Dr. Scott Schaefer Clipping Lines

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

3/94 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 min, y min ) (x max, y max ) (x,y)

4/94 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 min, y min ) (x max, y max ) (x,y) x min <=x<=x max ? y min <=y<=y max ?

5/94 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 1, y 1 ) (x 2, y 2 ) (x min, y min ) (x max, y max ) x min <=x<=x max ? y min <=y<=y max ?

6/94 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 min, y min ) (x max, y max ) x min <=x 1 <=x max Yes y min <=y 1 <=y max Yes (x 1, y 1 ) (x 2, y 2 )

7/94 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 min, y min ) (x max, y max ) x min <=x 2 <=x max No y min <=y 2 <=y max Yes (x 1, y 1 ) (x 2, y 2 )

8/94 Clipping Lines

9/94 Clipping Lines

10/94 Clipping Lines Given a line with end-points (x 0, y 0 ), (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 0, y 0 ) (x 1, y 1 ) (x min, y min ) (x max, y max )

11/94 Clipping Lines

12/94 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

13/94 Clipping Lines – Simple Algorithm

14/94 Clipping Lines – Simple Algorithm

15/94 Clipping Lines – Simple Algorithm

16/94 Clipping Lines – Simple Algorithm

17/94 Clipping Lines – Simple Algorithm

18/94 Clipping Lines – Simple Algorithm

19/94 Clipping Lines – Simple Algorithm

20/94 Clipping Lines – Simple Algorithm

21/94 Clipping Lines – Simple Algorithm

22/94 Clipping Lines – Simple Algorithm

23/94 Clipping Lines – Simple Algorithm

24/94 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 )

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

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

28/94 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 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

30/94 Cohen-Sutherland Algorithm Classify p 0, p 1 using region codes c 0, c 1 If, trivially reject If, trivially accept Otherwise reduce to trivial cases by splitting into two segments

31/94 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 4bit 3bit 2bit 1 leftrightbottomtop

32/94 Cohen-Sutherland Algorithm

33/94 Cohen-Sutherland Algorithm Classify p 0, p 1 using region codes c 0, c 1 If, trivially reject If, trivially accept Otherwise reduce to trivial cases by splitting into two segments

34/94 Cohen-Sutherland Algorithm Classify p 0, p 1 using region codes c 0, c 1 If, trivially reject If, trivially accept Otherwise reduce to trivial cases by splitting into two segments

35/94 Cohen-Sutherland Algorithm Classify p 0, p 1 using region codes c 0, c 1 If, trivially reject If, trivially accept Otherwise reduce to trivial cases by splitting into two segments Line is outside the window! reject

36/94 Cohen-Sutherland Algorithm Classify p 0, p 1 using region codes c 0, c 1 If, trivially reject If, trivially accept Otherwise reduce to trivial cases by splitting into two segments Line is inside the window! draw

37/94 Cohen-Sutherland Algorithm Classify p 0, p 1 using region codes c 0, c 1 If, trivially reject If, trivially accept Otherwise reduce to trivial cases by splitting into two segments

38/94 Cohen-Sutherland Algorithm

39/94 Cohen-Sutherland Algorithm

40/94 Cohen-Sutherland Algorithm

41/94 Cohen-Sutherland Algorithm

42/94 Cohen-Sutherland Algorithm

43/94 Cohen-Sutherland Algorithm

44/94 Cohen-Sutherland Algorithm

45/94 Cohen-Sutherland Algorithm

46/94 Cohen-Sutherland Algorithm

47/94 Cohen-Sutherland Algorithm

48/94 Cohen-Sutherland Algorithm

49/94 Cohen-Sutherland Algorithm

50/94 Cohen-Sutherland Algorithm

51/94 Cohen-Sutherland Algorithm

52/94 Cohen-Sutherland Algorithm

53/94 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

54/94 Liang-Barsky Algorithm Initialize interval to [t min, t max ]=[0,1] 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

55/94 Intersecting Two Parametric Lines

56/94 Intersecting Two Parametric Lines

57/94 Intersecting Two Parametric Lines

58/94 Intersecting Two Parametric Lines

59/94 Intersecting Two Parametric Lines

60/94 Intersecting Two Parametric Lines Substitute t or s back into equation to find intersection

61/94 Liang-Barsky: Classifying Lines

62/94 Liang-Barsky: Classifying Lines

63/94 Liang-Barsky: Classifying Lines

64/94 Liang-Barsky: Classifying Lines

65/94 Liang-Barsky: Classifying Lines

66/94 Liang-Barsky: Classifying Lines

67/94 Liang-Barsky: Classifying Lines

68/94 Liang-Barsky: Classifying Lines

69/94 Liang-Barsky Algorithm

70/94 Liang-Barsky Algorithm

71/94 Liang-Barsky Algorithm

72/94 Liang-Barsky Algorithm

73/94 Liang-Barsky Algorithm

74/94 Liang-Barsky Algorithm

75/94 Liang-Barsky Algorithm

76/94 Liang-Barsky Algorithm

77/94 Liang-Barsky Algorithm

78/94 Liang-Barsky Algorithm

79/94 Liang-Barsky Algorithm

80/94 Liang-Barsky Algorithm

81/94 Liang-Barsky Algorithm

82/94 Liang-Barsky Algorithm

83/94 Liang-Barsky Algorithm

84/94 Liang-Barsky Algorithm

85/94 Liang-Barsky Algorithm

86/94 Liang-Barsky Algorithm

87/94 Liang-Barsky Algorithm

88/94 Liang-Barsky Algorithm

89/94 Liang-Barsky Algorithm

90/94 Liang-Barsky Algorithm

91/94 Liang-Barsky Algorithm

92/94 Liang-Barsky Algorithm

93/94 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

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