Clipping Lines. Dr. Scott Schaefer

Transcription

1 Clipping Lines Dr. Scott Schaefer

2 Why Clip? We do not want to waste time drawing objects that are outside of viewing window (or clipping window) 2/94

3 Clipping Points Given a point (x, y) and clipping window (x min, y min ), (x max, y max ), determine if the point should be drawn (x max, y max ) (x,y) (x min, y min ) 3/94

4 Clipping Points Given a point (x, y) and clipping window (x min, y min ), (x max, y max ), determine if the point should be drawn (x max, y max ) (x,y) (x min, y min ) x min <=x<=x max? y min <=y<=y max? 4/94

5 Clipping Points Given a point (x, y) and clipping window (x min, y min ), (x max, y max ), determine if the point should be drawn (x max, y max ) (x 1, y 1 ) (x 2, y 2 ) (x min, y min ) x min <=x<=x max? y min <=y<=y max? 5/94

6 Clipping Points Given a point (x, y) and clipping window (x min, y min ), (x max, y max ), determine if the point should be drawn (x max, y max ) (x 1, y 1 ) (x 2, y 2 ) (x min, y min ) x min <=x 1 <=x max Yes y min <=y 1 <=y max Yes 6/94

7 Clipping Points Given a point (x, y) and clipping window (x min, y min ), (x max, y max ), determine if the point should be drawn (x max, y max ) (x 1, y 1 ) (x 2, y 2 ) (x min, y min ) x min <=x 2 <=x max No y min <=y 2 <=y max Yes 7/94

8 Clipping Lines P 6 P 1 P 3 P 4 P 8 P 9 P 5 P 7 P 2 P 1 8/94

9 Clipping Lines ˆP 6 ˆP 1 ˆP 9 P 8 P 5 P 7 9/94

10 Clipping Lines Given a line with end-points (x, y ), (x 1, y 1 ) and clipping window (x min, y min ), (x max, y max ), determine if line should be drawn and clipped end-points of line to draw. (x max, y max ) (x 1, y 1 ) (x, y ) (x min, y min ) 1/94

11 Clipping Lines P 6 P 1 P 3 P 4 P 8 P 9 P 5 P 7 P 2 P 1 11/94

12 Clipping Lines Simple Algorithm If both end-points inside rectangle, draw line Otherwise, intersect line with all edges of rectangle clip that point and repeat test 12/94

13 Clipping Lines Simple Algorithm ( x max, ymax ) ( x 1, y1) ( x, y ) ( x min, ymin ) 13/94

14 Clipping Lines Simple Algorithm ( x max, ymax ) ( x 1, y1) ( x, y ) ( x min, ymin ) 14/94

15 Clipping Lines Simple Algorithm ( x max, ymax ) ( x 1, y1) x ˆ, y ˆ ( ) ( x min, ymin ) 15/94

16 Clipping Lines Simple Algorithm ( x max, ymax ) ( x 1, y1) x ˆ, y ˆ ( ) ( x min, ymin ) 16/94

17 Clipping Lines Simple Algorithm ( x max, ymax ) ( x 1, y1) x ˆ, y ˆ ( ) ( x min, ymin ) 17/94

18 Clipping Lines Simple Algorithm ( x max, ymax ) ( x 1, y1) x ˆ, y ˆ ( ) ( x min, ymin ) 18/94

19 Clipping Lines Simple Algorithm ( x max, ymax ) ( x, y ) ( x min, ymin ) ( x 1, y1) 19/94

20 Clipping Lines Simple Algorithm ( x max, ymax ) x ˆ, y ˆ ( ) ( x min, ymin ) ( x 1, y1) 2/94

21 Clipping Lines Simple Algorithm ( x max, ymax ) x ˆ, y ˆ ( ) ( x min, ymin ) ( x 1, y1) 21/94

22 Clipping Lines Simple Algorithm ( x max, ymax ) x ˆ, y ˆ ( ) ( x min, ymin ) xˆ 1, yˆ ( 1 ) 22/94

23 Clipping Lines Simple Algorithm ( x max, ymax ) x ˆ, y ˆ ( ) ( x min, ymin ) xˆ 1, yˆ ( 1 ) 23/94

24 Window Intersection (x 1, y 1 ), (x 2, y 2 ) intersect with vertical edge at x right y intersect = y 1 + m(x right x 1 ) where m=(y 2 -y 1 )/(x 2 -x 1 ) (x 1, y 1 ), (x 2, y 2 ) intersect with horizontal edge at y bottom x intersect = x 1 + (y bottom y 1 )/m where m=(y 2 -y 1 )/(x 2 -x 1 ) 24/94

25 Clipping Lines Simple Algorithm Lots of intersection tests makes algorithm expensive Complicated tests to determine if intersecting rectangle Is there a better way? 25/94

26 Trivial Accepts Big Optimization: trivial accepts/rejects How can we quickly decide whether line segment is entirely inside window Answer: test both endpoints 26/94

27 Trivial Accepts Big Optimization: trivial accepts/rejects How can we quickly decide whether line segment is entirely inside window Answer: test both endpoints 27/94

28 Trivial Rejects How can we know a line is outside of the window Answer: both endpoints on wrong side of same edge, can trivially reject the line 28/94

29 Trivial Rejects How can we know a line is outside of the window Answer: both endpoints on wrong side of same edge, can trivially reject the line 29/94

30 Cohen-Sutherland Algorithm Classify p, p 1 using region codes c, c 1 If, trivially reject If c c c1 c1, trivially accept Otherwise reduce to trivial cases by splitting into two segments 3/94

31 Cohen-Sutherland Algorithm Every end point is assigned to a four-digit binary value, i.e. Region code Each bit position indicates whether the point is inside or outside of a specific window edge bit 4 bit 3 bit 2 bit 1 top bottom right left 31/94

32 Cohen-Sutherland Algorithm /94

33 Cohen-Sutherland Algorithm Classify p, p 1 using region codes c, c 1 If, trivially reject If c c c1 c1, trivially accept Otherwise reduce to trivial cases by splitting into two segments 33/94

34 Cohen-Sutherland Algorithm Classify p, p 1 using region codes c, c 1 If, trivially reject If c c c1 c1, trivially accept Otherwise reduce to trivial cases by splitting into two segments 34/94

35 Cohen-Sutherland Algorithm Classify p, p 1 using region codes c, c 1 If, trivially reject If c c c1 c1, trivially accept Otherwise reduce to trivial cases by splitting into two segments Line is outside the window! reject /94

36 Cohen-Sutherland Algorithm Classify p, p 1 using region codes c, c 1 If, trivially reject If c c c1 c1, trivially accept Otherwise reduce to trivial cases by splitting into two segments Line is inside the window! draw /94

37 Cohen-Sutherland Algorithm Classify p, p 1 using region codes c, c 1 If, trivially reject If c c c1 c1, trivially accept Otherwise reduce to trivial cases by splitting into two segments /94

38 Cohen-Sutherland Algorithm /94

39 Cohen-Sutherland Algorithm /94

40 Cohen-Sutherland Algorithm /94

41 Cohen-Sutherland Algorithm /94

42 Cohen-Sutherland Algorithm /94

43 Cohen-Sutherland Algorithm /94

44 Cohen-Sutherland Algorithm /94

45 Cohen-Sutherland Algorithm /94

46 Cohen-Sutherland Algorithm /94

47 Cohen-Sutherland Algorithm /94

48 Cohen-Sutherland Algorithm /94

49 Cohen-Sutherland Algorithm /94

50 Cohen-Sutherland Algorithm /94

51 Cohen-Sutherland Algorithm /94

52 Cohen-Sutherland Algorithm /94

53 Liang-Barsky Algorithm Uses parametric form of line for clipping Lines are oriented Classify lines as moving inside to out or outside to in Don t find actual intersection points Find parameter values on line to draw 53/94

54 Liang-Barsky Algorithm Initialize interval to [t min, t max ]=[,1] For each boundary Find parametric intersection t with boundary If moving in to out, t max = min(t max, t) else t min = max(t min, t) If t min >t max, reject line 54/94

55 Intersecting Two Parametric Lines ( x, y 3 3) ( x 1, y1) ( x, y ) ( x 2, y2 ) 55/94

56 Intersecting Two Parametric Lines x ( t) x ( x x ) t 1 y ( t) y ( y y ) t 1 t 1 ( x, y 3 3) ( x, y ) ( x 1, y1) ( x 2, y2 ) 56/94

57 Intersecting Two Parametric Lines x ( t) x ( x x ) t 1 y ( t) y ( y y ) t 1 x( ) x y( ) y x( 1) x 1 y( 1) y 1 ( x, y 3 3) ( x, y ) ( x 2, y2 ( x 1, y1) ) 57/94

58 Intersecting Two Parametric Lines ( x, y 3 3) ( x 1, y1) ( x, y ) x ( x x ) t x ( x x y y y ) t y ( y y ( 2 ) s ) s ( x 2, y2 ) 58/94

59 59/94 Intersecting Two Parametric Lines ), ( x y ), ( 2 x 2 y ), ( 1 x 1 y y y x x s t y y y y x x x x ), ( 3 x 3 y

60 6/94 Intersecting Two Parametric Lines ), ( x y ), ( 2 x 2 y ), ( 1 x 1 y y y x x s t y y y y x x x x ), ( 3 x 3 y Substitute t or s back into equation to find intersection

61 Liang-Barsky: Classifying Lines p(t) p() p(1) x1 x moving out 61/94

62 Liang-Barsky: Classifying Lines p(t) p(1) p() x x 1 moving in 62/94

63 Liang-Barsky: Classifying Lines p(t) p() p(1) x1 x moving in 63/94

64 Liang-Barsky: Classifying Lines p(t) p(1) p() x x 1 moving out 64/94

65 Liang-Barsky: Classifying Lines p(t) p() p(1) y1 y moving in 65/94

66 Liang-Barsky: Classifying Lines p(t) p(1) p() y y 1 moving out 66/94

67 Liang-Barsky: Classifying Lines p(t) p() p(1) y1 y moving out 67/94

68 Liang-Barsky: Classifying Lines p(t) p(1) p() y y 1 moving in 68/94

69 Liang-Barsky Algorithm p(t) p() p(1) 69/94

70 Liang-Barsky Algorithm p(t) p() p(1) 7/94

71 Liang-Barsky Algorithm p(t) p() p(1) 71/94

72 Liang-Barsky Algorithm p(t) p() p(.8) 72/94

73 Liang-Barsky Algorithm p() p(1) 73/94

74 Liang-Barsky Algorithm p(1) p() 74/94

75 Liang-Barsky Algorithm p() p(1) 75/94

76 Liang-Barsky Algorithm p() p(.7) p(1) 76/94

77 Liang-Barsky Algorithm p(1) p() 77/94

78 Liang-Barsky Algorithm p(1) p(.2) 78/94

79 Liang-Barsky Algorithm p(1) p() 79/94

80 Liang-Barsky Algorithm p(1) p() 8/94

81 Liang-Barsky Algorithm p(1) p(.1) 81/94

82 Liang-Barsky Algorithm p(1) p(.1) 82/94

83 Liang-Barsky Algorithm p(.8) p(.1) 83/94

84 Liang-Barsky Algorithm p(.8) p(.1) 84/94

85 Liang-Barsky Algorithm p(.8) p(.1) 85/94

86 Liang-Barsky Algorithm p() p(1) 86/94

87 Liang-Barsky Algorithm p() p(1) 87/94

88 Liang-Barsky Algorithm p() p(1) 88/94

89 Liang-Barsky Algorithm p() p(.6) 89/94

90 Liang-Barsky Algorithm p() p(.6) 9/94

91 Liang-Barsky Algorithm p() p(.6) 91/94

92 Liang-Barsky Algorithm p(.2) p(.6) 92/94

93 Comparison Cohen-Sutherland Repeated clipping is expensive Best used when trivial acceptance and rejection is possible for most lines Liang-Barsky Computation of t-intersections is cheap (only one division) Computation of (x,y) clip points is only done once Algorithm doesn t consider trivial accepts/rejects Best when many lines must be clipped 93/94

94 Line Clipping Considerations Just clipping end-points does not produce the correct results inside the window Must also update sum in midpoint algorithm Clipping against non-rectangular polygons is also possible but seldom used 94/94