2. Unit – IV 2D Viewing.

3. 2 of 30 Contents Windowing Concepts The viewing pipeline viewing coordinate reference frame, window to view-port coordinate transformation, viewing functions Line Clipping –Introduction –Cohen-Sutherland Clipping Algorithm –Cyrus-beck line clipping algorithms Polygon Clipping –Sutherland-Hodgman Area Clipping Algorithm

4. 3 of 30 Where we require clipping? An architect may have a graphics program to draw an entire building but be interested in only the swimming pool.

5. 4 of 30 Viewing in 2D - Viewport Window in world coordinates. 45  250  Viewport in Device coords 250 x 250 Pixels. Display Device

6. 5 of 30 The computer is used in design applications because it can easily and accurately create, store and modify very complex drawings. When drawings are too complex, however, they become difficult to read. In such situations it is useful to display only those portions of the drawing that are of immediate interest. Sometimes, it is desirable to enlarge these portions to take full advantage of the available display surface. The method for selecting and enlarging portions of a drawing is called windowing. The technique for not showing the part of the drawing which one is not interested is called a clipping.

7. 6 of 30 Windowing I A scene is made up of a collection of objects specified in world coordinates World Coordinates

8. 7 of 30 Window A world coordinate area selected for display is called a window. wy max wy min wx min wx max Window World Coordinates

9. 8 of 30 View port An area on a display device to which a window is mapped is called a view port. Vy max Vy min Vx min Vx max Device Coordinates View port

10. 9 of 30 VIDEOCON Display Screen View Port

11. 10 of 30 The window defines what is to be viewed. And the view port defines where it is to be displayed. Window and view ports are rectangles in standard positions, with rectangle edges parallel to the coordinate axes.

12. 11 of 30 Window to viewport mapping or windowing transformation. The mapping of a part of a world coordinate scene to device coordinates is referred to as a viewing transformation. 2-d viewing transformation is also referred as window to view port or windowing transformation. Y w max Y w min X w min X w max window Y v max Y v min X v min X v max View port

13. 12 of 30 The Viewing Pipeline Window A world-coordinate area selected for display. defines what is to be viewed Viewport An area on a display device to which a window is mapped. defines where it is to be displayed Viewing transformation The mapping of a part of a world-coordinate scene to device coordinates. A window could be a rectangle to have any orientation.

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16. 15 of 30 2D Viewing Pipeline To make the viewing process independent of the requirements of any output device, graphics systems convert object descriptions to normalized coordinates range from 0 to 1

17. 16 of 30 Window to viewing coordinate transformation  First, we construct the scene in world coordinates using the output primitives and attributes.  Next, to obtain a particular orientation for the window, we can set up a 2-dimenstional viewing coordinate system in the world coordinate plane and define a window in the viewing coordinate system.  The viewing coordinate reference frame is used to provide a method for setting up arbitrary orientation of rectangular windows.  Once the viewing reference frame is established, we can transform descriptions in world coordinates to viewing coordinates( in the range from 0 to 1) and map viewing coordinate description of the scene to normalized coordinates.  At the final step, all parts of the picture that lie outside the view port are clipped and the contents of the view port are transferred to device coordinates.

18. 17 of 30 Cntd.. By changing the position of the view port, we can view objects at different positions on the display area of an output device. Also, by varying the size of view port, we can change the size and proportions of displayed objects. We achieve zooming effects by successively mapping different sized windows on a fixed size view port. As the windows are made smaller, we zoom in on some part of a scene to view details that are not shown with larger windows. Similarly, more overview is obtained by zooming out from a section of a scene with successively larger windows.

19. 18 of 30 The Viewing Pipeline M wc,vc = R. T

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22. 21 of 30 (0,0) (100,50) (0,25) (50,50) So S x = 50/100 = ½ And S y = 25/50 = ½ XW min,YW min XW max,YW max XV min,YV min XV max,YV max XV max -XV min XW max -XW min Sx=Sx= YV max -YV min YW max -YW min Sy=Sy=

23. 22 of 30 Window to View port coordinate Transformation

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25. 24 of 30 Window-to-Viewport Coordinate Transformation To map a point in the window at world position (xw, yw ) in the associated view port (xv, yv ) we use the concept of relative distance.

26. 25 of 30 Workstation transformation Mapping selected parts of a scene in normalized coordinates to different video monitors with workstation transformations

27. 26 of 30 Clipping Operations Clipping Identify those portions of a picture that are either inside or outside of a specified region of space. Clip window The region against which an object is to be clipped. The shape of clip window Applications of clipping World-coordinate clipping

28. 27 of 30 Clipping Operations Viewport clipping It can reduce calculations by allowing concatenation of viewing and geometric transformation matrices. Types of clipping Point clipping Line clipping Area (Polygon) clipping Curve clipping Text clipping Point clipping (Rectangular clip window)

29. 28 of 30 Point clipping Point clipping: Remove points outside window. –A point is either entirely inside the region or not Assuming that the clip window is a rectangle in standard position, we save a point p=(x,y) for display if the following inequalities are satisfied: Where the edges of the clip window ( xw min, xw max, yw min, yw max ) can be either world coordinate window boundaries or viewport boundaries. If any one of these four inequalities is not satisfied, the point is clipped (not saved for display)

30. 29 of 30 For a point (x,y) to be inside the clip rectangle: Point Clipping

31. 30 of 30 Line Clipping Basic calculations: –Is an endpoint inside or outside the clipping window? –Find the point of intersection, if any, between a line segment and an edge of the clipping window. Both endpoints inside: trivial accept One inside: find intersection and clip Both outside: either clip or reject

32. 31 of 30 Line Clipping For the image below consider which lines and points should be kept and which ones should be clipped wy max wy min wx min wx max Window P1P1 P2P2 P3P3 P6P6 P5P5 P7P7 P 10 P9P9 P4P4 P8P8

33. 32 of 30 Point Clipping Easy - a point ( x,y ) is not clipped if: wx min ≤ x ≤ wx max AND wy min ≤ y ≤ wy max otherwise it is clipped wy max wy min wx min wx max Window P1P1 P2P2 P5P5 P7P7 P 10 P9P9 P4P4 P8P8 Clipped Points Within the Window are Not Clipped Clipped

34. 33 of 30 Line Clipping Harder - examine the end-points of each line to see if they are in the window or not SituationSolutionExample Both end-points inside the window Don’t clip One end-point inside the window, one outside Must clip Both end-points outside the window Must clip

35. 34 of 30 Cohen-Sutherland Clipping Algorithm An efficient line clipping algorithm The key advantage of the algorithm is that it vastly reduces the number of line intersections that must be calculated Dr. Ivan E. Sutherland co-developed the Cohen- Sutherland clipping algorithm. Sutherland is a graphics giant and includes amongst his achievements the invention of the head mounted display.

36. 35 of 30 Cases for clipping lines Line Clipping

37. 36 of 30 Cases for clipping lines Line Clipping

38. 37 of 30 Cases for clipping lines Line Clipping

39. 38 of 30 Cases for clipping lines Line Clipping

40. 39 of 30 Cases for clipping lines Line Clipping

41. 40 of 30 Cases for clipping lines Line Clipping

42. 41 of 30 Cohen-Sutherland: World Division World space is divided into regions based on the window boundaries –Each region has a unique four bit region code –Region codes indicate the position of the regions with respect to the window 100110001010 0001 0000 Window 0010 010101000110

43. 42 of 30 1.End-points pairs are checked for trivial acceptance or rejection using outcode; 2.If not trivially accepted or rejected, divide the line segment into two at a clip edge; 3.Iteratively clipped by test trivial-acceptance or trivial-rejection, and divided into two segments until completely inside or trivial-rejection. A B C D E F G H I 1001 0001 0101 0100 0000 10001010 0010 0110 Steps for Cohen-Sutherland alg.

44. 43 of 30 Cohen-Sutherland: Labelling Every end-point is labelled with the appropriate region code wy max wy min wx min wx max Window P 3 [0001] P 6 [0000] P 5 [0000] P 7 [0001] P 10 [0100] P 9 [0000] P 4 [1000] P 8 [0010] P 12 [0010] P 11 [1010] P 13 [0101] P 14 [0110]

45. 44 of 30 Cohen-Sutherland: Lines In The Window Lines completely contained within the window boundaries have region code [0000] for both end-points so are not clipped wy max wy min wx min wx max Window P 3 [0001] P 6 [0000] P 5 [0000] P 7 [0001] P 10 [0100] P 9 [0000] P 4 [1000] P 8 [0010] P 12 [0010] P 11 [1010] P 13 [0101] P 14 [0110]

46. 45 of 30 Cohen-Sutherland: Lines Outside The Window Any lines with a common set bit in the region codes of both end-points can be clipped –The AND operation can efficiently check this wy max wy min wx min wx max Window P 3 [0001] P 6 [0000] P 5 [0000] P 7 [0001] P 10 [0100] P 9 [0000] P 4 [1000] P 8 [0010] P 12 [0010] P 11 [1010] P 13 [0101] P 14 [0110]

47. 46 of 30 Cohen-Sutherland: Other Lines Lines that cannot be identified as completely inside or outside the window may or may not cross the window interior These lines are processed as follows: –Compare an end-point outside the window to a boundary (choose any order in which to consider boundaries e.g. left, right, bottom, top) and determine how much can be discarded –If the remainder of the line is entirely inside or outside the window, retain it or clip it respectively

48. 47 of 30 Cohen-Sutherland: Other Lines (cont…) –Otherwise, compare the remainder of the line against the other window boundaries –Continue until the line is either discarded or a segment inside the window is found We can use the region codes to determine which window boundaries should be considered for intersection –To check if a line crosses a particular boundary we compare the appropriate bits in the region codes of its end-points –If one of these is a 1 and the other is a 0 then the line crosses the boundary

49. 48 of 30 Cohen-Sutherland Examples Consider the line P 9 to P 10 below –Start at P 10 –From the region codes of the two end-points we know the line doesn’t cross the left or right boundary –Calculate the intersection of the line with the bottom boundary to generate point P 10 ’ –The line P 9 to P 10 ’ is completely inside the window so is retained wy max wy min wx min wx max Window P 10 [0100] P 9 [0000] P 10 ’ [0000] P 9 [0000]

50. 49 of 30 Cohen-Sutherland Examples (cont…) Consider the line P 3 to P 4 below –Start at P 4 –From the region codes of the two end-points we know the line crosses the left boundary so calculate the intersection point to generate P 4 ’ –The line P 3 to P 4 ’ is completely outside the window so is clipped wy max wy min wx min wx max Window P 4 ’ [1001] P 3 [0001] P 4 [1000] P 3 [0001]

51. 50 of 30 Cohen-Sutherland Examples (cont…) Consider the line P 7 to P 8 below –Start at P 7 –From the two region codes of the two end-points we know the line crosses the left boundary so calculate the intersection point to generate P 7 ’ wy max wy min wx min wx max Window P 7 ’ [0001] P 7 [0001] P 8 [0010] P 8 ’ [0010]

52. 51 of 30 Cohen-Sutherland Examples (cont…) Consider the line P 7 ’ to P 8 –Start at P 8 –Calculate the intersection with the right boundary to generate P 8 ’ –P 7 ’ to P 8 ’ is inside the window so is retained wy max wy min wx min wx max Window P 7 ’ [0001] P 7 [0001] P 8 [0010] P 8 ’ [0010]

53. 52 of 30 Cohen-Sutherland Worked Example wy max wy min wx min wx max Window

54. 53 of 30 Calculating Line Intersections Intersection points with the window boundaries are calculated using the line- equation parameters –Consider a line with the end-points ( x 1, y 1 ) and ( x 2, y 2 ) –The y-coordinate of an intersection with a vertical window boundary can be calculated using: y = y 1 + m (x boundary - x 1 ) where x boundary can be set to either wx min or wx max

55. 54 of 30 Calculating Line Intersections (cont…) –The x-coordinate of an intersection with a horizontal window boundary can be calculated using: x = x 1 + (y boundary - y 1 ) / m where y boundary can be set to either wy min or wy max –m is the slope of the line in question and can be calculated as m = (y 2 - y 1 ) / (x 2 - x 1 )

56. 55 of 30 Initial Condition Sutherland-Hodgeman Algo. Clip Against Right Clipping Boundary Clip Against Top Clipping Boundary The Clipped Polygon Clip Against Bottom Clipping Boundary Clip Against Left Clipping Boundary

57. 56 of 30 Polygon Clipping v1 v2 v1’ out in Output: v1’ and v2 v1 v2 in Output: v2 v1v2 v1’ in out Output: v1’ v1 v2 out Output: none outside inside Sutherland-Hodgeman Polygon Clipping against a rectangle (or a window) For each stage, the boundary clipper divides the plane into an inside half-plane And outside half-plane Inside half-plane contains the clipping window; Outside half-plane doesn’t. Given an edge going from vertex V[i] to vertex V[i+1], during the transversal, there are four cases (assuming the V[i] has been checked in the previous edge):

58. 57 of 30 Case 1 4 Cases of Polygon Clipping Case 2 Case 3 Case 4

59. 58 of 30 Cohen-Sutherland – polygons s p InsideOutside s p InsideOutside s p InsideOutside s p InsideOutside 1 st Output i OutputNo Output i 2 nd Output Output

60. 59 of 30 Area Clipping Similarly to lines, areas must be clipped to a window boundary Consideration must be taken as to which portions of the area must be clipped

61. 60 of 30 Sutherland-Hodgman Area Clipping Algorithm A technique for clipping areas developed by Sutherland & Hodgman Put simply the polygon is clipped by comparing it against each boundary in turn Original AreaClip LeftClip RightClip BottomClip Top Sutherland turns up again. This time with Gary Hodgman with whom he worked at the first ever graphics company Evans & Sutherland

62. 61 of 30 Sutherland-Hodgman Area Clipping Algorithm (cont…) To clip an area against an individual boundary: –Consider each vertex in turn against the boundary –Vertices inside the boundary are saved for clipping against the next boundary –Vertices outside the boundary are clipped –If we proceed from a point inside the boundary to one outside, the intersection of the line with the boundary is saved –If we cross from the outside to the inside intersection point and the vertex are saved

63. 62 of 30 Sutherland-Hodgman Example Each example shows the point being processed (P) and the previous point (S) Saved points define area clipped to the boundary in question S P Save Point P S P Save Point I I P S No Points Saved S P Save Points I & P I

64. 63 of 30 Other Area Clipping Concerns Clipping concave areas can be a little more tricky as often superfluous lines must be removed Clipping curves requires more work –For circles we must find the two intersection points on the window boundary Window

65. 64 of 30 Summary Objects within a scene must be clipped to display the scene in a window Because there are can be so many objects clipping must be extremely efficient The Cohen-Sutherland algorithm can be used for line clipping The Sutherland-Hodgman algorithm can be used for area clipping

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67. 66 of 30 Cohen-Sutherland Clipping Algorithm VI Let’s consider the lines remaining below wy max wy min wx min wx max Window P 3 [0001] P 6 [0000] P 5 [0000] P 7 [0001] P 10 [0100] P 9 [0000] P 4 [1000] P 8 [0010] P 12 [0010] P 11 [1010] P 13 [0101] P 14 [0110]

68. 67 of 30 Cohen-Sutherland Clipping Algorithm Easy - a point ( x,y ) is not clipped if: wx min ≤ x ≤ wx max AND wy min ≤ y ≤ wy max otherwise it is clipped wy max wy min wx min wx max Window P1P1 P2P2 P5P5 P7P7 P 10 P9P9 P4P4 P8P8 Clipped Points Within the Window are Not Clipped Clipped

69. 68 of 30 Clipping Point clipping is easy: –For point ( x,y ) the point is not clipped if wx min ≤ x ≤ wx max AND wy min ≤ y ≤ wy max wy max wy min wx min wx max Before Clipping Window P1P1 P2P2 P3P3 P6P6 P5P5 P7P7 P 10 P9P9 P4P4 P8P8

70. 69 of 30 Cohen-Sutherland – polygons s p InsideOutside s p InsideOutside s p InsideOutside s p InsideOutside 1 st Output i OutputNo Output i 2 nd Output Output

71. Polygon Clipping.

72. 71 of 30 Ex: Simple Convex

73. 72 of 30 Ex: Multiple Components

74. 73 of 30 Nonconvex with Many Edges

75. 74 of 30 Polygons : Before Clipping

76. 75 of 30 Polygons: Clip on Top Top Clip Boundary

77. 76 of 30 Polygons: Clip on Right Right Clip Boundary

78. 77 of 30 Polygons: Clip on Bottom Bottom Clip Boundary

79. 78 of 30 Polygons: Clip on Left Left Clip Boundary

80. 79 of 30 Polygons: Clipped Polygon

81. 80 of 30 Begin Animated Recap

82. 81 of 30 Polygons: Before Clipping

83. 82 of 30 Polygons: Clip on Top Top Clip Boundary

84. 83 of 30 Polygons: Clip on Top Top Clip Boundary

85. 84 of 30 Polygons: Clip on Right Right Clip Boundary

86. 85 of 30 Polygons: Clip on Right Right Clip Boundary

87. 86 of 30 Polygons: Clip on Bottom Bottom Clip Boundary

88. 87 of 30 Polygons: Clip on Bottom Bottom Clip Boundary

89. 88 of 30 Polygons: Clip on left Left Clip Boundary

90. 89 of 30 Polygons: Clip on left Left Clip Boundary

91. 90 of 30 Polygons: Fully Clipped

92. 91 of 30 Polygons: Clip on Left Left Clip Boundary

93. 92 of 30 Polygons: Clipped Polygon

94. 93 of 30 End Animated Recap

95. Hodgeman-Sutherland

96. 95 of 30 Polygons: Case 1 INOUT S i output P

97. Case 1: S out; P in Action: Output i and P

98. 97 of 30 Polygons: Case 2 (no output) P S IN OUT

99. Case 2: S & P both in Action: output p

100. 99 of 30 Polygons: Case 3 P : second output i: first output S IN OUT

101. Case 3: S IN ; P OUT Action: Output i

102. 101 of 30 Polygons: Case 4 Polygon being clipped P: output Clip boundary S IN OUT

103. Case 4: S & P both out Action: None

104. 103 of 30 Case 2 : S & P both IN Action: Output P Case 3: S IN ; P OUT Action: Output i ( S already output under Case 1) Case 4: S & P both OUT Action: None Case 1: S OUT ; P IN Action: Output i and P

105. 104 of 30 Hodgeman-Sutherland

106. 105 of 30 Hodgeman-Sutherland

107. 106 of 30 Hodgeman-Sutherland: Top

108. 107 of 30 Hodgeman-Sutherland: Top

109. 108 of 30 Hodgeman-Sutherland: Bottom

110. 109 of 30 Hodgeman-Sutherland: Bottom

111. 110 of 30 Hodgeman-Sutherland: Bottom

112. 111 of 30 Hodgeman-Sutherland: Right

113. 112 of 30 This case is trivial: do nothing Hodgeman-Sutherland: Right

114. 113 of 30 Hodgeman-Sutherland: Left

115. 114 of 30 Hodgeman-Sutherland: Left

116. 115 of 30 Hodgeman-Sutherland: Left

117. 116 of 30 Hodgeman-Sutherland: Done

118. 117 of 30 Begin Animated Recap

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120. 119 of 30 Done Top

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122. 121 of 30 Done Bottom

123. 122 of 30 Nothing needs to be clipped against Right

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125. 124 of 30 End Animated Recap

126. 125 of 30 Hodgeman-Sutherland: Done

127. 126 of 30 Hodgeman-Sutherland Problem Extra Lines

128. 127 of 30 Correct Result How to Fix Up?

129. 128 of 30 Distinguish Between Case 2 S i output P INOUT

130. 129 of 30 And, Case 4 P : second output i : first output S INOUT

131. 130 of 30 Distingish Between Case 2 S P INOUT

132. 131 of 30 And, Case 4 P : second output S INOUT

133. 132 of 30 Extended H-S Algorithm Tag the different new vertices Call the one generated on transition from IN to OUT Call the one generated on transition from OUT to IN Starting from OUT, connect as soon as is generated

134. 133 of 30 Example of Extended Algo INOUT

135. 134 of 30 Result INOUT

136. 135 of 30 Extended H-S Algorithm Connect and as soon as the. is generated Generate next polygon beginning with next, i.e.,

137. Another Polygon Clipping Algorithm Weiler-Atherton Clipping

138. 137 of 30 Another approach to polygon clipping No extra clipping outside window Works for arbitrary shapes Avoids degenerate polygons Weiler-Atherton Polygon Clipping

139. 138 of 30 Weiler-Atherton Clipping 13 11 12 6 5 4 14 3 2 7 9 8 10 1 Clip Polygon (ep) Subject Polygon (sp)

140. 139 of 30 Gives “Right” Answer

141. 140 of 30 Start at first (inside) vertex Traverse polygon until hitting a window boundary Output intersection point i Turn right Follow window boundary until next intersection Weiler-Atherton Clipping

142. 141 of 30 Output second intersection Turn right, again, and follow subject polygon until closed Continue on subject polygon from first intersection point. Repeat processing until complete Weiler-Atherton Clipping

143. 142 of 30 Approximate description Can be improved Weiler-Atherton Clipping

144. 143 of 30 Generalizations of W-A Can be extended to complex situations, arbitrary windows Stability issues can arise for such cases

145. General Boolean Operations via Boundary Representations (B-reps)

146. General Clipping with abitrary polygons

147. 146 of 30 General Boolean Operations A B

148. 147 of 30 A B

149. 148 of 30 B

150. 149 of 30 A B

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152. 151 of 30 A B A

153. 152 of 30 A

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155. 154 of 30 A B

156. 155 of 30 A B

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158. 157 of 30 A B

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160. 159 of 30 B on

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162. 161 of 30 A B on

163. 162 of 30

164. 163 of 30 ((anti-)shared ) anti-sharedshared

165. 164 of 30 A B

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167. 166 of 30 A B

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169. 168 of 30 A B

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171. The End Polygon Clipping Lecture Set 5