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. Review: Drawing General Polygons
How to draw every interior pixels? ?
Scanline conversion of polygons!

4. Review: Curved Boundaries
How to deal with curved boundaries?
Boundary fill algorithm!

5. Review: How to deal with this?
Multiple color boundaries?
Flood fill algorithm!

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

7. 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)

8. 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?

9. 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?

10. 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

11. 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 No

12. Clipping Lines

13. Clipping Lines

14. 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)

15. Clipping Lines

16. Outline
Simple line clipping algorithm
Cohen-Sutherland
Liang-Barsky

17. Clipping Lines

18. 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

19. Clipping Lines – Simple Algorithm

20. Clipping Lines – Simple Algorithm

21. Intersecting Two Lines

22. Intersecting Two Lines

23. Intersecting Two Lines

24. Intersecting Two Lines

25. Intersecting Two Lines

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

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

40. Clipping Lines – Simple Algorithm

41. Clipping Lines – Simple Algorithm

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

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

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

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

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

47. 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

48. 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

49. 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

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

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

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

53. Cohen-Sutherland Algorithm

54. Cohen-Sutherland Algorithm

55. 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

56. Cohen-Sutherland Algorithm
Bitwise and

57. Cohen-Sutherland Algorithm
Bitwise and
Bitwise or

58. Cohen-Sutherland Algorithm
Bitwise and
Bitwise or

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

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
Line is outside the window! reject

61. 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

62. 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

63. 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

64. 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

65. Example 1

66. Example 1

67. Example 1

68. Example 1

69. Example 2

70. Example 2

71. Example 2

72. Example 2

73. Example 2

74. Example 2

75. Example 2

76. Example 3

77. Example 3

78. Example 3

79. Example 3

80. Cohen Sutherland
Java applet: click here

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

82. 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

83. 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

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

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

86. 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.

87. 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.

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)
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) // 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

90. 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

91. 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

92. 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

93. 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

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

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
line starts outside this boundary
r1 = q1/p1=0.2

97. 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

98. 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

99. 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

100. 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)

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

102. 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

103. 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

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
p2 = Δx =2.5>0
(-.5,-.5)
q2 = (xmax-x1)=1.5
r2 = q2/p2=0.6

105. 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)

106. 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

107. 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

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
p3 = -Δy =-1.5<0
(-.5,-.5)
q3 = (y1-ymin)=0.5
r3 = q3/p3=0.333

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

110. 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

111. 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

112. 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

113. 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

114. 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

115. 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

116. Curve Clipping

117. Curve Clipping

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

119. Next Lecture
Polygon fill-area clipping

120. Next Lecture
Polygon fill-area clipping