Topology of the Intersection of Two Parameterized Surfaces, Using Computations in 4D Space

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1 Topology of he Inersecion of Two Parameerized Srfaces, Using Compaions in 4D Space Séphane Cha, André Galligo To cie his ersion: Séphane Cha, André Galligo. Topology of he Inersecion of Two Parameerized Srfaces, Using Compaions in 4D Space. Tor Dokken Georg Mningh. SAGA Adances in ShApes, Geomery, and Algebra, 10, Springer, pp , 2014, Geomery and comping, < / _7>. <hal > HAL Id: hal hps://hal.archies-oeres.fr/hal Sbmied on 25 Mar 2016 HAL is a mli-disciplinary open access archie for he deposi and disseminaion of scienific research docmens, wheher hey are pblished or no. The docmens may come from eaching and research insiions in France or abroad, or from pblic or priae research ceners. L archie oere plridisciplinaire HAL, es desinée a dépô e à la diffsion de docmens scienifiqes de niea recherche, pbliés o non, émanan des éablissemens d enseignemen e de recherche français o érangers, des laboraoires pblics o priés.

2 Topology of he Inersecion of Two Parameerized Srfaces, Using Compaions in 4D Space Séphane Cha and André Galligo Uniersié de Nice Sophia-Anipolis and INRIA (GALAAD projec) FRANCE Résmé The inersecion cre of wo parameerized srfaces is characerized by 3 eqaions F i(s, ) = G i(, ), i = 1, 2, 3 of 4 ariables. So, i is he image of a cre in for dimensional space. We proide a mehod o draw sch cre wih a garaneed opology. 1 Inrodcion 1.1 Ineres of he problem In Comper Aided Geomeric Design (CAGD), parameerized srfaces are sed for delimiing olmes. The compaion of he inersecion cre beween sch wo srfaces is hs crcial for he descripion of he CAGD objecs. A simple sed mehod o address his problem consiss in sing a mesh for each srface, and hen proceed o heir inersecion ia inersecion of riangles. A drawback is insabiliy creaed by inersecing almos parallel riangles. A more sable mehod relies on global represenaions of he srfaces by B-splines ; howeer he sal CAGD procedres (offseing, drafing,...) do no consere his model. In pracice, so-called procedral srfaces (i.e. gien by ealaion) are sed, in CAGD sysems, for represening seqences of consrcions indicaed by he ser. Then a B-spline approximaion is comped for frher deelopmens. So, een if he inersecion mehod is exac, in is final sep, i only proides an approximaion of he real inersecion cre. Idealisically, approximaions of he srfaces shold no be separaed from he inersecion process. An inermediae sraegy is o approximae he gien srfaces by meshes of algebraic shapes more complex han he riangles ; hence he inersecion locs will be more precise. A good choice is o approximae by Bézier srface paches of small degree (see secion 1.2). Then, i is crcial o be able o efficienly inersec sch wo polynomial parameerized srfaces. In his paper, we aim o conribe o a robs solion of his problem which 1

3 (2n + 1) poins = n paches (2m + 1) poins = m paches 2 (p 2) m = 2 q 2 (p 1) 2 (q 2) 2 (q 1) n = 2 p Figre 1 Grid of biqadraic paches on he lef. Grid of boxes wih n = 2 p and m = 2 q on he righ. aoid some drawbacks as large inermediae algebraic expressions ha appear in projecion mehods. The inersecion cre of wo sch parameerized srfaces is characerized by 3 eqaions F i (s, ) = G i (, ), i = 1, 2, 3 of 4 ariables. So, i is he image of a cre in for dimensional space. We proide a mehod o draw sch cre wih a garaneed opology. 1.2 An example of biqadraic meshing of a procedral srface Le S be a general parameerized srface gien by ealaions. We consider a grid of poins on S of size (2m + 1, 2n + 1). This is sed o consrc a grid of biqadraic paches of size (m, n). Figre 1 lef illsraes his grid in he 2D parameer space. Ths he coefficiens are shared beween adjacen paches. An example of sch kind of approximaion is gien in figre 2. In his example, we hae on he lef a shape composed by hree B-spline srfaces, hen we consider an offse, which canno be represened by a B-spline, and we approximae i by a grid of 144 biqadraic paches (he resl is shown on he righ). In order o see he offse, a clipped picre is also gien (figre 3). Now, we consider wo sch grids and hierarchies on S 1 and S 2 wo srfaces o be inerseced. We prodce anoher grid of m n 3D boxes aking minmax ales of he pach coefficiens, each box conains he pach hanks o he conex hll propery of he Bézier srfaces. Then, we bild a qadree hierarchy coering his grid. Figre 1 righ illsrae his consrcion. Using hese qadrees we search for inersecing boxes and we obain a se of pairs of inersecing boxes associaed o paches. This process is efficien and, as we will 2

4 Figre 2 Approximaion of an offse by a grid of 144 biqadraic paches. see, proides a good descripion of he inersecion cre. Howeer, i reqires an efficien and robs algorihm for he inersecion of wo Bézier srface paches. Remark 1 If m and n as powers of 2, hen he daa srcre is simplified. In he seqel of he paper, we concenrae on he presenaion of or sbdiision algorihm for he inersecion of algebraic paches. 1.3 Organizaion of he paper In secion 2, a brief descripion of preios work on opology compaion is gien. Especially an inrodcion on sbdiision approach for he plane cres is illsraed. Then, secion 3 deals wih he opology of an implici for dimension cre. A complee descripion of is compaion, by a sbdiision mehod, is gien in his case. In fac, his case corresponds o he inersecion cre beween wo polynomial parameerized paches. Some implemenaion aspecs are addressed in secion 4 and some examples are presened. The las secion (5) is abo he opology in R 3. I shows ha he link beween he inersecion problem in R 4 and he corresponding geomeric siaion in R 3 is no riial. 3

5 Figre 3 Approximaion of an offse by a grid of 144 biqadraic paches (clipped picre). 2 Preios work on opology compaion of a cre 2.1 Isoopic cre The opology of an algebraic cre C in R n (n 2) can be represened by a lis of line segmens whose concaenaion forms a cre isoopic o C. Seeral consrcions make his definiion effecie. Sweeping mehods rely on parallel lines or planes and deec opological eens (criical poins) sch as angen poins o he sweeping planes or singlariies ; we refer o [1, 2] for planar cres and [3, 4] for spaial cres. Wih hese algebraic approaches, he precise deerminaion of he criical poins generally reqires o compe sb-reslan seqences and is ofen ime consming. Sbdiision and exclsion echniqes (see [5, 6]) rely on (simple) crieria o remoe nnecessary domains hen resric o domains where he siaion is ame. Polynomial represenaion in Bernsein bases is generally preferred (see [7, 8]). 4

6 y x Figre 4 Topology ia reglariy es in 2D case. 2.2 Reglariy es and sbdiision mehod A sbdiision approach for comping he opology of he inersecion cre beween wo algebraic srfaces is gien in [5]. I consiss in sbdiiding he domain nil a reglariy es is saisfied. Le s briefly recall i. Le f(x, y) be a polynomial and B = [a, b] [c, d] R 2 a box, consider he implici cre associaed o f in he box B by he eqaion f(x, y) = 0. A reglariy es will allow o deermine niqely he opology of he cre in he box from is inersecion wih he bondary. A collecion of segmens is proided, which realizes an isoopy. Proposiion 1 If y f(x, y) 0 for all (x, y) B = [a, b] [c, d], hen for all x [a, b] here exiss a mos one y [c, d] sch ha f(x, y) = 0. Proof. Le x 0 be a ale in [a, b]. If here were wo differen ales y 0 < y 1 in [c, d] sch ha f(x 0, y 0 ) = f(x 0, y 1 ) = 0 hen by Rolle s heorem, i wold exis y 2 [y 0, y 1 ] sch ha y f(x 0, y 2 ) = 0. Remark 2 This crierion considers y f(x, y) for all ales (x, y) in he box and no only for all poins of he cre, so i is raher resricie. To implemen his crierion, he polynomial y f(x, y) is expressed in Bernsein basis and he coefficiens are reqired o share he same sign. A similar saemen holds replacing he condiion y f(x, y) 0 by x f(x, y) 0 (for all (x, y) B). If f saisfies his es, hen he opology of he cre {(x, y) B f(x, y) = 0} can be deermined niqely knowing he inersecion poins beween he cre and he border of B. Hence, a firs sep is o compe all hese inersecion 5

7 poins (a poin is repeaed if is mlipliciy is een) and sor hem by heir x componen o obain a lis of poins p 1, p 2,..., p 2s 1, p 2s. Then, in he box, he cre is isoopic o he se of segmens : [p 1, p 2 ],..., [p 2s 1, p 2s ] (see he illsraion in figre 4). The crierion can be checked recrsiely sbdiiding he iniial cre (sing De Caselja s algorihm) nil a family of boxes is obained where he es is erified. The approach is exended (in [5]) o he case of 3D cre defined implicily by 2 eqaions. This proides an elegan and efficien solion o he opology compaion problem of an inersecion cre beween wo implici srfaces. 3 Topology of a parameerized srface/parameerized srface inersecion 3.1 Eqaions Le F and G be wo polynomial srface paches ( ) [0, 1] 2 R F : 3 (s, ) F (s, ) ( ) [0, 1] 2 R G : 3 (, ) G(, ) We sppose ha he inersecion F G is a cre : C = { (s,,, ) [0, 1] 4 F (s, ) G(, ) = 0 }. Or aim is o compe he opology of C by a sbdiision mehod generalizing he approach described in secion 2. An injeciiy crierion which says ha for all s 0 [0, 1] here exis a mos one ( 0, 0, 0 ) [0, 1] 3 sch ha F (s 0, 0 ) G( 0, 0 ) = 0 is needed. So, for a fixed s 0 [0, 1], le s sdy he map : ( ) [0, 1] 3 R φ : 3 (,, ) F (s 0, ) G(, ) Thereafer, we se he noaion φ(,, ) = F (s 0, ) G(, ) = (φ 1, φ 2, φ 3 ). 3.2 Topology of a 4 dimension implici cre (reglariy crierion) Consrcion of he injeciiy crierion for φ. A necessary condiion of injeciiy is he local injeciiy of φ. By he inerse fncion heorem, i is saisfied when he jacobian of φ is non zero oer [0, 1] 3 : (,, ) [0, 1] 3, de ( φ(,, ), φ(,, ), φ(,, )) 0. 6

8 If φ is no injecie, here exis wo differen poins A and B in [0, 1] 3 sch ha for all i {1, 2, 3}, φ i (A) = φ i (B). Or analysis relies on he inrodcion of he wo following sbses of [0, 1] 3 : S 1 := { M [0, 1] 3 φ 1 (M) = φ 1 (A) } (1) C 1,2 := { M [0, 1] 3 φ 1 (M) = φ 1 (A) and φ 2 (M) = φ 2 (A) }. (2) We assme local injeciiy and look for sfficien condiions of injeciiy of φ. Firs case : A and B are on a same conneced componen of C 1,2 denoed by Γ. As Γ is a conneced cre, local injeciiy of φ implies ha Γ can be parameerized (by he implici fncion heorem). So φ 3 resriced o Γ is differeniable and akes he same ale a A and B. Hence, φ 3 admis an exremm on C 1,2. This wold conradics local injeciiy of φ, so i canno happen. Second case : A and B are on wo differen conneced componens of C 1,2 denoed by C A and C B. None of hese wo cres can describe a loop becase his wold conradic he local injeciiy of φ. Therefore C A (respeciely C B ) inersecs wo imes he border of he cbe [0, 1] 3 ; in for disinc poins P 1, P 2, P 3 and P 4. So we ge a sfficien condiion of injeciiy if we can rle o his las possibiliy. Or sraegy is o impose sfficien monoony condiions on φ 1 and φ Monoony condiion on φ 1 Firs, we impose monoony condiions on φ 1 resriced o he edges of [0, 1] 3. For example, we can reqire ha φ 1 increases on each edges of [0, 1] 3 as indicaed in figre 5. So φ 1 anishes a mos once on each pah going from he erex O o he erex I following he ordered egdes. This condiion implies ha he implici srface S 1 (of eqaion φ 1 (,, ) = φ 1 (A)) is conneced. Indeed, if S 1 admied wo conneced componens in he cbe, hey wold inersec he edges a he same poins which is impossible. Moreoer we classify all possible configraions by he nmber of he inersecion poins (3, 4, 5 or 6) beween S 1 and he edges as illsraed in figre 6. Noe ha as C 1,2 S 1 and S 1 [0, 1] 3, he eqaliy #(C 1,2 S 1 ) = #(C 1,2 [0, 1] 3 ) holds (see figre 7) Monoony condiion on φ 2 along S 1 Now, we sdy each configraion. We impose monoony condiions on φ 2 along he border of S 1 o force C 1,2 o hae a mos wo inersecion poins wih his border. The following lemma will be sefl : 7

9 I O Figre 5 Example of monoony of φ 1 on he edges of [0, 1] 3. Lemma 1 Le f be a C 1 real fncion oer an open conex se U R 2 and h be a nonzero ecor in R 2. If for all U we hae f() h > 0, hen f is increasing in he direcion h on U i.e U and ɛ > 0 sch ha (+ɛ h) U, we hae f( + ɛ h) > f(). Proof. Le 0 U and ɛ > 0 sch ha (+ɛ h) U. Then f( 0 +ɛ h) f( 0 ) = 1 0 ϕ() d wih ϕ() = f ( 0 + ɛh) h which is posiie. Replacing f by f, we ge similarly ha f() h < 0 implies f is decreasing in he direcion h on U. Recall ha in he plane, for a nonzero ecor = (a, b), he ecor := ( b, a) is normal o and he oriened angle (, ) is eqal o π/2. So, in order o ensre monoony of φ 2 along he border of S 1 (see figre 7), we orien S 1 by he ecor field φ 1. This indces an orienaion on he border S 1 of S 1 ; S 1 is he inersecion of S 1 wih he faces of he cbe. This orienaion of he border of S 1 in each face is gien by where is he projecion of φ 1 on he faces. Then, we impose a monoony direcion of φ 2 resriced o S 1 on each face of he cbe. To illsrae his procedre, figre 8 represens, in he hree coordinaes planes, he monoonies shown on figre 7 : he desired monoony in he (, )-plane (picred in he middle of figre 8) is obained by projecing he ecor φ 1 on his plane and we hae = ( φ 1 (0, ), φ 1 (0, )). Then, we force he decreasing of (, ) φ 2 (0,, ) in he direcion. Applying Lemma 1, we reqire : ( ) ( ) (, ) [0, 1] 2 φ, 2 (0,, ) φ 1 (0,, ) < 0. φ 2 (0,, ) φ 1 (0,, ) 8

10 Inersecion wih 3 faces of he box Inersecion wih 4 faces of he box Inersecion wih 5 faces of he box Inersecion wih 6 faces of he box Figre 6 Configraions of he srface S 1 in [0, 1] 3 nder he monoony consrain on φ 1. This preios do prodc is a polynomial of bi-degree (3, 3) wih respec o he ariables (, ). Considered also as a polynomial in s (ha we fixed a he beginning of his secion) i is of degree 4 in s Choice of monoony consrains Here, we presen or choice of sfficien condiion sch ha #(C 1,2 S 1 ) 2. Firs, we consider he case where S 1 inersecs he 6 faces of he cbe [0, 1] 3. In he oher cases, we js skip he condiion corresponding o missing segmens conraced o a poin (see figre 9). The border of S 1 is isoopic o an hexagon {M 1,..., M 6 } as shown on figre 9

11 = 0 = 0 C 1,2 φ 1 Variaions of φ 2 = 0 Figre 7 Example of monoony of φ 2 along he border of S 1. w w Figre 8 Traces of S 1 on he faces of he box wih orienaions. 11. A sfficien monoony condiion is gien by a choice of an iniial poin M I and a final poin M F among {M 1,..., M 6 } wih he possible choice M I = M F sch ha φ 2 is monoonic on he pahs on S 1 joining M I o M F. This clearly implies ha φ 2 anishes a mos wice on S 1. Now, we can exend or choice of sfficien condiions simply by remarking ha he 4 ariables {s,,, } play similar roles. 1. Insead of fixing s, we can fix, or and consider he corresponding maps. 2. Also, he roles of φ 1, φ 2,and φ 3 can be exchanged. 10

12 φ 1 Variaions of φ Figre 9 Exemple of monoony of φ 2 along he border of S 1 in he case where S 1 inersec 6 faces of he box and he resling configraions in he oher cases. All hese opions will be considered o speed p he implemenaion. 4 Algorihms and daa srcre sed for implemenaion In his secion, we presen some implemenaion aspecs of he inersecion algorihm described in secion 3. They are implemened in Axel 1 which is an algebraic geomeric modeler. 1. hp ://axel.inria.fr 11

13 Figre 10 Traces of S 1 on he faces of he box (i corresponds o he case represened in figre 9). M 2 M 1 M 2 M 1 M 3 M 6 M 3 M 6 M 5 M 4 M 5 M 4 Figre 11 Traces of S 1 on he faces of he box (i corresponds o he case represened in figre 9). 4.1 Hexaree daa srcre and opology A sbdiision algorihm on a box in [a 1, b 1 ] [a 2, b 2 ] [a 3, b 3 ] [a 4, b 4 ] R 4 explores sb-boxes consrced by considering inermediae ales c i beween 12

14 a i and b i for i {1,..., 4} ; here we choose c i = a i + b i. So a box has 16 sbboxes. Ieraing his consrcion, an hexaree is bild ; i.e. each node of he 2 ree has 16 children nmbered from 0 o 15. In binary expression, his nmber is wrien α 1 α 2 α 3 α 4 wih α i = 0 or 1 ; for i {1,..., 4}, if α i = 0, he sb-boxe is consrced oer [a i, c i ] and if α i = 1 i is consrced oer [c i, b i ]. For example, he child wele is wrien 1100 and corresponds o he sb-box [c 1, b 1 ] [c 2, b 2 ] [a 3, c 3 ] [a 4, c 4 ]. This is called an hexaree daa srcre, i generalizes he qadrees which are widely sed o represen planar shapes. To each node of he ree is associaed a label which sores he needed informaion. Here, he informaion will be he descripion of he opology of he inersecion cre C ino he corresponding sb-boxe. More precisely, we reqire ha, a he leaes of he ree, his inersecion is empy or is dimensions are below some hreshold or i is isoopic o a collecion of disjoin segmens ; each segmen connecs wo inersecion poins of he cre C wih he border of he considered sb-box. Each sch segmen is represened by he coordinaes of is exremal poins. Noe ha in R 4, all he 16 children sb-boxes of a gien box are adjacen. Or injeciiy crierion described in 3 is implemened in a es fncion (called reglar) if i rerns false on a sb-box hen he sb-box is sbdiided. 4.2 Sbdiision algorihm The algorihm 4.1 describes he sbdiision mehod for he opology compaion. Some oher fncions are needed and are described in he seqel. 13

15 Algorihm 4.1: Sbdiision algorihm for opology in 4D. opology(c, B, ɛ) Inp: The cre C, a box B = [a 1, b 1 ] [a 2, b 2 ] [a 3, b 3 ] [a 4, b 4 ] and a olerance ɛ. Op: A lise of segmens in R 4 represening he opology. Creae he hexaree H; Iniialize he roo of H by B and he inersecion poins C B; Creae a lis of nodes L; L rooof(h); while L do Take he firs iem n of L (and remoe i from L); if reglar(c, n) hen n reglartopology(c, n); else if he crren box has a size ɛ hen L sbdiision(c, n); else Gie an arbirary opology by connecing all he border poins o he cener of he box (his applies when we sop he sbdiision); end end rern fsion(h); Now we describe he oher fncions called by opology. Fncion reglar : This fncion is he injeciiy crierion described in secion 3. In fac here are 4 differen ess and each of hem corresponds o he fixed ariable choice s,, or (see algorihm 4.2). If one is erified, hen we call he corresponding fncion reglartopology. Fncion reglartopology : If one of he for reglariy ess reglar is erified, hen he opology of C is known. In he fncion reglartopology, we js hae o connec he border poins in he crren node. In fac, we also hae 4 differen reglartopology fncions corresponding o he fixed ariable s,, or. For example, if s = s 0 is fixed, hen we hae, in he crren node, a lis of een nmber of border poins p 1, p 2,..., p 2k 1, p 2k (by repeaing a poin if is mlipliciy is een) sored by heir s componen. Then, he opology is described by he lis of segmens : [p 1, p 2 ],..., [p 2k 1, p 2k ]. Fncion sbdiision : This fncion sbdiides he crren box creaing 16 children as described in secion 4.1. I allocaes he inheried inersecion poins and compe he new 14

16 Algorihm 4.2: Injeciiy crierion. reglar(c, n) if φ is locally injecie in n hen if φ 1 has he waned monoony on he edges of n hen if φ 2 or φ 3 has he waned monoony on S 1 hen rern re; else rern false; end else if φ 2 has he waned monoony on he edges of n hen if φ 1 or φ 3 has he waned monoony on S 2 hen rern re; else rern false; end else if φ 3 has he waned monoony on he edges of n hen if φ 1 or φ 2 has he waned monoony on S 3 hen rern re; else rern false; end else rern false; end else rern false; end inersecion poins ha appear wih he faces of hese sb-boxes. Fncion fsion : This fncion is called when he consrcion of he hexaree H is finished. More precisely each leaf of H conains he opology in he corresponding sbbox. fsion proides he opology of C in he iniial box B. Is implemenaion (see algorihm 4.3) consiss in merging recrsiely he opology beween he children of each node. For a gien node n and an ineger i {0,..., 15}, we denoe by child(i, n) he i-h child as described in secion 4.1. Besides, if l 1 and l 2 are wo lis of segmens in R 4, merge(l 1, l 2 ) will be he lis of segmens in R 4 formed by all he segmens of l 1 and l 2. 15

17 Algorihm 4.3: Topology by sbdiision. fsion(n) Inp: A node of hexaree as i is described in secion 4.1 Op: A lis of segmens in R 4 if n is a leaf hen rern he lis of segmens in n; else rern merge( merge( merge( merge(fsion(child(0, n)), fsion(child(1, n))) merge(fsion(child(2, n)), fsion(child(3, n)))) merge( merge(fsion(child(4, n)), fsion(child(5, n))) merge(fsion(child(6, n)), fsion(child(7, n))))) merge( merge( merge(fsion(child(8, n)), fsion(child(9, n))) merge(fsion(child(10, n)), fsion(child(11, n)))) merge( merge(fsion(child(12, n)), fsion(child(13, n))) merge(fsion(child(14, n)), fsion(child(15, n)))))); end 4.3 Conneced componens and loops The algorihm 4.1 allows o idenify he conneced componens easily. Indeed, he resled opology of C is a lis of segmens (in R 4 ) of he form {[p 1, p 2 ],..., [p 2k 1, p 2k ]} where k is a posii ineger. If here exis i {1,..., k 1}, sch ha p 2i p 2i+1, hen he wo segmens [p 2i 1, p 2i ] and [p 2i+1, p 2i+2 ] are on wo differen conneced componens of he opology. A similar simple argmen allows o deec he loops (conneced) componens. 4.4 Examples We illsrae he algorihm on some examples. Firs we gies wo inersecion siaions of wo polynomial paches shown on figres 12 and 13. Anoher classical example (he eapo) is gien. Figre 14 shows an approximaion of he eapo by 32 biqadraic paches wih inersecion loci. The resling opology of hese loci is shown on figre

18 Figre 12 Example of inersecion beween wo polynomial paches. Figre 13 Example of inersecion beween wo polynomial paches. 5 Topology in R 3 Secions 3 and 4 presened an algorihm for comping he opology of a cre C in R 4 defined by 3 eqaions F (s, ) = G(, ) (wih (s,,, ) [0, 1] 4 17

19 Inersecion loci Figre 14 Teapo inersecion loci. Figre 15 Topology of he eapo inersecion. 18

20 for example). We inrodce he following noaions : ( ) [0, 1] 4 R π 1 : 2 (s,,, ) (s, ) ( ) [0, 1] 4 R π 2 : 2. (s,,, ) (, ) The inersecion Γ in R 3 of he wo parameerized srface paches F and G is he image of C by F π 1 (or G π 2 ) of C. Or algorihm garanee (p o he olerance ɛ) he opology of C, which is isoopic o a collecion of segmens in [0, 1] 4. This implies ha he image by F π 1 of a conneced componen C 1 of C is conneced. Howeer, if C 1 is a loop in ]0, 1[ 4 (a closed pah) hen is image is also a loop in R 3 b we do no know is kno srcre. Moreoer, if C admis seeral conneced componens which are loops in [0, 1] 4, heir images by F π 1 in R 3 may be inerlaced (like he olympic rings). If C 1 is deermined by a segmen discreizaion which is oo coarse, he kno srcre (and he inerlacemens) can be missed in he image by F π 1 of his piecewise approximaion. We may hae he siaion depiced in figre 16. Similarly, he opology of he projecion Ĉ of C [0, 1]4 on [0, 1] 2 by π 1, may no be deermined by a coarse discreizaion of C, een if his discreizaion is sfficien o deermine he opology of C in [0, 1] 4, see figre 17 : he selfinersecion poin is missed. In order o capre hese feares, he algorihm described in secion 3 and 4 shold be exended and he sbdiision crieria refined. As described in secions 3 and 4, we chose a hreshold ɛ sch ha he singlar poins of he cre Γ will be conained in boxes of size smaller han ɛ. We aim o deermine he opology of he cre Γ p o his indeerminaion, i.e. wo F π 1 C 1 [0, 1] 4 inersecion cre in R 3 Figre 16 Image of a loop wih kno srcre. 19

21 π 1 C [0, 1] 4 Ĉ [0, 1] 2 Figre 17 Missing a self-inersecion poin by projecion. segmens enering a box of size smaller han ɛ are spposed o inersec and form a singlar poin. All oher poins are considered smooh. Sppose ha C has k loops conneced componens (where k is a posii ineger) denoed respeciely by C 1,..., C k (we can deec hem by sing he crierion described in secion 4.3). We denoe Γ i = (F π 1 )(C i ) for all i {1,..., k}. 5.1 One cre box Recall ha each node of he hexaree H (described in secion 4.1) sores a box in R 4 and he opology of C in his box. Le n be a node of H, B n be he corresponding box and B n = B F B G, where B F (respeciely B G ) is he bonding box consrced wih he conrol poins of F (s, ) (respeciely G(, )) wrien in he Bernsein basis wih respec o π 1 (B n ) (respeciely π 2 (B n )). Then, by he conex hll propery of he Bézier paches, he bonding box B n conains he par of Γ corresponding o B n i.e. he image of C B n by F π 1 (or G π 2 ). The discreizaion of C is refined, by sbdiiding all he leafs of H, sch ha each box (in R 4 ) inersecing one of he loops C 1,..., C k conains a mos one segmen, i.e. is border inersecs C in wo poins. Noe ha in he preios 20

22 secion or algorihm allowed more inersecion poins. Afer his sep some ambigiies of he node and inerlacemen srcre of Γ may remain. One can see in figre 18 wo bonding boxes (in R 3 ) sharing inerior poins. Joining he pairs of poins on he borders, he red cre segmen may (or may no) pass behind he oher green cre segmen. So we need o refine frher he discreizaion. Lemma 2 Le γ 1, γ 2 Γ be wo disjoin segmens of cres. Afer a finie nmber of sbdiisions of (F π 1 ) 1 (γ 1 ) and (F π 1 ) 1 (γ 2 ), he boxes conaining γ 1 are disjoin from he boxes conaining γ 2. Proof. Indeed by sbdiision, he boxes can be made nearer o he cres han he disance beween he wo cres. The sbdiision on he leafs of H is refined by sing lemma 2. Then, we rle o poenial ambigiy on inerlacemens beween wo loops (siaion corresponding he o righ picre on figre 16) becase we aoid he siaion depiced on figre 18. So i remains o analyze he ambigiy on a possible node ha is no a loop. Figre 18 Two boxes sharing inerior poins. 21

23 5.2 Node and discreizaion Lemma 3 Le γ Γ be a segmen of cre conained in a bonding box obained afer he sbdiision process described in secion 5.1. Then, he border of his box has js wo poins p 1 and p 2 of γ. Afer a finie nmber of sbdiisions of (F π 1 ) 1 (γ), we hae de(n F, N G, p 1 p 2 ) 0 (in he corresponding box) wih N F = s F F and N G = G G. Proof. The condiion de(n F, N G, p 1 p 2 ) 0 means ha he angen ecor of γ is neer orhogonal o p 1 p 2. As γ is smooh par hypohesis, he lemma is a conseqence of he implici fncion heorem. If we sbdiide he leafs of H by sing lemma 3, hen we rle o poenial inerlacemens ambigiies inside each bonding box. Howeer, i remains o aoid inerlacing from wo adjacen branches. Proposiion 2 Assme he discreizaion saisfies lemma 2 and lemma 3. Sppose also ha de(n F, N G, p 1 p 2 ) 0, de(n F, N G, p 2 p 3 ) 0 and de(n F, N G, p 1 p 3 ) 0 for wo adjacen branches [p 1, p 2 ] and [p 2, p 3 ]. If he image (by F π 1 or G π 2 ) of a loop conneced componen of C admis a node, hen i shows p on he discreizaion i.e. he seqence of segmens obained by sbdiision also describes a node isoopic o ha of Γ. Proof. Indeed, we will hae he siaion depiced on figre 19. By he condiions de(n F, N G, p 1 p 2 ) 0, de(n F, N G, p 2 p 3 ) 0 and de(n F, N G, p 1 p 3 ) 0, we canno hae a node wih formed by wo adjacen segmens (depiced on figre 20). So, we js hae o inesigae he case where we hae a leas hree segmens [p 1, p 2 ], [p 2, p 3 ] and [p 3, p 4 ]. Lemma 3 ensre ha each of hese segmens does no inerlace. If he hree segmens are inerlacing, hen he bonding boxes conaining respeciely [p 1, p 2 ] and [p 3, p 4 ] inersec each oher so i conradics lemma 2. 22

24 p 3 p 1 p 4 p 2 Figre 19 Inerlacemen siaion. y y p 1 p 1 z Projecion p x x 2 p 3 p 2 p 3 Figre 20 Inerlacemen wih wo adjacen segmens. 23

25 Références [1] L. González-Vega, I. Necla, Efficien opology deerminaion of implicily defined algebraic plane cres, Comp. Aided Geom. Design 19 (9) (2002) [2] T. A. Grandine, F. W. Klein, A new approach o he srface inersecion problem, Comper Aided Geomeric Design 14 (1997) [3] J. G. Alcázar, J. R. Sendra, Comping he opology of real algebraic space cres, J. Symbolic Comp. 39 (2005) [4] G. Gaellier, A. Labrozy, B. Morrain, J.-P. Técor, Comping he opology of 3-dimensional algebraic cres, in : Compaional Mehods for Algebraic Spline Srfaces, Springer-Verlag, 2004, pp [5] C. Liang, B. Morrain, J.-P. Paone, Sbdiision mehods for he opology of 2d and 3d implici cres, in : B. J.. R. Piene (Ed.), Compaional Mehods for Algebraic Spline Srfaces, 2005, Compaional Mehods for Algebraic Spline Srfaces, Springer, Oslo, Norway, 2007, pp URL hp://hal.inria.fr/inria /en/ [6] S. Planinga, G. Veger, Isoopic approximaion of implici cres and srfaces, in : SGP 04 : Proceedings of he 2004 Erographics/ACM SIGGRAPH symposim on Geomery processing, ACM Press, New York, NY, USA, 2004, pp [7] G. Farin, Cres and Srfaces for Comper Aided Geomeric Design : A Pracical Gide, 3rd Ed., Academic Press, [8] T. W. Sederberg, Algorihm for algebraic cre inersecion, Comp. Aided Des. 21 (9) (1989) [9] S. Cha, M. Oberneder, A. Galligo, B. Jler, Inersecing biqadraic bézier srface paches, in : B. J.. R. Piene (Ed.), Compaional Mehods for Algebraic Spline Srfaces, 2005, Compaional Mehods for Algebraic Spline Srfaces, Springer, Oslo, Norway, 2007, pp URL hp://hal.inria.fr/inria /en/ 24