1. CSCE 441 Computer Graphics: Clipping Lines Jinxiang Chai

2. OpenGL Geometric Primitives
All geometric primitives are specified by vertices
GL_LINE_STRIP
GL_LINE_LOOP
GL_POLYGON
GL_LINES
GL_POINTS
GL_TRIANGLE_STRIP
GL_QUAD_STRIP
GL_TRIANGLES
GL_QUADS
GL_TRIANGLE_FAN

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

4. Clipping Points
Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn
(xmax,ymax)
(x,y)
(xmin,ymin)

5. Clipping Points
Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn
(xmax,ymax)
(x,y)
xmin=<x<=xmax?
(xmin,ymin)
ymin=<y<=ymax?

6. Clipping Points
Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn
(xmax,ymax)
(x2,y2)
(x1,y1)
xmin=<x<=xmax?
(xmin,ymin)
ymin=<y<=ymax?

7. Clipping Points
Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn
(xmax,ymax)
(x2,y2)
(x1,y1)
xmin=<x1<=xmax Yes
(xmin,ymin)
ymin=<y1<=ymax Yes

8. Clipping Points
Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn
(xmax,ymax)
(x2,y2)
(x1,y1)
xmin=<x2<=xmax No
(xmin,ymin)
ymin=<y2<=ymax Yes

9. Clipping Lines

10. Clipping Lines

11. Clipping Lines Given a line with end-points (x0, y0), (x1, y1)
and clipping window (xmin, ymin), (xmax, ymax),
determine if line should be drawn and clipped end-points of line to draw.
(xmax,ymax)
(x1, y1)
(x0, y0)
(xmin,ymin)

12. Clipping Lines

13. Outline Simple line clipping algorithm Cohen-Sutherland Liang-Barsky
Required readings: HB 8-5, 8-6, 8-7 and 8-9

14. Clipping Lines

15. Clipping Lines – Simple Algorithm
If both end-points inside rectangle, draw line
If one end-point outside,
intersect line with all edges of rectangle
clip that point and repeat test

16. Clipping Lines – Simple Algorithm

17. Clipping Lines – Simple Algorithm

18. Intersecting Two Lines

19. Intersecting Two Lines

20. Intersecting Two Lines

21. Intersecting Two Lines

22. Intersecting Two Lines

23. Intersecting Two Lines
Substitute t or s back into equation to find intersection

24. Clipping Lines – Simple Algorithm

25. Clipping Lines – Simple Algorithm

26. Clipping Lines – Simple Algorithm

27. Clipping Lines – Simple Algorithm

28. Clipping Lines – Simple Algorithm

29. Clipping Lines – Simple Algorithm

30. Clipping Lines – Simple Algorithm

31. Clipping Lines – Simple Algorithm

32. Clipping Lines – Simple Algorithm

33. Clipping Lines – Simple Algorithm

34. Clipping Lines – Simple Algorithm

35. Clipping Lines – Simple Algorithm

36. Clipping Lines – Simple Algorithm

37. Clipping Lines – Simple Algorithm

38. Clipping Lines – Simple Algorithm

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

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

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

42. Trivial Rejects How can we know a line is outside of the window
Answer:

43. Trivial Rejects How can we know a line is outside of the window
Answer:
Wrong side

44. 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

45. Cohen-Sutherland Algorithm
Classify p0, p1 using region codes c0, c1
If , trivially reject
If , trivially accept
Otherwise reduce to trivial cases by splitting into two segments

46. 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 edges
bit 4
bit 3
bit 2
bit 1
top
bottom
right
left

47. Cohen-Sutherland Algorithm
bit 4
bit 3
bit 2
bit 1
top
bottom
right
left
top
right
left
Region code?
bottom

48. Cohen-Sutherland Algorithm
bit 4
bit 3
bit 2
bit 1
top
bottom
right
left
top
right
left
0010
bottom ?

49. Cohen-Sutherland Algorithm
bit 4
bit 3
bit 2
bit 1
top
bottom
right
left
top
right
left
0010
bottom
0100

50. Cohen-Sutherland Algorithm

51. Cohen-Sutherland Algorithm

52. Cohen-Sutherland Algorithm
Classify p0, p1 using region codes c0, c1
If , trivially reject because both points located the wrong side of one edge.
If , trivially accept because both points are not in the wrong side of any edge.
Otherwise reduce to trivial cases by splitting into two segments

53. Cohen-Sutherland Algorithm
Bitwise and

54. Cohen-Sutherland Algorithm
Bitwise and
Bitwise or

55. Cohen-Sutherland Algorithm
Bitwise and
Bitwise or

56. Cohen-Sutherland Algorithm
Classify p0, p1 using region codes c0, c1
If , trivially reject
If , trivially accept
Otherwise reduce to trivial cases by splitting into two segments

57. Cohen-Sutherland Algorithm
Classify p0, p1 using region codes c0, c1
If , trivially reject
If , trivially accept
Otherwise reduce to trivial cases by splitting into two segments
Line is outside the window! reject

58. Cohen-Sutherland Algorithm
Classify p0, p1 using region codes c0, c1
If , trivially reject
If , trivially accept
Otherwise reduce to trivial cases by splitting into two segments
Line is outside the window! reject

59. Cohen-Sutherland Algorithm
Classify p0, p1 using region codes c0, c1
If , trivially reject
If , trivially accept
Otherwise reduce to trivial cases by splitting into two segments
Line is inside the window! draw

60. Cohen-Sutherland Algorithm
Classify p0, p1 using region codes c0, c1
If , trivially reject
If , trivially accept
Otherwise reduce to trivial cases by splitting into two segments

61. Window Intersection
(x1, y1), (x2, y2) intersect with vertical edge at xright
yintersect = y1 + m(xright – x1)
where m=(y2-y1)/(x2-x1)
(x1, y1), (x2, y2) intersect with horizontal edge at ybottom
xintersect = x1 + (ybottom – y1)/m

62. Example 1

63. Example 1

64. Example 1

65. Example 1

66. Example 2

67. Example 2

68. Example 2

69. Example 2

70. Example 2

71. Example 2

72. Example 2

73. Example 3

74. Example 3

75. Example 3

76. Example 3

77. Cohen-Sutherland Algorithm
Extends easily to 3D line clipping
27 regions
6 bits

78. Cohen-Sutherland Algorithm
Use region codes to quickly eliminate/include lines
Best algorithm when trivial accepts/rejects are common
Must compute viewing window clipping of remaining lines
Non-trivial clipping cost
More efficient algorithms exist

79. Liang-Barsky Algorithm
Parametric definition of a line:
x = x1 + uΔx
y = y1 + uΔy
Δx = (x2-x1), Δy = (y2-y1), 0<=u<=1
Lines are oriented: classify lines as moving inside to out or outside to in
Goal: find range of u for which x and y both inside the viewing window

80. Liang-Barsky Algorithm
Current clipped line (u1=0,u2=1)
(2,1)
(-.5,-.5)

81. Liang-Barsky Algorithm
Current clipped line (u1=0.333,u2=0.6)
(2,1)
(-.5,-.5)

82. Liang-Barsky Algorithm
For lines starting outside of boundary, update its starting point (u1)
For lines starting inside of boundary, update end point (u2)
For lines paralleling the boundaries and outside window, reject it.

83. Liang-Barsky Algorithm
For lines starting outside of boundary, update its starting point (u1)
For lines starting inside of boundary, update end point (u2)
For lines paralleling the boundaries and outside window, reject it.

84. Liang-Barsky Algorithm
Mathematically:
xmin <= x1 + uΔx <= xmax
ymin <= y1 + uΔy <= ymax
Rearranged
1: u*(-Δx) <= (x1 – xmin)
2: u*(Δx) <= (xmax – x1)
3: u*(-Δy) <= (y1 – ymin)
4: u*(Δy) <= (ymax – y1)
gen: u*(pk) <= (qk), k=1,2,3,4

85. Liang-Barsky Algorithm
Mathematically:
xmin <= x1 + uΔx <= xmax
ymin <= y1 + uΔy <= ymax
Rearranged
1: u*(-Δx) <= (x1 – xmin) // left edge
2: u*(Δx) <= (xmax – x1)
3: u*(-Δy) <= (y1 – ymin)
4: u*(Δy) <= (ymax – y1)
gen: u*(pk) <= (qk), k=1,2,3,4

86. Liang-Barsky Algorithm
Mathematically:
xmin <= x1 + uΔx <= xmax
ymin <= y1 + uΔy <= ymax
Rearranged
1: u*(-Δx) <= (x1 – xmin)
2: u*(Δx) <= (xmax – x1) // right edge
3: u*(-Δy) <= (y1 – ymin)
4: u*(Δy) <= (ymax – y1)
gen: u*(pk) <= (qk), k=1,2,3,4

87. Liang-Barsky Algorithm
Mathematically:
xmin <= x1 + uΔx <= xmax
ymin <= y1 + uΔy <= ymax
Rearranged
1: u*(-Δx) <= (x1 – xmin)
2: u*(Δx) <= (xmax – x1)
3: u*(-Δy) <= (y1 – ymin) // bottom edge
4: u*(Δy) <= (ymax – y1)
gen: u*(pk) <= (qk), k=1,2,3,4

88. Liang-Barsky Algorithm
Mathematically:
xmin <= x1 + uΔx <= xmax
ymin <= y1 + uΔy <= ymax
Rearranged
1: u*(-Δx) <= (x1 – xmin)
2: u*(Δx) <= (xmax – x1)
3: u*(-Δy) <= (y1 – ymin)
4: u*(Δy) <= (ymax – y1) // top edge
gen: u*(pk) <= (qk), k=1,2,3,4

89. Liang-Barsky Algorithm
Mathematically:
xmin <= x1 + uΔx <= xmax
ymin <= y1 + uΔy <= ymax
Rearranged
1: u*(-Δx) <= (x1 – xmin)
2: u*(Δx) <= (xmax – x1)
3: u*(-Δy) <= (y1 – ymin)
4: u*(Δy) <= (ymax – y1)
Gen: u*(pk) <= (qk), k=1,2,3,4

90. Liang-Barsky Algorithm
Rules:
pk = 0: the line is parallel to boundaries
If for that same k, qk < 0, it’s outside
Otherwise it’s inside
pk < 0: the line starts outside this boundary
rk = qk/pk
u1 = max(0, rk, u1) /**update starting point**/
pk > 0: the line starts inside the boundary
u2 = min(1, rk, u2) /** update end point**/
If u1 > u2, the line is completely outside

91. Liang-Barsky Algorithm
Current clipped line (u1=0,u2=1)
(2,1)
(-.5,-.5)

92. Liang-Barsky Algorithm
Current clipped line (u1=0,u2=1)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p1 = -Δx =-2.5<0
(-.5,-.5)
q1 = (x1-xmin)=-.5
r1 = q1/p1=0.2

93. Liang-Barsky Algorithm
Current clipped line (u1=0,u2=1)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p1 = -Δx =-2.5<0
(-.5,-.5)
q1 = (x1-xmin)=-.5
line starts outside this boundary
r1 = q1/p1=0.2

94. Liang-Barsky Algorithm
Current clipped line (u1=0,u2=1)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p1 = -Δx =-2.5<0
(-.5,-.5)
q1 = (x1-xmin)=-.5
r1 = q1/p1=0.2

95. Liang-Barsky Algorithm
Current clipped line (u1=0,u2=1)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p1 = -Δx =-2.5<0
(-.5,-.5)
q1 = (x1-xmin)=-.5
r1 = q1/p1=0.2

96. Liang-Barsky Algorithm
Current clipped line (u1=0,u2=1)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p1 = -Δx =-2.5<0
(-.5,-.5)
q1 = (x1-xmin)=-.5
u1 = max(0, r1, u1)
r1 = q1/p1=0.2

97. Liang-Barsky Algorithm
Current clipped line (u1=0.2,u2=1)
(2,1)
u1<u2?
yes: continue
no: the line is completely outside window
(-.5,-.5)

98. Liang-Barsky Algorithm
Current clipped line (u1=0.2,u2=1)
(2,1)
(-.5,-.5)

99. Liang-Barsky Algorithm
Current clipped line (u1=0.2,u2=1)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p2 = Δx =2.5>0
(-.5,-.5)
q2 = (xmax-x1)=1.5
r2 = q2/p2=0.6

100. Liang-Barsky Algorithm
Current clipped line (u1=0.2,u2=1)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p2 = Δx =2.5>0
(-.5,-.5)
q2 = (xmax-x1)=1.5
u2 = min(1, r2, u2)
r2 = q2/p2=0.6

101. Liang-Barsky Algorithm
Current clipped line (u1=0.2,u2=0.6)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p2 = Δx =2.5>0
(-.5,-.5)
q2 = (xmax-x1)=1.5
r2 = q2/p2=0.6

102. Liang-Barsky Algorithm
Current clipped line (u1=0.2,u2=0.6)
(2,1)
u1<u2?
yes: continue
no: the line is completely outside window
(-.5,-.5)

103. Liang-Barsky Algorithm
Current clipped line (u1=0.2,u2=0.6)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p3 = -Δy =-1.5<0
(-.5,-.5)
q3 = (y1-ymin)=0.5
r3 = q3/p3=0.333

104. Liang-Barsky Algorithm
Current clipped line (u1=0.2,u2=0.6)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p3 = -Δy =-1.5<0
(-.5,-.5)
q3 = (y1-ymin)=0.5
u1 = max(0, r3, u1)
r3 = q3/p3=0.333

105. Liang-Barsky Algorithm
Current clipped line (u1=0.333,u2=0.6)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p3 = -Δy =-1.5<0
(-.5,-.5)
q3 = (y1-ymin)=0.5
r3 = q3/p3=0.333

106. Liang-Barsky Algorithm
Current clipped line (u1=0.333,u2=0.6)
(2,1)
(-.5,-.5)

107. Liang-Barsky Algorithm
Current clipped line (u1=0.333,u2=0.6)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p3 = -Δy =-1.5<0
(-.5,-.5)
q3 = (y1-ymin)=0.5
r3 = q3/p3=0.333

108. Liang-Barsky Algorithm
Current clipped line (u1=0.333,u2=0.6)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p4 = Δy =1.5>0
(-.5,-.5)
q4 = (ymax-y1)=1.5
r4 = q4/p4=1

109. Liang-Barsky Algorithm
Current clipped line (u1=0.333,u2=0.6)
(2,1)
Δx = (x2-x1)=2.5
Δy = (y2-y1)=1.5
p4 = Δy =1.5>0
(-.5,-.5)
q4 = (ymax-y1)=1.5
u2 = min(1, r4, u2)
r4 = q4/p4=1

110. Liang-Barsky Algorithm
Check the left edge (u1=0.333,u2=0.6)
(2,1)
u1<u2
(-.5,-.5)
xnew1=x1+Δx*u1
xnew2=x2+Δx*u2
ynew1=y1+Δy*u1
ynew2=y2+Δy*u2

111. Liang-Barsky Algorithm
Faster than Cohen-Sutherland
Extension to 3D is easy
- Parametric representation for 3D lines
- Compute u1,u2 based on the intersection between line and plane

112. Comparison Cohen-Sutherland Liang-Barsky
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

113. Curve Clipping

114. Curve Clipping

115. Curve Clipping
Approximate a curve using a set of straight-line segments
Apply line clipping for curve clipping

116. Next Lecture
Polygon fill-area clipping

117. Next Lecture
Polygon fill-area clipping