerior of he objec while raversing he curve along he increasing direcion of parameerizaion. The orienaion consrain purges away poins on he unrimmed bisecor ha do no lie on he desired side ha is, a lef-lef consrain generaes r-poins corresponding o bisecor poins ha lie o he lef of boh curves. Hence i can also be ermed as LL-consrain where LL implies Lef-Lef. In a similar manner, we can consruc he oher orienaion consrains as LR/RL/RR consrains. Figure 2(a), in fac, shows he LL, LR, RL, and RR componens of he bisecor beween he wo C 1 planar closed curves. The LL-consrain for wo regular C 1 raional curves is reduced o he following wo equaions: P(,r) C0 (),N L 0 () > 0, (8) P(,r) C1 (r),n L 1 (r) > 0, (9) where N0 L() is he normal field defined by ( y 0 (),x 0 ()). NL 1 (r) has a similar expression and P(,r) is as defined in Equaion (5). Noe ha N0 L() and NL 1 (r) do no flip direcions a heir inflecion poins. The LL-consrain can also be formulaed in he following way. Since a poin on he bisecor is obained by he inersecion of normal lines a he foopoins of he raional curves [Elber and Kim Figure 4: The bisecor (in gray) beween wo closed C 1 planar curves afer applying he LL-consrain (see also Figure 2(a)). The bisecor exends o on boh sides 1998a], he inersecion poin saisfies he following equaion: +αn0 L () = C 1(r)+βN1 L (r), (10) where α and β represen he parameer on he normals of and, respecively, whose signs deermine wheher he inersecion poin is o he lef of he curves. Posiive values for boh α and β imply ha he inersecion poin is o he lef of boh curves. Thus, he LL-consrain can also be wrien as follows: α(,r) > 0, (11) β(,r) > 0, (12) where α and β are given by (refer o [Elber and Kim 1998a]): x 1(r) x 0 () y 1 (r) y 1 (r) y 0 () x 1 α = α(,r) = (r) y 0 () y 1 (r), x 0 () x 1 (r) y 0 () x 1(r) x 0 () x 0 β = β(,r) = () y 1(r) y 0 () y 0 () y 1 (r). (13) x 0 () x 1 (r) Figure 4 shows he porions of he bisecor of Figure 2(a) afer applying he LL-consrain. 5.2 Curvaure consrain The curvaure consrain is based on he compuaion of he curvaure fields of he inpu raional curves and disk a a poin on he bisecor. This consrain eliminaes some (hough no all) poins on he bisecor ha do no saisfy he minimal disance requiremen of he Voronoi diagram. This consrain, in he form of a lemma, is as follows: Lemma 1 The radius (of curvaure) of he disk a any poin on he Voronoi diagram/cell should be less han or equal o he minimum of he local radii of curvaure of he boundary segmens. For he proof of he lemma, please refer o [Chou 1995]. Figure 5 illusraes he curvaure consrain. Figure 5(a) shows ha he radius of he disk is smaller han he radius of curvaure of he curves a he foopoins, implying ha he curvaure consrain is fully saisfied and he cener of he disk (marked a in he figure) becomes a possible valid poin on he Voronoi diagram. Violaion of his consrain implies ha he radius of he disk a ha bisecor poin is greaer han he radius of curvaure of a curve a he foopoin. Figure 5(b) shows a disk ha violaes he curvaure consrain. Consequenly, he cener of such a disk (marked b in he figure), hough i is a poin on he bisecor, is no a poin on he Voronoi
a b (a) (b) Figure 6: The bisecor (in gray) afer applying he curvaure consrain, for he wo curves shown in Figure 1(a). Figure 5: (a) The radius of he disk is smaller han he radius of curvaures a he foopoins of he curves. (b) The radius of he disk is greaer han he radius of curvaure of a he foopoin implying he violaion of he consrain. diagram, as i violaes he minimaliy of he disance. Hence, poins such as b have o be purged away. To formulae he consrain, a disance funcion D(, r) is inroduced ha corresponds o he disance beween he bisecor poin and he foopoin (acually o boh foopoins as hey are equidisan from he bisecor poin) and is defined as follows: D 0 (,r) = P(,r), (14) D 1 (,r) = P(,r). (15) The disance funcions correspond o he radius of he disk ha is angen o he foopoins, and eiher D 0 (,r) or D 1 (,r) can be used. Since he curvaure of he disk of a bisecor poin is smaller han he (posiive) curvaure of one of he curves a he foopoin, we have: 1/D 0 (,r) > ( sign C 0 0 () ) κ 0 (), (16) 1/D 1 (,r) > ( sign C 1 1 (r) ) κ 1 (r), (17) where he sign funcion denoes he sign of he expression wihin he brackes and κ 0 () and κ 1 (r) are he curvaure funcions of he respecive curves. Consrains (16) and (17) appeared o be oo slow in our preliminary implemenaion since hey are applied over all he poins on he bisecor. Moreover, hey are no raional expressions as hey conain square roos and hence mus be squared. In conras, vecor funcions ˆN 0 ()/κ 0 (), ˆN 1 (r)/κ 1 (r), κ 0 () ˆN 0 (), and κ 1 (r) ˆN 1 (r) are raional, provided and are raional curves, where ˆN 0 () and ˆN 1 (r) denoe he uni normal vecors. Then, he curvaure consrains can be reformulaed in he following alernae way: P(,r) C0 (), ˆN 0 ()/κ 0 () < ˆN 0 ()/κ 0 (), ˆN 0 ()/κ 0 (), (18) P(,r) C1 (r), ˆN 1 (r)/κ 1 (r) < ˆN 1 (r)/κ 1 (r), ˆN 1 (r)/κ 1 (r). (19) N0 L() (resp. NL 1 (r)) can eiher be in he same or he opposie direcion wih respec o ˆN 0 () (resp. ˆN 1 (r)). If hey are in he opposie direcions, he lef hand side of inequaliies (18) and (19) will be negaive and, herefore, always hold. If N0 L() (resp. NL 1 (r)) is in he same direcion as ˆN 0 () (resp. ˆN 1 (r)), hen hese inequaliies will hold only when he radius of he disk is smaller han he radius of curvaure 1/κ 0 () (resp. 1/κ 1 (r)) of he boundary curve (see Figure 5). The curvaure consrain also implies ha a poin on he Voronoi cell canno be closer o is foopoin han he evolue poin corresponding o ha foopoin, since + ˆN 0 ()/κ 0 () (resp. + ˆN 1 (r)/κ 1 (r)) races he evolue of a curve (resp. ) and is raional, provided (resp. ) is raional. The curvaure consrains (18) and (19) can be reduced o P(,r) C0 (), ˆN 0 ()/κ 0 () or 1/κ 0 () 2 ˆN 0 (), ˆN 0 () < 1, (20) P(,r) C1 (r), ˆN 1 (r)/κ 1 (r) 1/κ 1 (r) 2 < 1; (21) ˆN 1 (r), ˆN 1 (r) P(,r) C0 (),κ 0 () ˆN 0 () < 1, (22) P(,r) C1 (r),κ 1 (r) ˆN 1 (r) < 1, (23) since ˆN 0 (), ˆN 0 () = ˆN 1 (r), ˆN 1 (r) = 1. Figure 6 shows he porions of he bisecor obained afer applying he curvaure consrain (and afer subjecing i o he LL-consrain) for he unrimmed bisecor shown in Figure 1(a). Noe ha some poins on he bisecor ha do no correspond o he minimaliy of he disance condiion of he Voronoi diagram have been removed. However, furher processing o remove all remaining poins ha do no correspond o he minimaliy of he disance is required. This is achieved by applying he lower envelope algorihm ha is described in he following secion. 6 Voronoi Cell Exracion In his secion, he exracion of he Voronoi cell for a C 1 - coninuous closed curve, wih respec o he oher C 1 - coninuous closed curves C i (r i ), i > 0 is described. The exracion sars wih he compuaion of he unrimmed bisecors using he bivariae funcion F 3 (,r 1 ) described in Secion 3 and is followed by spliing F 3 (,r 1 ) = 0 ino monoone pieces described in Secion 4. The process of rimming he zero-se of F 3 (,r 1 ) using he LL-consrain described in Secion 5.1 is hen carried ou. Subsequenly, he curvaure consrains described in Secion 5.2 rim he zero-se of F 3 (,r 1 ) furher. The above process is repeaed for all pairs of curves and C i (r i ), i > 1. The lower envelope
algorihm (described in his secion) is used a he end o compue he Voronoi cell of. The problem of consrucing he Voronoi cell is reduced o he compuaion of he lower envelopes in he following way: Given a curve and he se of curves C i (r i ), le D i () denoe he disance funcion D i (,r i ), described in Secion 5.2. Then, he lower envelope of he se {D i ()},i = 1,...,n idenifies he Voronoi cell of. In oher words, he problem of compuing he Voronoi region of is reduced o he problem of compuing he lower envelope of a se of disance funcions, D i (), over he -domain of. Poins on he Voronoi cell are hen compued by mapping he poins on he lower envelope ono he r i -domain. The concep of compuing he lower envelopes has been exensively used for arrangemens of line segmens [Halperin 1997; Sharir and Agarwal 1995]. For he benefi of readers, he basic lower envelope algorihm is inroduced in Secion 6.1, following [Sharir and Agarwal 1995]. The descripion of he basic predicaes needed o implemen he algorihm over raional curves is hen presened in Secion 6.2, followed by he deails of using he predicaes o generae he Voronoi cell, in Secion 6.3. 6.1 General Lower Envelope Algorihm Given a se of -monoone curves, hey are iniially divided ino wo subses of size a mos n/2, recursively compuing he lower envelope for each of he subses. Then, hese subenvelopes are merged back o obain he overall lower envelope. The erminaion condiion of he recursion is a single -monoone curve ha is he lower envelope of iself. The main par of he algorihm is he merging sep, which is performed in a sweep-like manner. Le us assume ha we have wo liss of curves, which are hemselves lower envelopes ha are o be merged. The merging process of he wo liss sars from he lefmos parameer value. A sweep-like algorihm, along he - domain, idenifies he se of inervals [a,b] where he wo liss overlap. Therefore, we end up wih merges of he inervals [a i,b i ] in he -domain where boh curves are defined. All he inersecion poins of he curves from he wo liss lying on he inerval [a i,b i ] are idenified. The parameer values, where inersecions occur, are idenified. Beween a pair of inersecion poins, he porions ha belong o he merged lower envelope have o be idenified. This can be done by comparing he D i ()-values of he wo curves a he middle -parameer beween he inersecion poins. Because of coninuiy, he curve wih he smaller D i ()-value a he mid-parameer has smaller D i ()-values over he inerval [a,b] and, herefore, belongs o he merged lower envelope. D 2 1 1 2 a b D 2 1 1 2 a c b I 1 2 (a) (b) (c) Figure 7: Illusraion of he lower envelope algorihm. D The lower envelope algorihm is illusraed for line segmens in Figure 7(a). Consider he wo segmens 1-1 and 2-2, as shown in he figure. The firs sep involves he idenificaion of overlapping - parameric regions and he spliing of he wo segmens a hese parameric values. For example, for he wo curves 1-1 and 2-2 shown in Figure 7(a), he overlapping parameer region is ab and hence hey are spli a parameers a and b. Furher processing is required only for he porions where he overlapping of parameers occurs. The porion of overlapping parameric regions is esed for inersecions. If inersecion poins exis, hey are again spli a hose parameer values (e.g., c in Figure 7(b)). The spli porions beween inersecion poins are hen esed o idenify he ones wih minimal disance and o eliminae oher porions. The D i () comparison check is performed only a he mid-parameer value beween he inersecions (marked as dos in Figure 7(b)). The resul of he merging process in he example is shown in Figure 7(c). 6.2 Lower envelopes of raional curves Analyzing he algorihm, i is easy o see ha i requires only a few basic geomeric operaions. However, when raional curves are involved, he possible compuaion for each of hese funcions may differ. Then, he main funcions needed are A funcion ha idenifies all -parameers where inersecions occur. Formulaion of he inersecion parameers can vary from one problem o anoher and also depends on he ools ha are available for ha paricular problem. For example, in he case of a Voronoi cell, equaing wo disance funcions, compued symbolically, will deermine he parameers of he inersecions. A funcion ha compares he D i ()-values of wo curves a a given -parameer. In he Voronoi cell deerminaion, his funcion amouns o a comparison of he disance funcion values a he corresponding middle -parameer value. A funcion for spliing a -monoone curve ino wo subcurves a a new -parameer. For line segmens or for univariae funcions, hese basic funcions can easily be implemened. In he following secion, he deails of generaing he Voronoi cell via lower envelopes is described. 6.3 Voronoi Cell via Lower Envelope Given a r i -monoone curve F 3 (,r i ) = 0, for every value here is a single corresponding r i value. Furhermore, here is a correspondence beween F 3 (,r i ) = 0 and D i (). If F 3 (,r i ) = 0 is monoone over an inerval [a,b], hen D i () is well defined over [a,b] since for every value here is a single r i value and hence a single corresponding disance value. Therefore, we can apply he lower envelope algorihm on he disance funcions defined over monoone bisecors of he form F 3 (,r i ) = 0, provided we can compue he needed basic predicaes for D i () as described in Secion 6.2. Unforunaely, a raional represenaion of D i () is no available. The square of he disance funcion D i (,r i ) may be used insead. Figure 8(b) shows he squared disance funcions D 2 1 (,r 1) and D 2 2 (,r 2) for he bisecor beween pairs of planar C 1 raional curves (,) and (,) shown in Figure 8(a). Then, he required basic predicaes can be compued symbolically using F 3 (,r i ) and D 2 i (,r i) funcions. The inersecion poins can be idenified using he following se of
D 2 D 2 1 (,r 1) D 2 2 (,r 2) (a) Figure 8: (a) The bisecors (in gray) beween he pairs (,) and (,) of C 1 planar curves. (b) The squared disance funcions D 2 1 (,r 1) and D 2 2 (,r 2) of he bisecor of he respecive pairs of curves. equaions (i j): (b) D i (,r i ) 2 = D j (,r j ) 2, (24) F 3 (,r i ) = 0, (25) F 3 (,r j ) = 0. (26) Figure 9: Voronoi cell (in gray) of. has Deg = 3, and CP = 7, and have Deg = 5, CP = 5. DegF 3max = 18. more es cases where all objecs have convex profiles. The algorihm works well for non-convex curves as well (see Figures 13 and 14). Comparison of wo squared disance funcions, D 2 i () and D2 j (), a a given parameer is performed by iniially obaining he corresponding r i and r j parameer values (a single soluion for r i - monoone F 3 (,r i ) = 0 segmens) and hen comparing he funcion values of D 2 i (,r i) and D 2 j (,r j) a he respecive parameer values. To spli a D 2 i () funcion ino wo subcurves a a given -parameer, all ha is needed is o spli is corresponding r i -monoone implici curve F 3 (,r i ) = 0 a ha parameer. The resul of he lower envelope algorihm is hen a lis of r i - monoone implici curves F 3 (,r i ) = 0 ha are spli a -parameers of equidisan bisecor poins. The union of he curves F 3 (,r i ) = 0 represens he Voronoi cell of. C 3 (r 3 ) 7 Resuls and Discussion We have implemened he proposed algorihm for exracing he Voronoi cell of a closed curve using he IRIT [Elber 2002] modeling environmen. In his secion, resuls from some es cases are presened. The inpu raional curves are represened as Bézier or B-spline curves. A no ime hese (piecewise) raional curves are approximaed ino line segmens, circular arcs, or a differen represenaion. Hence, we are working wih only a few (piecewise) raional inpu eniies as opposed o a large number of (linear and/or circular) inpu generaors which may be he case when approximaion schemes are employed. In wha follows, he closed curve for which he Voronoi cell is o be generaed is denoed as and he oher closed curves are denoed appropriaely as C i (r i ), i > 0. The degree of he inpu curves (denoed as Deg), number of conrol poins used (denoed as CP) and he degree of F 3 (,r) (denoed as DegF 3 ) compued as given in [Elber and Kim 1998a] are also provided for each of he figures. They can someimes denoe he maximum value of he inpu curves which can be idenified by he ag max. Figure 9 shows he Voronoi cell of a closed curve. Figure 10 shows he Voronoi cell of, where he inpu geomery consiss of four closed curves. Figures 11 and 12 show wo Figure 10: Voronoi cell (in gray) of. All he inpu curves have Deg = 5, and CP = 5. DegF 3 = 18. 7.1 Discussion The following are he major seps of our algorihm: 1. formulae he bivariae funcion F 3 (,r); 2. spli he zero-ses of F 3 (,r) s ino monoone pieces; 3. apply he rimming consrains; and 4. compue he lower envelope. Our experimenal resuls indicae ha all he above seps are reasonably efficien. Compuaion of he symbolic funcion F 3 (,r) ook from a fracion of a second o a few seconds. Compuaion of he monoone pieces and applicaion of he consrains ook from several seconds o a minue. The lower envelope algorihm ook from a few seconds o a few minues. All he experimens were carried ou
C 4 (r 4 ) C 3 (r 3 ) C 4 (r 4 ) Figure 11: Voronoi cell (in gray) of. All he inpu curves have Deg = 3, and CP = 7. Here, he bisecor curves degenerae ino lines. DegF 3 = 10. C 4 (r 4 ) C 3 (r 3 ) Figure 12: Voronoi cell (in gray) of. All he inpu curves have Deg = 3, CP = 7. DegF 3 = 10. C 3 (r 3 ) Figure 14: Voronoi cell (in gray) of. All he inpu curves have Deg = 3, CP = 7. DegF 3 = 10. Even hough he algorihm has been implemened using floaing poin arihmeic, i has been found o be reasonably robus. Spliing he zero-se of a bivariae funcion F 3 (,r) ino monoone pieces involves he compuaion of urning poins, which can be compued o arbirary precision. Though he spliing procedure described in [Keyser e al. 2000] uses exac arihmeic, our experimens indicae ha he spliing worked well wih floaing poin arihmeic in all cases ha have been esed so far. Exac arihmeic ends o slow down he algorihm considerably. I is also possible ha alernae algorihms ha spli a curve ino monoone pieces could be used. The lower envelope algorihm uses symbolic compuaion of disance funcions and hence i is fas, reliable and robus. The inpu raional free-form curves are used direcly wihou approximaing hem by linear or circular arcs. As a resul, he daa ha are generaed on he Voronoi cell can be considered precise o wihin he machine precision. Moreover, he algorihm does no require a pos-processing sage o remove he arifacs ha would have been creaed by an approximaion of he raional curves. The algorihm generaes Voronoi cells ha are opologically correc and geomerically accurae o whaever precision desired. No all bisecors, generaed beween pairs of closed curves, are going o play a role in generaing he Voronoi cell of some closed curve. For example, for he closed curve in Figure 10, he Voronoi cell of does no conain pars of he bisecor generaed for he pair and C 3 (r 3 ), even hough he unrimmed bisecor is generaed for hem and processed by he algorihm. Since he bivariae funcion F 3 (,r) can be used o generae selfbisecors (bisecors of a curve wih iself), he presened algorihm can also be used o generae self-voronoi edges. Since he definiion of a Voronoi diagram precludes self-voronoi edges (radiionally, Voronoi diagram is defined only for disinc eniies), i is no shown in he curren work. However, he self-voronoi edges will be useful when consrucing he MAT of free-form curves. Since he bisecor beween a poin and a raional curve is shown o be raional [Farouki and Johnsone 1994] and he bisecor beween wo raional curves can be represened implicily as shown in [Elber and Kim 1998a], his work can also be used o generae Voronoi diagrams or MATs for domains bounded by piecewise C 1 raional curves and verices ha are concave or reflex. This is currenly under invesigaion. Figure 13: Voronoi cell (in gray) of. All he inpu curves have Deg = 3. However, and have CP = 7, whereas has CP = 8. DegF 3max = 10. on an Inel Penium 4 1.8GHz compuer wih 256MB RAM using he IRIT [Elber 2002] environmen. 8 Conclusions This paper presened an algorihm for generaing he Voronoi cells of a se of C 1 -coninuous raional planar closed curves. I is shown ha he symbolically compued bisecors in suiable parameer space for pairs of raional curves can be used o generae he Voronoi cells of planar curves by employing a lower envelope algo-
rihm. The approach in his paper also shows ha he inpu raional curves do no need o be preprocessed and approximaed wih linear or circular segmens, hereby eliminaing he pos-processing sage required o rim he arifacs generaed by his preprocessing. Anoher promising imporan advanage of his approach is he opion of applying he same basic sraegy o generae Voronoi diagram and medial axis of an objec bounded by piecewise raional curves. This is currenly under invesigaion. In conclusion, i should be noed ha he proposed algorihm does no approximae he bisecor segmens and he oupu is he boundary of a Voronoi cell ha is represened as piecewise implici forms in a r-parameer space ha is arbirarily precise. The presened approach can also be implemened using exac arihmeic [Yap 1997], using packages such as bignums [GMP 2002], and using long floas [CORE 2004]. Acknowledgmens This research was suppored in par by he Israel Science Foundaion (gran No. 857/04) and in par by an European FP6 NoE gran 506766 (AIM@SHAPE), and in par by he Korean Minisry of Science and Technology (MOST) under he Korean-Israeli Binaional Research Gran. References ALT, H., AND SCHWARZKOPF, O. 1995. The voronoi diagram of curved objecs. 11h Symposium on Compuaional Geomery, 89 97. ARMSTRONG, C. G. 1994. Modeling requiremens for finie elemen analysis. Compuer Aided Design 26, 7 (July), 573 578. AURENHAMMER, F. 1991. A survey of fundamenal geomeric daa srucure. ACM Compuing Surveys 23, 3 (Sepember), 345 405. BAJA, G. S., AND THIEL, E. 1994. (3-4)-weighed skeleon decomposiion for paern represenaion and descripion. Paern Recogniion 27, 8, 1039 1049. BLUM, H. 1967. A ransformaion for exracing new descripors of shape. Models for he Percepion of Speech and Visual Form, 362 381. BLUM, H. 1973. Biological shape and visual science (par i). 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