Continuous Flattening of Convex Polyhedra

Transcription

1 Continuous Flttnin o Conv Polr Jin-ii Ito 1, Ci Nr 2, n Costin Vîlu 3 1 Fult o Eution, Kummoto Univrsit, Kummoto, , Jpn. j-ito@kummoto-u..jp 2 Lirl Arts Eution Cntr, Aso Cmpus, Toki Univrsit, Aso, Kummoto, , Jpn. nr@ktmil.toki-u.jp 3 Institut o Mtmtis Simion Stoilow o t Romnin Am, P.O. Bo 1-764, Burst, Romni. Costin.Vilu@imr.ro Astrt. A lt olin o polron is olin rss into multilr plnr sp. It is n opn prolm o E. Dmin t l., tt vr lt ol stt o polron n r ontinuous olin pross. Hr w prov tt vr onv polron posssss ininitl mn ontinuous lt olin prosss. Morovr, w iv suiint onition unr wi vr lt ol stt o onv polron n r ontinuous olin pross. 1 Introution W us t trminolo polron or los polrl sur wi is prmitt to tou itsl ut not sl-intrst (n so oul ovr polon is polron). A lt olin o polron is olin rss into multilr plnr sp. T rsults prsnt r r rlt to t ollowin prolm propos Erik Dmin t l. (s Opn Prolm 18.1 in [5]): Cn vr lt ol stt o polron r ontinuous olin pross? Noti tt, i polron P is lttn ontinuous olin pross (s Dinition 1) wit polr {P t : 0 t 1}, tn t rs pttrn in P or {P t : 0 t 1} is n ininit st o lin smnts. Tis ollows rom Cu s riiit torm n Sitov s rsult on t volum invrin unr lin [9]-[10]. T istn o lt ol stts or polr omomorpi to t 2- spr ws prov t mto o isk pkin (s 18.3 in [5]), n or som spil lsss o onv polr ws lso prov t mto o strit skltons (s 18.4 in [5], n [4]). Stion 2 o tis work is vot to prliminris. W lso ril prsnt tr (Torm 1) t mto to ontinuousl lttn t Pltoni polr onto tir oriinl s, propos in [6]. Support Grn-in-Ai or Sintii Rsr (No ), JSPS. Support Grn-in-Ai or Sintii Rsr (No ), JSPS. Prtill support t rnt PN-II-ID-PCE o CNCS-UEFISCDI.

2 In St. 3 w propos mto to lttn nrl onv polr ontinuous olin prosss (Torm 2). W mplo Alnrov s luin torm n t strutur o ut loi (s Dinition 2) or t proos. In St. 4 w iv suiint onition, unr wi vr lt ol stt o onv polron n r ontinuous olin pross (Torm 3). W n t ppr wit w rmrks n opn qustions (St. 5). 2 Prliminris W strt wit t inition o ontinuous olin pross or polron. Dinition 1. Lt P polron in t Eulin sp IR 3. W s tt mil o polr {P t : 0 t 1} is ontinuous olin pross rom P = P 0 to P 1 i it stisis t ollowin onitions: (1) or 0 t 1, tr ists polron P t otin rom P suiviin som s o P (i.., som s o P t m inlu in t sm o P, ut P t is onrunt to P ) su tt P t is omintorill quivlnt to P t n t orrsponin s o P t n P t r onrunt, (2) t mppin [0, 1] τ P τ {P t : 0 t 1} is ontinuous. Morovr, i P 1 is lt ol polron, w s tt P is lttn ontinuous olin pross n w ll P 1 lt ol polron (or stt) o P. In t s o Pltoni polr, two o us prov t nt rsult [6], wi will srv s ontrst in Stion 4. Torm 1. For t iv Pltoni polr tr r ontinuous lt olin prosss onto tir oriinl s. Fiur 1 sows ow to ontinuousl lttn t u n t rulr otron on tir s (s [6] or tils or or t otr Pltoni polr). Torm 1 ws prov usin k lmm: n romus n ol into sp s sow in Fi.2 (2), wit istns ()() = l n ()( = m or n ivn 0 l n 0 m, wr w not t Eulin mtri istn twn, IR 3. Our min tools r r t Alnrov s luin torm (stt low) n t ut loi, to wi t rminin o tis stion is vot. Alnrov s luin torm. Consir topoloil spr S otin luin plnr polons (i.., nturll intiin pirs o sis o t sm lnt) su tt t most 2π nl is lu t point. Tn S, now wit t intrinsi mtri inu t istn in IR 2, is isomtri to polrl onv sur P IR 2, possil nrt. Morovr, P is uniqu up to rii motion n rltion in IR 3. (S [2], p.100.) 2

3 (1) (2) (3) (4) Fi. 1. (1) T u; (2) t ltt u on its ; (3) t rulr otron; (4) t ltt otron on its. q () (q)=(q ) () o q (o) (1) () (2) () Fi. 2. An mpl o ol romus. Dinition 2. Lt P onv polron. T ut lous C() = C(, P ) o t point on P is in s t st o npoints (irnt rom ) o ll nontnl sortst pts (on t sur P ) strtin t. Fiur 3 provis two mpls o ut loi or points on t u. W summriz in our irst lmm vrious known proprtis o ut loi. Lmm 1. (i) C() is tr wos lvs (npoints) r vrtis o P, n ll vrtis o P, ptin (i t s), r inlu in C(). Noti tt w llow vrtis o r two in C(). (ii) T juntion points in C() r join to s mn sortst pts s tir r in t tr. (iii) T s o C() r sortst pts on P. (iv) Assum t sortst pts γ n γ rom to C() r ounin omin D o P, wi intrsts no otr sortst pt rom to. Tn t r o C() t towrs D ists t nl o D t. (v) I P s n vrtis tn C() is tr wit O(n) vrtis, n it n onstrut in tim O(n 2 ). Proo. (i)-(ii) n (iv) Ts r wll known. 3

4 = = 1 2 (1) (2) Fi. 3. (1) T ut lous o t vrt = on u; (2) t ut lous on u wit rspt to t mipoint o t. (iii) Tis is Lmm 2.4 in [1]. (v) T irst prt is lr. For t son prt, w us t loritm o J. Cn n Y. Hn [3] (s [7] or puli implmnttion). W will us t u to illustrt our mto. Tis mto pns upon sortst pts rom point on t u to prtiulr points on C(), in Fi. 4(1), n 1 n 2 in Fi. 4(2). = = 1 2 (1) (2) Fi. 4. (1) Sortst pts joinin = to = ; (2) sortst pts joinin to 1, 2, n to t u vrtis tt r intrior points o C(). W ll n o C() n l i it is inint to l o C(). 3 Continuous Flttnin Prosss or Conv Polr In tis stion w provi mto to ontinuousl lttn n onv polron P, s on ut loi n Alnrov s luin torm. Towr tis ol, w urtr sri t strutur o ut loi. 4

5 = = =p = = =p = ==q ==q ==q (1) (2) (3) = = ==q =p ==q = =p ==q = ==q (4) (5) (6) Fi. 5. (1) T ut lous o = ; (2) two sortst pts γ 1 n γ 2 joinin to, nlosin prisl on l E = uv = ; (3) two prts o t u, sprt γ 1 γ 2; (4) two rsultin surs otin luin (t ims o) γ 1 n γ 2; (5) t rsultin polron rom t u tr luin γ 1 to γ 2 ; (6) t lt ol stt o t u inll otin (1) (2) (3) (4) Fi. 6. (1) T ut lous o t mipoint o ; (2) two sortst pts γ 1 n γ 2 joinin to 1, n nlosin prisl on l E = uv = 1 ; (3) t rsultin polron tr luin γ 1 to γ 2 ; (4) t lt ol stt o t u inll otin. 5

6 W irst iv i-lvl viw o t mto, prsntin two irnt ws to lttn t u, illustrt in Fis. 5 n 6. W strt wit n ritrr point on P, n trmin its ut lous n ll smnts rom to t juntion points o C(), s Fis 5 (1) n 6 (1). Evr l E o C() is inlu in som rion T o P oun two onsutiv smnts rom to juntion point o C() (E = in Fiur 5 (2/3) n E = 1 in Fiur 6 (2)). T n lttn to oul ovr trinl T E, n P \ T n zipp to som onv polron Q E ( Alnrov s luin torm). Tror, P is isomtri to P E = T E Q E, onsistin o nt oul ovr trinl T E (s in t iurs) tt to som onv polron Q E (Lmm 2), s Fis. 5 (4/5) n 6 (3). T ut lous C(, Q E ) o on Q E is prisl t truntion o t ut lous C(, P ) (Lmm 3), n w n itrt t pross until C() is mrl pt, in wi s t rsultin polron is lr lttn (Lmm 4), s Fis. 5 (6) n 6 (4). Lmm 5 sows t ontinuit o tis prour. Lt γ 1 n γ 2 sortst pts on P rom P to C(); ut lon γ 1 γ 2 n kp on l-sur P. B luin γ 1 to γ 2 w mn to inti t points on γ 1 n rsptivl γ 2 t qul istn to. Lmm 2. Lt point on onv polron P. E l E = uv o t ut lous C(), strtin t t l v o C(), is oun two sortst pts γ 1 n γ 2 rom to u, wos union nloss prisl on l, v, o C(). T rion T o P, nlos γ 1 γ 2 n ontinin v, n lttn to oul ovr trinl T E, n t rminin prt o P orrspons to onv polron Q E luin γ 1 to γ 2. T oriinl polron P is isomtri to t polron P E = Q E T E, wr w tt T E to Q E, su tt γ 1 n γ 2 r touin otr ut r inlu in istint lrs. Proo. Lt E = uv n l wit l v o t ut lous C(). Tn tr r sortst pts joinin to u on P, wr is t r o t point u in t tr C(), n tl two o tm, s γ 1 n γ 2, nlos prisl on l v. Fis. 5(2) n 6(2) sow t rions o t u orrsponin to t l s n 1 in t rsptiv ut loi. T rion T o P, oun γ 1 γ 2 n ontinin v, s no otr vrt o P insi, n it onsists o two lt onrunt trinls wit s v, uv n γ i (i = 1, 2). Tror, T n lttn to som oul ovr trinl T E, luin γ 1 to γ 2. T rminin prt o P orrspons to onv polron Q E Alnrov s luin torm. Hn P is isomtri to P E = Q E T E, wr w tt T E to Q E su tt γ 1 n γ 2 r touin otr ut r inlu in istint lrs in P E. Noti tt, ltou lttn, T E m not li in pln. For n l E o C(, P ) w will us t nottion Q E n P E or t polr introu in Lmm 2. Lmm 3. Lt point in onv polron P, n lt E = uv n l o t ut lous C(), inint to t t l v o C(). T ut lous 6

7 C(, Q E ) o on Q E is (isomtri to) t truntion o t ut lous C(, P ) wit rspt to t uts lon γ 1 n γ 2 in in Lmm 2, n t luin lon tm. Proo. Tis ollows rom t inition o ut lous n t proprt (iv) in Lmm 1. Lmm 4. I t ut lous C() o point in onv polron P is pt, tn P is oul ovr polon. Proo. Sin ll vrtis o P pt possil r inlu in C(), P is oul ovr polon, Lmms 2 n 3. Lmm 5. Lt point in onv polron P, wos ut lous C() is not pt, n lt E = uv n l o C(), inint to t t l v o C(). Tr is ontinuous olin pross rom P to P E. Proo. Lt p t point movin ontinuousl rom v to u lon t E = uv, s t inrss rom 0 to 1, n not E t = p t v. Tr r two sortst pts γ t,1 n γ t,2 joinin to p t, nlosin prisl on l v o C(). B uttin lon γ t,1 γ t,2 n luin γ t1 to γ t2, w otin oul ovr trinl T t = T Et n onv polron Q t = Q Et, Alnrov s luin torm. Lt P t = P Et = Q t T t in similrl to P E in Lmm 2. Hr T t is lipp lokwis out t point n it tous Q t, in orr to voi n onlit ltr. W stlis low proprt o Q t. T strutur o Q t is ivn to us vi Alnrov s luin torm. W know t vrtis o Q t (t r, p t, n vrtis o P ), ut not its s. Nvrtlss, (i) t s o Q t twn vrtis (orrsponin to vrtis) o P r sust o t olltion o ll sortst pts twn pirs o su vrtis; (ii) t s o Q t rom its vrt to vrtis (orrsponin to vrtis) o P r sust o t olltion o ll sortst pts rom to su vrtis; (iii) t s o Q t rom its vrt p t to vrtis (orrsponin to vrtis) o P r sust o t olltion o ll sortst pts rom p t to su vrtis; (iv) t o Q t twn n p t orrspons to γ t,1 γ t,2. Dnot G t sust o P onsistin o ll sortst pts joinin pirs o vrtis o P, or joinin to vrtis o P. Consir now nioroo N o u, n points 0, z 0 N su tt t trinl 0 = u 0 z 0 intrsts G C() onl t u. Tis is possil, us ot G n C() r ompos o initl mn sortst pts on P, n N (G C()) onsists o initl mn sortst pts on N. Lt now G t P onsist o ll sortst pts rom p t to t vrtis o P, n in G t = G G t γ t,1 γ t,2. It ollows tt G t N is t union o initl mn lin-smnts, n tr ist points t, z t N \ G t su tt t trinl t = u t z t intrsts G t C() onl t u (s Fi. 7). 7

8 t t v p t u C() z t t Fi. 7. A trinl u tz t su tt u tz t G t = {u}. W lim tt tr r t most initl mn vlus τ 1, τ 2,..., τ m ]0, 1[ su tt, or n numr t in ]τ i, τ i+1 [, n n vrt v o P, t numr o sortst pts rom p t to v os not pn on t (i = 1,..., m 1). To prov t lim, rll tt o C(v) is sortst pt on P, n so is uv (s Lmm 1). Tror, or n E o C(v), E uv is itr, or point, or n r A. Morovr, ll points in uv \ E r join to v prisl on sortst pt, n i E uv = A tn ll intrior points to A r join to v prisl two sortst pts. B t ov rumnt, t numr o sortst pts rom v to n ptionl point p τi is t lst qul to t numr o sortst pts rom v to p t, or t suiintl los to τ i. T ov lim n t uppr smi-ontinuit o sortst pts sow tt, or n t [τ i, τ i+1 [, t n z t n oosn su tt t pns ontinuousl on t. Morovr, i t rs n ptionl vlu τ i, 0 < t < τ i, tn w m oos t n z t su tt τi t. In onlusion, t ists n pns lowr smi-ontinuousl on t [0, 1]. W sow nt ow to rliz P t in IR 3, or 0 t 1. O ours, it suis to sow ow to rliz Q t. For t = 0, w rliz Q 0 (n n P 0 = P ) in IR 3, stisin t onitions: (i) u = (0, 0, 0), 0 = ( 0,1, 0, 0), n z 0 = (z 0,1, z 0,2, 0), or som rl numrs 0,1 0, z 0,1, z 0,2 ; n (ii) Q 0 is inlu in t l-sp z 0. For 0 < t < 1, w rliz Q t in t l-sp z 0 su tt t is rliz s t orrsponin sust o N. Assum now tt t = 1. Lt {t n } n 1 squn onvrin to 1, wit 0 < t n < 1. Sin t mil {Q tn } n 1 is oun wit rspt to t Husor mtri on t sp o ll ompt sts in IR 3, tr ists susqun wi onvrs to ompt st R 1, Blsk s onvrn torm. T uniit in Alnrov s luin torm sows now tt R 1 os not pn on t oi o t onvrin squn t n 1, n w m rliz Q 1 R 1. 8

9 Conluin, i Q t is los to Q s tn tir orrsponin s r los to otr, in prtiulr tos inluin t n s, n n tir rliztions in IR 3 r los to otr. Finll, w noti tt t mppin rom 0 t 1 to t 1-prmtr mil o ompt sts {Q t : 0 t 1} is ontinuous wit rspt to t Husor mtri. Lt s rl numr wit 0 s 1, n {s n } n 1 squn onvrin to s, wit 0 < s n < 1. T mil {Q sn } n 1 is oun wit rspt to t Husor mtri on t sp o ll ompt sts in IR 3, n tr ists susqun wi onvrs to ompt st R s, Blsk s onvrn torm, n t uniit in Alnrov s luin torm ns t proo. Torm 2. For vr onv polron tr ist ininitl mn ontinuous lt olin prosss. Proo. Lt P onv polron n lt point in P. Stp 1. Dtrmin t ut lous C(), wi is tr (s Lmm 1). Stp 2. Flttn t rion T o P orrsponin to n l E o C() (s Lmm 2). T rminin prt o P, tr lttnin T s ov, is rliz s onv polron Q E, Alnrov s luin torm. Tror, t rsult P E tr tis lttnin is isomtri to P, n onsists o t polron Q E tt to t oul ovr trinl T E. T E soul li lokwis out t point in orr to voi onlit. Stp 3. Itrt Stp 2 or Q E inst o P, until C(, Q E ) is ru to pt; i.., until Q E is oul ovr polon. Lmm 3 urnts t itrtions r possil, wil Lmm 4 stliss t inl orm o Q. Fis. 5(6) n 6(4) sow t lt ol stts o t u tr lttnin ll su rions orrsponin to l s o C(, P ). Sin tr r O(n) vrtis in C(), wr n is t numr o vrtis o P (s Lmm 1), w v to lttn O(n) rions o P orrsponin to l s o C(, Q) on on, n tror t lttnin pross ns tr O(n) itrtions. All olin prosss orrsponin to l s r ontinuous Lmm 5, so P is ontinuousl ol to lt ol stt. 4 Continuous Flttnin Prosss or Simpl Flt Fol Stts In tis stion w iv suiint onition or lt ol stt o onv polron, to r ontinuous olin pross. Dinition 3. A 2-ovr onv polon onsists o two opis o onv polon lu lon som o tir orrsponin s (t otr s r ut ). Fiur 8(3) provis mpls o 2-ovr onv polons. W will lws rr su surs wit ounr s vin two onrunt lrs touin t tir orrsponin points, ut lu lon onl som s. 9

10 u =z (1) v (2) w = u =z v (3) w (4) = Fi. 8. (1) A rtnulr o B; (2) simpl lt ol stt o B otin pusin in two si s o B; (3) omposition o t lt ol stt into our onrunt 2- ovr trpzois n on 2-ovr rtnl; (4) t rion nlos two sortst pts rom to, orrsponin to t oul ovr trinl. Dinition 4. A lt ol stt P o onv polron is ll simpl i it njos t ollowin proprtis: (i) P n ompos into init numr o 2-ovr onv polons {R i : 1 i k}, (ii) or n 1 i k, i R 1, R 2,, R i 1 wr ut o rom P, tn R i n lso ut o rom t rminin prt o P prisl on ut lon n o R i. Noti tt ll lt ol stts otin in Torm 2 r simpl, wil Torm 1 provis mpls o non-simpl lt ol stts; or mpl, t lt ol stts sown in Fis. 1(2) n 1(4) r not simpl us t o not stis t onition (ii). T ol stt P sow in Fi. 1(2) or Fi. 1(4) n ompos into twlv 2-ovr trinls R i, ut no R i n ut o rom P prisl on ut lon n o R i. Fiur 8(2) sows lt ol stt o rtnulr o B, wr two si s o B r pus in. It is simpl, us it n ompos into iv 2-ovr onv polons s sown in Fi. 8(3), nml our onrunt ut oul trpzois n on 2-ovr rtnl. Howvr, it nnot otin t ut lous mto sri in t prvious stion. Torm 3. Evr simpl lt ol stt o onv polron n r ontinuous olin pross. 10

11 Proo. Lt P simpl lt ol stt o onv polron P, ompos into init olltion o 2-ovr onv polons {R i : 1 i k} uttin P lon som o R i, on on. B suiviin R i (1 i k) i nssr, w n ssum, witout loss o nrlit, tt ll R i r 2-ovr trinls. W prov t rsult inution ovr k. I k = 1 tn P = P n t onlusion ols. Suppos now tt t sttmnt is tru n, n ssum k = n + 1 or P. Lt R 1 2-ovr trinl z wit t ut n s z n z lu. Consir t ut lous C() = C(, P ). Tn E = z is l o C(). B Lmm 5, tr ists ontinuous olin pross rom P to t polron P E = Q E R 1. Noti tt Q E s t lt ol stt Q E, onsistin o 2-ovr onv trinls {R i : 2 i n + 1}, n Q E, stisis t onition (ii) in Dinition 4, wit t o R 2 lu. B t inution s ssumption, Q E, n r ontinuous olin pross rom Q E. Tror, P n r ontinuous olin pross rom P. 5 Rmrks n Opn Qustions Our irst rsult in tis ppr proposs n loritmi mto to ontinuousl lttn onv polr. Hr, t ssumption o onvit is ssntil t two points. First, or t istn o t lt ol stt w us t t tt o t ut lous is sortst pt (Lmm 1 (iii)), proprt wi is not tru on non-onv surs. Son, or t ontinuit o t pross, w us t uniqunss in Alnrov s luin torm, wi ils or non-onv surs. Sin ssntill t sm rumnt is mplo to prov our son rsult, suiint onition or t istn o ontinuous lttnin prosss, our proo tr lso ils or non-onv surs. Our ppro riss svrl qustions onrnin t strutur o ut loi. Our lttnin prour strts wit t ut lous C() o t point in P, n t stp it trts (mor prisl, it limints) ll l s o C(). Qustion 1. For wi polr n on in points wos ut lous s prisl on rmiition point (i.., C() is omomorpi to str rp)? Gnrll, wt is t miniml numr o stps (i.., o rmiition points in C()) to n t prour on onv polron P wit n vrtis, i t point vris on P? Consir now t ontinuous lttnin pross. It is s on point ontinuousl movin lon C(). Suppos t movmnt is t onstnt sp. Tn t lttnin tim is proportionl to t lnt λc() o C(), so it sms o prtiulr intrst to in lowr n uppr ouns on λc(). Qustion 2. Cn on lot on P point wit miniml lnt ut lous? T strtin point o our invstition is t qustion o Erik Dmin t l. (s Opn Prolm 18.1 in [5]), on t istn o ontinuous olin prosss 11

12 or ll lt ol stts o (not nssril onv) polr. Tis prolm rmins opn, n n rprs n win in irnt rmwork, s ollows. Consir n strt onv polron P (i.., on otin orin to Alnrov s luin torm). It s uniqu isomtri min in IR 3 s onv sur, ut mn otr non-onv rliztions in IR 3 (s [8] or t pris initions). Qustion 3. Lt R not t sp o ll rliztions o P in IR 3, wit t topolo inu t Husor mtri on t sp o ll ompt sts in IR 3. Is R rwis onnt? Aknowlmnt. T utors r int to Josp O Rourk or is rul rin o prliminr vrsion o tis ppr n is vlul sustions, prtiulrl improvin t proo o Lmm 5. Rrns 1. P. K. Arwl, B. Aronov, J. O Rourk n C. A. Svon, Str unolin o poltop wit pplitions, SIAM J. Comput. 26 (1997), A. D. Alnrov, Conv Polr, Sprinr-Vrl, Brlin, Monorps in Mtmtis. Trnsltion o t 1950 Russin ition N. S. Dirkov, S. S. Kuttlz, n A. B. Sossinsk. 3. J. Cn n Y. Hn, Sortst pt on polron, Pro. 6t Ann. ACM Smpos. Comput. Gom. (1990), E. D. Dmin, M. D. Dmin, n A. Luiw, Flttnin polr, Unpulis mnusript, E. D. Dmin n J. O Rourk, Gomtri olin loritms, Lins, Orimi, Polr, Cmri Univrsit Prss, J. Ito n C. Nr, Continuous lt olins o Pltoni polr. In: Akim t l. (s) CGGA2010, LNCS 7033, , Sprinr, Hirr (2011). 7. B. Knv n J. O Rourk, An implmnttion o Cn n Hn s sortst pt loritm, Pro. 12t Cnin Con. Comput. Gom. (2000), I. Pk, Inltin polrl surs, ttp:// pk/pprs/pillow4.p, I. Sitov, T volum o polron s untion o its mtri, Funm. Prikl. Mt. 2(4) (1996), I. Sitov, T volum s mtri invrint o polr, Disrt Comput. Gom. 20 (1998),