1. CSCE 441 Computer Graphics: Clipping Polygons Jinxiang Chai

2. Review: 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?

3. Review: Clipping Lines

4. Review: Clipping Lines

5. Review Clipping Simple line clipping algorithm
- compute intersection points
- clip the line segment between the outside point and intersection point
Cohen-Sutherland
- region coding (9 regions, 4bits)
- 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

6. Clipping Polygons
Clipping polygons is more complex than clipping the individual lines
- Input: polygon

7. Clipping Polygons
Clipping polygons is more complex than clipping the individual lines
- Input: polygon
- Output: original polygon, new polygon, or nothing

8. Why Is Polygon Clipping Hard?
What happens to a triangle during clipping?
- possible outcomes:
triangle->triangle
triangle->quad
triangle->5-gon

9. Why Is Polygon Clipping Hard?
What happens to a triangle during clipping?
- possible outcomes:
triangle->triangle
triangle->quad
triangle->5-gon
How many sides a clipped triangle might have? 9 9

10. How Many Sides?
Seven

11. Why Is Clipping Hard? A really tough case
Concave polygon -> Multiple polygons

12. Why Is Clipping Hard?
A really tough case

13. Outline Sutherland-Hodgman Clipping - convex polygons
Weiler-Atherton Algorithm
- general polygons
Required readings
- HB 8-8

14. Any idea?
Clipping polygon edges? 14 14

15. Any idea?
Clipping polygon edges?

16. Any idea?
Clipping polygon edges? 16 16

17. Any idea?
Clipping polygon edges? 17 17

18. Any idea?
Clipping polygon edges?
Unconnected set of lines!

19. Any idea? Clipping polygon edges?
Wrong even with connected line segments! 19 19

20. Sutherland-Hodgman Clipping
Basic idea: similar to line clipping

21. Sutherland-Hodgman Clipping
Basic idea: similar to line clipping
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation 21 21

22. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation

23. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation

24. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation

25. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation

26. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation

27. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation

28. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation

29. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation

30. Sutherland-Hodgman Clipping
Basic idea:
- consider each edge of the view window individually
- clip the polygon against the view window edge’s equation
- after doing all edges, the polygon is fully clipped

31. Sutherland-Hodgman Clipping
Input/output for algorithm:
- input: list of polygon vertices in order
{abcdef} a f b e d c

32. Sutherland-Hodgman Clipping
Input/output for algorithm:
- input: list of polygon vertices in order
- output: list of clipped polygon vertices consisting of old vertices (maybe) and new vertices (maybe)

33. Sutherland-Hodgman Clipping
Input/output for algorithm:
- input: list of polygon vertices in order
- output: list of clipped polygon vertices consisting of old vertices (maybe) and new vertices (maybe)
- exactly what expect from the clipping operation against each edge!

34. Sutherland-Hodgman Clipping
How to clip the polygon against the window edge?
- Go around polygon one vertex (i.e. polygon edge) at a time
- Current vertex has position E
- Previous vertex had position S, and it has been added to the output if appropriate a f b e d
{abcdef} c

35. Sutherland-Hodgman Clipping
How to clip the polygon against the window edge?
- Go around polygon one vertex (i.e. polygon edge) at a time
- Current vertex has position E
- Previous vertex had position S, and it has been added to the output if appropriate a f b e d
{abcdef} c

36. Sutherland-Hodgman Clipping
How to clip the polygon against the window edge?
- Go around polygon one vertex (i.e. polygon edge) at a time
- Current vertex has position E
- Previous vertex had position S, and it has been added to the output if appropriate a f b e d
{abcdef} c
Need to determine output vertices for each polygon line! 36 36

37. Sutherland-Hodgman Clipping
Polygon edge from S to E takes one of four cases:
inside
outside
inside
outside
inside
outside
inside
outside E S E S E S E S

38. Sutherland-Hodgman Clipping
Polygon edge from S to E takes one of four cases:
inside
outside
inside
outside
inside
outside
inside
outside E S E S E S E S
? output
? output
? output
? output 38 38

39. Sutherland-Hodgman Clipping
Polygon edge from S to E takes one of four cases:
inside
outside
inside
outside
inside
outside
inside
outside E S E S E S E S
I output E output
E output
I output
no output 39 39

40. Sutherland-Hodgman Clipping
Four cases:
S inside plane and E inside plane
Add E to output
Note: S has already been added
S inside plane and E outside plane
Find intersection point I
Add I to output
S outside plane and E outside plane
Add nothing
S outside plane and E inside plane
Add I to output, followed by E

41. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

42. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E
Input vertices
after clipping against left edge
after clipping against right edge
after clipping against bottom edge
after clipping all the window edges

43. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

44. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

45. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

46. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

47. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E b

48. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

49. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

50. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

51. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

52. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

53. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

54. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

55. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

56. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

57. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

58. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

59. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

60. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

61. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

62. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

63. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

64. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

65. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

66. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

67. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

68. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

69. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

70. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

71. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

72. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

73. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

74. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

75. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

76. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

77. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

78. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

79. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

80. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

81. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

82. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

83. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E

84. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E
Possible error for non-convex polygons

85. Sutherland-Hodgman Algorithm
Out In
No output
Output E
Output I
Output I,E
Possible error for non-convex polygons
#1: split concave polygon into two or more convex polygons (see Section 4-7)
#2: use more general polygon clipper. 85 85

86. Sutherland-Hodgman Algorithm
How to deal with more general clipping boundaries (convex clip polygons) 86 86

87. Sutherland-Hodgman Algorithm
How to deal with more general clipping boundaries (convex clip polygons) 87 87

88. Sutherland-Hodgman Algorithm
Work for convex input polygons
Work for convex clipping boundaries
Easy to pipeline for parallel processing
Polygon from one boundary does not have to be completed before next boundary starts
Leads to substantial performance gains 88 88

89. Weiler-Atherton Algorithm
+ Work for general input polygons (concave & convex)
+ Handles a clipping window with any polygon shape (concave & convex)
+ Can be extended to 3D
- Not as efficient as Sutherland-Hodgman
- Not easy to parallelize

90. Weiler-Atherton Algorithm
Key idea: tracing around the perimeter of the fill polygon searching for the borders that enclose a clipped fill region.

91. Weiler-Atherton Algorithm
Advantage: multiple regions can be identified and displayed as separate, unconnected polygons. 91 91

92. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order 92 92

93. Weiler-Atherton Algorithm

94. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

95. Weiler-Atherton Algorithm

96. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

97. Weiler-Atherton Algorithm

98. Weiler-Atherton Algorithm

99. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

100. Weiler-Atherton Algorithm

101. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

102. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

103. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

104. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

105. Weiler-Atherton Algorithm

106. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

107. Weiler-Atherton Algorithm

108. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

109. Weiler-Atherton Algorithm

110. Weiler-Atherton Algorithm

111. Weiler-Atherton Algorithm

112. Weiler-Atherton Algorithm

113. Weiler-Atherton Algorithm

114. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

115. Weiler-Atherton Algorithm

116. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

117. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

118. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

119. Weiler-Atherton Algorithm

120. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

121. Weiler-Atherton Algorithm

122. Weiler-Atherton Algorithm
Process the edges of the polygon fill area in a counterclockwise order until an inside-outside pair of vertices is encountered
Follow the window boundaries in a counterclockwise direction from the exit-intersection point to another intersection point with the polygon
- previously processed?
- yes, go to the next step
- no, continue processing polygon until a previously processed vertex is encountered
Form the vertex list for this section of the clipped fill area
Return to the exit-intersection point and continue processing the polygon edges in a counterclockwise order

123. Weiler-Atherton Algorithm

124. Weiler-Atherton Algorithm

125. Nonlinear Clipping-window Boundaries
Approximate the boundaries with straight-line sections.
Use the existing polygon clipping algorithms for clipping against a general polygon-shaped clipping window.

126. 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

127. Summary Polygon drawing Polygon clipping
You need to implement both algorithms in assignment #3.

128. Next Lecture
2D transformations y y x x y x

129. Next Lecture
2D transformations
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