INTERSECTIONS OF LINE SEGMENTS AND POLYGONS

Transcription

1 INTERSECTIONS OF LINE SEGMENTS AND POLYGONS PETR FELKEL FEL CTU PRAGUE Bas on [Brg], [Mount], [Kukral], an [Drtina] Vrsion rom

2 Talk ovrviw Intrsctions o lin sgmnts (Bntly-Ottmann) Motivation Swp lin algorithm rcapitulation Swp lin intrsctions o lin sgmnts Intrsction o polygons or planar suivisions S assignmnt [1] or [Brg, Sction.] Intrsction o axis paralll rctangls S assignmnt [6] ( / 71)

3 Gomtric intrsctions what ar thy or? On o th most asic prolms in computational gomtry Soli moling Intrsction o ojct ounaris in CSG Ovrlay o suivisions,.g. layrs in GIS Brigs on intrsctions o roas an rivrs Maintnanc rsponsiilitis (roa ntwork X county ounaris) Rootics Collision tction an collision avoianc Computr graphics Rnring via ray shooting (intrsction o th ray with ojcts) ( / 71)

4 Lin sgmnt intrsction Intrsction o complx shaps is otn ruc to simplr an simplr intrsction prolms Lin sgmnt intrsction is th most asic intrsction algorithm Prolm statmnt: Givn n lin sgmnts in th plan, rport all points whr a pair o lin sgmnts intrsct. Prolm complxity Worst cas I = O(n ) intrsctions Practical cas only som intrsctions Us an output snsitiv algorithm O(n log n + I) optimal ranomiz algorithm O(n log n + I log n ) swp lin algorithm -% [Brg] ( / 71)

5 Plan swp lin algorithm rcapitulation Horizontal lin (swp lin, scan lin) l movs top-own (or vrtical lin: lt to right) ovr th st o ojcts Th mov is not continuous, ut l jumps rom on vnt point to anothr Evnt points ar in priority quu or sort list (~y) Th (lt) top-most vnt point is rmov irst Nw vnt points may crat (usually as intraction o nighors on th swp lin) an insrt in th quu Scan-lin status Stors inormation aout th ojcts intrsct y l It is upat whil stopping on vnt point ( / 71)

6 Lin sgmnt intrsction - Swp lin alg. Avoi tsting o pairs o sgmnts ar apart Comput intrsctions o nighors on th swp lin only O(n log n + I log n ) tim in O(n) mmory n stps or n points, I stps or intrsctions, log n sarch th status tr Ignor nasty cass (most o thm will solv latr on) No sgmnt is paralll to th swp lin Sgmnts intrsct in on point an o not ovrlap No thr sgmnts mt in a common point (6 / 71)

7 Lin sgmnt intrsctions Status = orr squnc o sgmnts intrscting th swp lin l Evnts (waiting in th priority quu) = points, whr th algorithm actually os somthing Sgmnt n-points known at algorithm start Sgmnt intrsctions twn nighoring sgmnts along SL Discovr as th swp xcuts (7 / 71)

8 Dtcting intrsctions Intrsction vnts must tct an insrt to th vnt quu or thy occur Givn two sgmnts a, intrscting in a point p, thr must a placmnt o swp lin l prior to p, such that sgmnts a, ar ajacnt along l (only ajacnt will tst or intrsction) sgmnts a, ar not ajacnt whn th alg. starts sgmnts a, ar ajacnt just or p => thr must an vnt point whn a, com ajacnt an thror ar tst or intrsction (8 / 71) [Brg]

9 Data structurs Swp lin l status = orr o sgmnts along l Balanc inary sarch tr o sgmnts Coors o intrsctions with l vary as l movs => stor pointrs to lin sgmnts in tr nos Position o l is plugg in th y=mx+ to gt th x-ky [Brg] (9 / 71)

10 Data structurs Evnt quu (postupový plán, časový plán) y top-own Din: Orr (top-own, lxicographic) p q i p y > q y or p y = q y an p x < q x top-own, lt-right approach (points on l trat lt to right) x Oprations Insrtion o comput intrsction points Ftching th nxt vnt (highst y low l ) Tst, i th sgmnt is alray prsnt in th quu (Locat an lt intrsction vnt in th quu) (10 / 71)

11 Prolm with uplicitis o intrsctions 1 x tct intrsction (11 / 71)

12 Data structurs Evnt quu ata structur Hap Prolm: can not chck uplicat intrsction vnts (rinvnt mor than onc) Intrsctions procss twic or vn mor Mmory complxity up to O(n ) Orr ictionary (alanc inary tr) Can chck uplicat vnts (as just constant actor) Nothing insrt twic I non-nighor intrsctions ar lt i.., i only intrsctions o nighors along l ar stor thn mmory complxity just O(n) (1 / 71) 1 x tct intrsction

13 Lin sgmnt intrsction algorithm FinIntrsctions(S) Input: Output: A st S o lin sgmnts in th plan Th st o intrsction points + pointrs to sgmnts in ach 1. init an mpty vnt quu Q an insrt th sgmnt npoints. init an mpty status structur T. whil Q in not mpty. rmov nxt vnt p rom Q. hanlevntpoint(p) Uppr npoint Intrsction Lowr npoint Not: Uppr-n-point vnts stor ino aout th sgmnt (1 / 71)

14 hanlevntpoint() principl Uppr npoint U(p) insrt p (on s j ) to status T a intrsctions with lt an right nighors to Q Intrsction C(p) switch orr o sgmnts in T a intrsctions o lt an right nighors to Q Lowr npoint L(p) rmov p (on s l ) rom T a intrsctions o lt an right nighors to Q [Brg] (1 / 71)

15 Mor than two sgmnts incint U(p)= {s } C(p)= {s 1, s } start hr cross on l L(p) = {s, s } n hr [Brg] (1 / 71)

16 Hanl Evnts [Brg, pag ] hanlevntpoint(p) U(p) 1. Lt U(p) = st o sgmnts whos Uppr npoint is p. Ths sgmts ar stor with th vnt point p (will a to T). Sarch T or all sgmnts S(p) that contain p (ar ajacnt in T): Lt L(p) S(p) = sgmnts whos Lowr npoint is p Lt C(p) S(p) = sgmnts that Contain p in intrior. i( L(p) U(p) C(p) contains mor than on sgmnt ). rport p as intrsction togthr with L(p), U(p), C(p) p p. Dlt th sgmnts in L(p) C(p) rom T 6. Insrt th sgmnts in U(p) C(p) into T Rvrs orr o C(p) in T (orr as low l, horizontal sgmnt as th last) 7. i( U(p) C(p) = ) thn innwevnt(s l, s r, p) // lt & right nighors 8. ls s = ltmost sgmnt o U(p) C(p); innwevnt(s l, s, p) s = rightmost sgmnt o U(p) C(p); innwevnt(s, s r, p) s l p s r p L(p) C(p) (16 / 71)

17 Dtction o nw intrsctions innwevnt(s l, s r, p) // with hanling o horizontal sgmnts Input: two sgmnts (lt & right rom p in T) an a currnt vnt point p Output: upat vnt quu Q with nw intrsction 1. i [ ( s l an s r intrsct low th swp lin l ) or ( intrsct on l an to th right o p ) ] an // horizontal sgmnts ( th intrsction is not prsnt in Q ). thn insrt intrsction as a nw vnt into Q s = ltmost sgmnt o U(p) C(p); s l p s r s l p s s r s l p s r s s l an s r intrsct low s = ltmost rom C(p) s r an s intrsct on l an to th right o p s (17 / 71)

18 Lin sgmnt intrsctions Mmory O(I) = O(n ) with uplicitis in Q or O(n ) with uplicitis in Q lt Oprational complxity n + I stops log n ach => O( I + n) log n total Th algorithm is y Bntly-Ottmann Bntly, J. L.; Ottmann, T. A. (1979), "Algorithms or rporting an counting gomtric intrsctions", IEEE Transactions on Computrs C-8 (9): 6-67, oi: /tc S also (18 / 71)

19 Intrsction o axis paralll rctangls Givn th collction o n isothtic rctangls, rport all intrscting parts r 6 Altrnat sis Ovrlap long to two r 7 r r pncils o lins (trsy přímk) (otn us with r 8 Inclusion r points in ininity = axis paralll) r 1 r r 9 Answr: (r 1, r ) (r 1, r ) (r 1, r 8 ) (r, r ) (r, r ) (r, r 9 ) (r, r ) (r 7, r 8 ) [?] (19 / 71)

20 Brut orc intrsction Brut orc algorithm Input: st S o axis paralll rctangls Output: pairs o intrsct rctangls 1. For vry pair (r i, r j ) o rctangls S, i j. i (r i r j ) thn. rport (r i, r j ) Analysis Prprocssing: Non. Qury: = ( ). Storag: (0 / 71)

21 Plan swp intrsction algorithm Vrtical swp lin movs rom lt to right Stops at vry x-coorinat o a rctangl (ithr its lt si or its right si). activ rctangls a st = rctangls currntly intrscting th swp lin lt si vnt o a rctangl => th rctangl is a to th activ st. right si => th rctangl is lt rom th activ st. Th activ st us to tct rctangl intrsction (1 / 71)

22 Exampl rctangls an swp lin y activ rctangl not activ rctangl swp lin [Drtina] ( / 71)

23 Intrval tr as swp lin status structur Vrtical swp-lin => Only y-coorinats along it Turn our viw in slis 90 right Swp lin (y-axis) will rawn as horizontal y activ rctangl not activ rctangl L R y swp lin [Drtina] ( / 71)

24 Intrsction tst twn pair o intrvals Givn two intrvals R = [y 1, y ] an R = [y 1, y ] th conition R R is quivalnt to on o ths mutually xclusiv conitions: y 1 y a) y 1 y 1 y y 1 y ) y 1 y 1 y y 1 y y 1 y Intrvals along th swp lin a) ) ) Intrsction (ork) ( / 71)

25 Static intrval tr stors all n points Lt v = y m th mian o n-points o sgmnts S l : sgmnts o S that ar compltly to th lt o y m S m : sgmnts o S that contain y m S r : sgmnts o S that ar compltly to th right o y m y m [Vignron] ( / 71)

26 Static intrval tr Exampl L(v) R(v) S m Lt ns ascning Right ns scning S l S r [Vignron] (6 / 71)

27 Static intrval tr [Elsrunnr80] Stors intrvals along y swp lin kins o inormation - n points - incint intrvals - activ nos, 6, [Kukral] (7 / 71)

28 Primary structur static tr or npoints v = mipoint o all sgmnt npoints H(v) = valu (y-coor) o v, 6, [Kukral] (8 / 71)

29 Sconary lists o incint intrval n-pts. ML(v) lt npoints o intrval containing v (sort ascning) MR(v) right npoints (scning) , 6, ML(v) MR(v) 6 [Kukral] (9 / 71)

30 Activ nos intrsct y th swp lin Sust o all nos at prsnt intrsct y swp lin (nos with intrvals) LPTR Activ no RPTR Activ no, 6, Activ no [Kukral] (0 / 71)

31 Qury = swp an rport intrsctions RctanglIntrsctions( S ) Input: St S o rctangls Output: Intrsct rctangl pairs 1. Prprocss( S ) // crat th intrval tr T (or y-coors) // an vnt quu Q (or x-coors). whil ( Q ) o. Gt nxt ntry (x i, y il, y ir, t) rom Q // t { lt right }. i ( t = lt ) // lt g. a) QuryIntrval ( y il, y ir, root(t)) // rport intrsctions 6. ) InsrtIntrval ( y il, y ir, root(t)) // insrt nw intrval 7. ls // right g 8. c) DltIntrval ( y il, y ir, root(t) ) (1 / 71)

32 Prprocssing Prprocss( S ) Input: St S o rctangls Output: Primary structur o th intrval tr T an th vnt quu Q 1. T = PrimaryTr(S) // Construct th static primary structur // o th intrval tr -> swp lin STATUS T. // Init vnt quu Q with vrtical rctangl gs in ascning orr ~x // Put th lt gs with th sam x aha o right ons. or i = 1 to n. insrt( ( x il, y il, y ir, lt ), Q) // lt gs o i-th rctangl. insrt( ( x ir, y il, y ir, right ), Q) // right gs ( / 71)

33 Intrval tr primary structur construction PrimaryTr(S) // only th y-tr structur, without intrvals Input: Output: St S o rctangls Primary structur o an intrval tr T 1. S y = Sort npoints o all sgmnts in S accoring to y-coorinat. T = BST( S y ). rturn T BST( S y ) 1. i( S y = 0 ) rturn null. ym = mian o S y. L = npoints p y ym. R = npoints p y > ym. t = nw IntrvalTrNo( ym ) 6. t.lt = BST(L) 7. t.right = BST(R) 8. rturn t ( / 71)

34 Intrval tr sarch th intrsctions QuryIntrval (,, T ) Input: Intrval o th g an currnt tr T Output: Rport th rctangls that intrsct [, ] H(v) Nw intrval ing tst or intrsction 1. i( T = null ) rturn. i=0; i( < H(v) < ) // orks at this no. whil ( MR(v).[i] >= ) && (i < Count(v)).. Rport all intrvals inm. RportIntrsction; i++ Othr nw intrval ing. QuryIntrval(,,T.LPTR ) tst or intrsction 6. QuryIntrval(,,T.RPTR ) 7. ls i (H(v) < ) // sarch RIGHT ( ) 8. whil (MR(v).[i] >= ) && (i < Count(v)) Crosss A,B 9. RportIntrsction; i++ Crosss A,B,C 10. QuryIntrval(,,T.RPTR ) Crosss A,B,C 11. ls // < H(v) //sarch LEFT( ) 1. whil (ML(v).[i] <= ) 1. RportIntrsction; i++ 1. QuryIntrval(,,T.LPTR ) Crosss nothing Crosss C Stor intrvals o activ rctangls C A B Cross.B T.LPTR ( / 71) T.RPTR

35 Intrval tr - intrval insrtion InsrtIntrval (,, T ) Input: Output: Intrval [,] an intrval tr T T atr insrtion o th intrval 1. v = root(t ). whil( v!= null ) // in th ork no. i (H(v) < < ). v = v.right // continu right. ls i ( < < H(v)) 6. v = v.lt // continu lt 7. ls // H(v) // insrt intrval 8. st v no to activ 9. connct LPTR rsp. RPTR to its parnt 10. insrt [,] into list ML(v) sort in ascning orr o s 11. insrt [,] into list MR(v) sort in scning orr o s 1. rak 1. nwhil 1. rturn T ( / 71) H(v) Nw intrval ing insrt

36 Exampl 1 (6 / 71)

37 Exampl 1 static tr on npoints Y H(v) valu o no v B A X 1 B A [Drtina] (7 / 71)

38 Intrval insrtion [1,] a) Qury Intrval Y B Sarch MR(v) or ML(v): MR(v) is mpty No activ sons, stop < H(v) < 1 < < A Activ rctangl Currnt no X A B Activ no [Drtina] (8 / 71)

39 Intrval insrtion [1,] ) Insrt Intrval Y B H(v)? 1? A Activ rctangl Currnt no X A B Activ no [Drtina] (9 / 71)

40 Intrval insrtion [1,] ) Insrt Intrval Y B H(v) 1 ork => to lists A Activ rctangl Currnt no X A B Activ no [Drtina] (0 / 71)

41 Intrval insrtion [,] a) Qury Intrval Y Sarch MR(v) only: H(v) < B MR(v)[1] =? => intrsction < A 1 1 R(v) Activ rctangl Currnt no X A B Activ no [Drtina] (1 / 71)

42 Y Intrval insrtion [,] ) Insrt Intrval B H(v) ork => to lists A 1 1,, 1 0 Activ rctangl Currnt no Activ no X 1 B A [Drtina] ( / 71)

43 Intrval lt [1,] Y B A 1 1,, Activ rctangl Currnt no X A B Activ no [Drtina] ( / 71)

44 Intrval lt [1,] Y B A Activ rctangl Currnt no X A B Activ no [Drtina] ( / 71)

45 Intrval lt [,] Y B A Activ rctangl Currnt no X A B Activ no [Drtina] ( / 71)

46 Intrval lt [,] Y B A X 1 B A [Drtina] (6 / 71)

47 Exampl RctanglIntrsctions( S ) // this is a copy o th sli or Input: St S o rctangls // just to rmmr th algorithm Output: Intrsct rctangl pairs 1. Prprocss( S ) // crat th intrval tr T an vnt quu Q. whil ( Q ) o. Gt nxt ntry (x il, y il, y ir, t) rom Q // t { lt right }. i ( t = lt ) // lt g. a) QuryIntrval ( y il, y ir, root(t)) // rport intrsctions 6. ) InsrtIntrval ( y il, y ir, root(t)) // insrt nw intrval 7. ls // right g 8. c) DltIntrval ( y il, y ir, root(t) ) (7 / 71)

48 Exampl Y c a X a c [Drtina] (8 / 71)

49 Insrt [,] mpty => ) Insrt Intrval H(v) Y Insrt th nw intrval to sconary lists?? ork no => activ => to lists c a X Activ rctangl Currnt no Activ no a c [Drtina] (9 / 71)

50 Insrt [,7] a) Qury Intrval H(v) < Y or (all in MR(v)) tst MR(v)[i] >= => rport intrsction c 1 7? < 7? 1 c a Activ rctangl Currnt no Activ no X a c [Drtina] (0 / 71)

51 Y Insrt [,7] ) Insrt Intrval Insrt th nw intrval to sconary lists H(v) ork no => activ => to lists, 7, c a X Activ rctangl a c Currnt no Activ no [Drtina] (1 / 71)

52 Y Insrt [0,] a) Qury Intrval c or (all in ML(v)) tst ML(v).[i] => rport intrsction c go lt, nil, stop a, 7, < H(v)? 0 <? 0 Activ rctangl Currnt no Activ no X a c [Drtina] ( / 71)

53 Insrt [0,] ) Insrt Intrval 1/ < <H(v) Y 8 7 6, 7, 1 7? 0 < <? => insrt lt 1 c a X Activ rctangl a c Currnt no Activ no [Drtina] ( / 71)

54 Y Insrt [0,] ) Insrt Intrval / c Insrt th nw intrval to sconary lists o th lt son LPTR link to parnt a, 7, H(v)? 0 1? ork no => activ => to lists 0 Activ rctangl Currnt no Activ no X a c [Drtina] ( / 71)

55 Y Insrt [1,] a) Qury Intrval 1/ c or (all in MR(v)) => rport intrsction c, go lt -> 1 go right - nil a, 7, < H(v) <? 1 < <? 0 Activ rctangl Currnt no Activ no X a c [Drtina] ( / 71)

56 Y Insrt [1,] a) Qury Intrval / c or (all in MR(v)) tst MR(v)[i] 1 => rport intrsction go right, nil, stop a, 7, H(v) <? 1 1 <? 0 Activ rctangl Currnt no Activ no X a c [Drtina] (6 / 71)

57 Y Insrt [1,] ) Insrt Intrval H(v) Insrt th nw intrval to sconary lists? 1? 1,, 7,, 1 7 c a 0 Activ rctangl Currnt no Activ no X a c [Drtina] (7 / 71)

58 Y Insrt [7,8] a) Qury Intrval c or (all in MR(v)) tst MR(v).[i] 7 => rport intrsction go right, nil, stop a 1,, 7,, H(v) <? 7 < 8? 0 Activ rctangl Currnt no Activ no X a c [Drtina] (8 / 71)

59 Y Insrt [7,8] ) Insrt Intrval c Insrt th nw intrval to sconary lists link to parnt a H(v) right <=? 7 < 8? right <=? 7 < 8? ,, 7,, Activ rctangl Currnt no Activ no X a c [Drtina] (9 / 71)

60 Y Dlt [,7] Dlt Intrval H(v) Dlt th intrval [,7] rom sconary lists? 7 8? 1,, 1 7 c a 0 Activ rctangl Currnt no Activ no X a c [Drtina] (60 / 71)

61 Y Insrt [,6] a) Qury Intrval c H(v) < or (all in MR(v)) tst MR(v).[i] => rport intrsction < 6? or (all in ML(v)) tst ML(v).[i] 6 < 6 7? => no intrsction 1,, 1 7 a Activ rctangl Currnt no Activ no X a c [Drtina] (61 / 71)

62 Y Insrt [,6] ) Insrt Intrval c Insrt th nw intrval to sconary lists a 1,, H(v) <? < 6?? 6? 0 Activ rctangl Currnt no Activ no X a c [Drtina] (6 / 71)

63 Y Dlt [1,] Dlt Intrval Dlt th intrval [1,] rom sconary lists H(v)? 1? c a 0 Activ rctangl Currnt no Activ no X a c [Drtina] (6 / 71)

64 Y Dlt [0,] Dlt Intrval 1/ Sarch or no with intrval [0,] < H(v)? 0 <? c a 0 Activ rctangl Currnt no Activ no X a c [Drtina] (6 / 71)

65 Y Dlt [0,] Dlt Intrval / Dlt th intrval [0,] rom sconary lists o no 1 H(v)? 0 1? c a 0 Activ rctangl Currnt no Activ no X a c [Drtina] (6 / 71)

66 Y Dlt [7,8] Dlt Intrval Sarch or an lt no with intrval [7,8] c a H(v)? 7 < 8?? 7 < 8?? 7 7 8? Activ rctangl Currnt no Activ no X a c [Drtina] (66 / 71)

67 Y Dlt [,] Dlt Intrval Sarch or an lt no with intrval [,] H(v)?? c 0 6 a 0 Activ rctangl Currnt no Activ no X a c [Drtina] (67 / 71)

68 Dlt [,6] Dlt Intrval H(v) Y Sarch or an lt no with intrval [,6]? 6? c 0 6 a 0 Activ rctangl Currnt no Activ no X a c [Drtina] (68 / 71)

69 Empty tr Y Sarch or an lt no with intrval [,6] 1 7 c 0 6 a 0 Activ rctangl Currnt no Activ no X a c [Drtina] (69 / 71)

70 Complxitis o rctangl intrsctions n rctangls, s intrsct pairs oun O(n log n) prprocssing tim to sparatly sort x-coorinats o th rctangls or th plan swp th y-coorinats or initializing th intrval tr. Th plan swp itsl taks O(n log n + s) tim, so th ovrall tim is O(n log n + s) O(n) spac This tim is optimal or a cision-tr algorithm (i.., on that only maks comparisons twn rctangl coorinats). (70 / 71)

71 Rrncs [Brg] Mark Brg, Otri Chong, Marc van Krvl, Mark Ovrmars: Computational Gomtry: Algorithms an Applications, Springr- Vrlag, r rv pags, 70 ig. ISBN: , Chaptrs an 9, [Mount] Davi Mount, - CMSC 7: Computational Gomtry, Lctur Nots or Spring 007, Univrsity o Marylan, Lcturs 7,, 1,1, an 0. [Rourk] Josph O Rourk:.: Computational Gomtry in C, Camrig Univrsity Prss, 199, ISBN [Drtina] Tomáš Drtina: Intrsction o rctangls. Smstral Assignmnt. Computational Gomtry cours, FEL CTU Pragu, 006 [Kukral] Ptr Kukrál: Intrsction o rctangls. Smstral Assignmnt. Computational Gomtry cours, FEL CTU Pragu, 006 [Vignron] Sgmnt trs an intrval trs, prsntation, INRA, Franc, (71 / 71)