AN EFFICIENT NEW ALGORITHM FOR 2-D LINE CLIPPING ITS DEVELOPMENT AND ANALYSIS

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1 AN EFFICIENT NEW ALGORITHM FOR 2-D LINE CLIPPING ITS DEVELOPMENT AND ANALYSIS Authors: T. Nicholl, D. T. Lee, and R. Nicholl Presentation by Marcus McCurdy

2 WHAT IS CLIPPING? Basically finding the intersection of two geometric entities. xright, ytop (x2,y2) Intersection (x1,y1) xleft, ybottom

3 OTHER ALGORITHMS Cohen-Sutherland Cyrus Beck Liang-Barsky SPY In all of these algorithms, intersection points not involved in the final output are calculated

4 CLIPPING INEFFICIENCIES SPY and CS will calculate IL before rejecting LB will compute all 4 IR IT IL IB

5 MACHINE INDEPENDENT ANALYSIS Evaluated efficiency by counting the number of executions of the following operations: Comparisons Addition Subtraction Multiplication Division

6 COSTS OF OPERATIONS Comparison Addition Subtraction Multiplication Division

7 INVISIBLE IR IT IL IB

8 VISIBLE

9 PARTIALLY VISIBLE

10 9 REGIONS Top left corner Top edge Top right corner Left edge Window Right edge Bottom left corner Bottom edge Bottom right corner

11 ALGORITHM Characterize P1 s location among the 9 regions (straightforward). Based on P1 characterize P2 among appropriate subdivisions (not quite so straightforward). Calculate intersection points according to the characterization. Only the intersection points required for output are calculated.

12 POSITIONS OF P1 AND P2 P1 is located in the window, an edge region, or a corner region. P2 may be at any point on the 2D Cartesian plane Divide all possible positions of P2 into regions each of which corresponds to intersection points at the same boundaries (or none) to be output The following 3 figures show the subdivisions. All other cases are the same as one of them up to a rotation of a multiple of 90º about the origin or a reflection about the line x = -y, or a combination of the two

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14 [ ~TLII'n_;IIXLTLTL'n..TL'rLTL'n..T~ i /ll.tltltl1ltltl'n.tltut.tltlti i,mtltltl~tltltltltl~tl~t I i Z L ' n ' ~ T L T L T L ' n : n ' ~ ' n " m ' n ' ~ T L T I i.ztltl'ii.tltltli~tltltltltltlil i Z~LTLYLTLTLTLTLTLIITLTLTLrLrLTLYL~ i /~LTL'IIILTLTLTLTLI~TLIITLll;It 12-'~.9.~R] i. ~TLT~TLT~rcTLTL~TLTL~.~,rS.LRLRLR~ I :LTLTLTLTLTLTLTLTLTbllL.t~. ~RLRLRLRL R L~ i :." LLLLLLUJ/..LLI.J/ - RLELRLRLRLRLRLI~i.i~! / LLLLE~I,~f~iLLLLLLLLI~ RLRLRLRLRLPJ-RLPJ- ~ /..~-~-~u~u-uu~u-u~u~w~u~u~r u~-~ i P 1 -'-":-... ~LilJ! r!! 7!! LLLLLLLLI.IRLRLRLRLRLRLRLPd..RI i "~::7---. LU t t t t t u..lllll RLRLRLRLRI.Pd..RLRLpj ~LB LB LBLB L B LBL B LB L~ L~ E~L~.L~LRL RL~ l "~BLBLBLBLBLBLBflBLBLB LBLBLB~ j "m~slblblblblb~blblblblblb LBLBLB] I "%~BLBLBLBII~LBLB LBLBLBLBLBLHI ",d~lblb[~blblblblblblblblb ; "~Bb~LBLBLBLBLB LBLB LB! I "N~BLBLBLBLBLBLBLBLB I! ["~'B'-LB LB L B LB LB L B LBI L... _/... Lm. ~ L. BLBLBLBLBLB~ Fi$. 4 Subdivision of the plane when Pl is in an edse region

15 , i i, ', i i ' I I I l i Pl! i i! ::-. I... I i I " i... ~ " I [ ~ $~J..l I I 1 I 1111 I 1 I I I I I 1 ~TI'rT~..... I i ~ [u.~;~.~... htrtrtrtrtr'm~! ~ LLLLLLU[~AI* lltt i~trtrtr'i~tr'i~trt~ i 5. I.LLLLD.U.J.I~[ i i,.j., ~ t,~rtrtrtrtrtriryrt~! :[LLLLLLLLLLLLLt L l *, i%1 i~ I'RTRTRTRTR~ ~I" ~ [... ~LBLBLSLBLBLSLSm~IR~~ i t B LB LB i B LB LB L B LB L~I~BTB'YCRTRTRTRTRTRT~ i! ~B LB LB L B LB LB LB LB L B I~W~TB TB?r/~TRTRTRTRT~! I ~LBLBLBLBLBLBLB LBL~L~TBTB~RTRTRTRT~ I : ~LBLBLBLBLBLBLBLBL~LB~';BTBTB~TRTRTII i! 'tblblblblblblblbl~lblbh~btbtbt~rtt i i ~BLBLBLBLBLBLBLBLt~LBLBLI~EP~TBTBT~p~ [ i ~LBLBLBLBLBLBLBL~LBLBLBLB~TBTBT~... i %--.~--B--L.B. L--B.L- B--L.B-Ir-B.L B--~.~ LB 5 BLB LB L 6~BTBTI~ L FiS. 5 Subdivision of the plane when P! is in a corner resion

16 GEOMETRIC TRANSFORMATIONS P1 may be in any one of 9 regions Different procedure could be written to handle each case However, writing similar but not identical code may be error-prone Exploit symmetry to prevent this

17 ALGORITHM Find where P1 is in relation to the Left and Right boundaries. May be beyond left, beyond right, or between the two. Beyond left is symmetrical to beyond right up to a rotation of 180º about the origin and they can be handled together

18 P1 IS BEYOND LEFT BOUNDARY If P2 is also beyond LB then line is invisible stop Now find if P1 is beyond top boundary, bottom boundary, or between the two. Again, beyond top is symmetrical with beyond bottom, reflection about the x-axis.

19 P1 IS IN TOP LEFT CORNER If P2 is above top stop line is invisible Compare P2 against the line joining P1 and the top-left corner The case where P2 is on one side of this line is symmetrical to the case in which it is on the other side. Handle one of the two cases.

20 P1 TOP LEFT CORNER P2 not beyond the left and to the right of vector from P1 to the top left corner If P2 is above the bottom then no more oblique boundaries need to be computed However, if P2 is below bottom 3 possibilities Boundary from line joining P1 to bottom left has to be computed no matter which side of right boundary P2 happens to be on. Do this comparison first If left no more oblique boundaries, else more comparison to decide which subdivision P2 is in

21 P1 IN LEFT EDGE If P2 is beyond left then line is invisible P2 may be above top, below bottom, or between top and bottom. Again above top is symmetric to below bottom

22 P1 IN LEFT EDGE & P2 BEYOND BOTTOM Comparing boundary joining P1 to the bottom left corner cannot be avoided so do this one first.

23 P1 BTW LEFT AND RIGHT If P1 above top or below bottom case is symmetrical to that of P1 in the left edge region so use the same procedure

24 P1 IN THE WINDOW If P2 is in an edge region then the intersection point can be seen from figure 3 If P2 is in a corner region then comparison with one oblique boundary is necessary before appropriate intersection calculation is possible Use symmetry to reduce different cases

25 -l~-l~r]q i i i i i 1 t 11 ITi ~ I I I I I I I I I I I I I I 1 1~ I 1 1 l-r~ l ~ 71~Iq I I I I I I I I I I I I-1"I~ I I l i i i I I I i I I l'l']'~tl~lq I [ I I i I I J~'~R~ [~ 'I IIIII tlll III I I~ tl II lllllll I III I I I I I~ I I III III I I I k~rrr~ ['FI~[TIq I I I L I I I I I FIt I I I I I I II II II Ill I II I I I I I I I I I I I I~RRRRR~ ~i i ii ii i ii i i ii II I~ II II i III II III II Ill I~I II I I ~(RRRRRR - 1~12'12~.L 1... ~ l t,~ I t,~rrrrrrrrrr~ ~LLLLLL[LE%.LI L t I t t~ I I I I I t L L t I L Z t t t A I I t I~ ~RRRRRRRRRRRRR~ ~LLLLLLLLLLLI~[~L~6.t L[i t I t t t L t t I t i t I i t t t ]71-I I~.RRRRRRRRRRRRRR [LLLLLLLLLLLLLLLLL~....." I~F.RRRRRRRRRRRRRR illlllllllllllllll... -'"... ra." RRRRRRRRRRRRRRRR illlllllllllllllll...":~" RRRRRRRRRRRRRRRR 'LLLLLLLLLLLLLLLL~... \ RRRRRRRRRRRRRRRR I ~LLLLLLLLLLLLLLLLLI] ""'" \ RR.RRRRRRR RRRRRRR! ILLLLLLLLLLLLLLLLL d... \ RRRRRRRRRRRRRRRRI illlllllllllllllll..."'" ~ RRRRRRRRRRRRRRRR i[ltl... i~... RRRRRRRRRRRRRRRR] LLLLLLLLLLLLLLi~i,:.~ BBBBB B BB B B BB BB BBf~ RR.R RRRRRRRRRRRR! illllllllllli~3~ B B~ B B B B B B B B B BBB BB BBB B pi~rrrrrr RRRR RRRR[ illlllllll[.a:,'i~ BBB B B~BBB BB B BB BBBB BBBBB B ~ BI~RRRRRRRRRRRRR] ~LLLLLL~..~'I~'BB B B B B B~BB B BB BBB BBB B B BBBB B ~ B B ~RR RRRRRRRRRR.ILLLi,~'BBBBBBBBBBB~BBBBBBBBBBBBBBBBBB ~BBB~RRRRRRRRRRRR! -I~ ~B BBBBBBBBBBBB~BBBBBBBBBBBBBBBBBB~BBBB~RRKRRRRRRRI LiBBBBBBBBBBBBBBBB~BBBBBBBBBBBBBBBBBB 3BBBBB~RRRRRRRRRR I ibbbbbbbbbbbbbbbb~bbbbbbbbbbbbbbbbbb 3BBBBBB~RRRRRRRRR LBBBB.BBBBBBBBBBBBJ~BBBBBBBBBBBp_BBBBBB ~BBBBBBBRRRRRRRRR Fig. 3 Subdivision of the plane when Pl is in the window

26 ANALYSIS Fewest divisions among the algorithms. Equal to the number of intersection points for output. Fewest comparisons. About 1/3 CS and about 1/2 LB. CS requires largest number of comparison in most cases With large window dominating case is when P2 is in window. CS does few set operations, and so LB is slowest Small window dominating cases are the four corner regions. CS is slower in this case,

27 I CS LB NLN I l CS LB NLN COMPARISONS + / CS i0 LB 20 NLN 8 P J FIG.6 Analysis of the three algorithms when P1 is in the window

28 CS 9.5 LB 10.5 NLN COMPARISONS + / i/ CS LB NLN F ~ CS.75 LB " NLN ~ i~.75 ~ I/ CS LB NLN 32 / / /~ CS LB 3b.5 20 NLN g / / / CS LB / NLN i ,0 0 0 /2.5 / 0 / /_ - n CS LB NLN l CS LB NLN L FIG.7 Analysis of the three algorithms when P1 is in an edge region

29 CS LB NLN [ CS LB NLN CS LB NLN ~ '-" - o -! I o,o I \ \ o ~o o o o \ \ , ~s ~ ~ c~ \ i 1.75 i TM \ 1.5\ 2 1 \ 4.5 " \ ~ \ ~ 0 CS LB NLN ~ CS LB NLN COMPARISONS * / 2.5\ CS LB NLN CS LB NLN i FIG.8 Analysis of the three algorithms when P1 is in a corner region

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