2. Transformations I CS5600Computer Graphics by Rich Riesenfeld 27 February 2002 Lecture Set 5

3. CS5600 2 Transformations and Matrices Transformations are functions Matrices are functions representations Matrices represent linear transf’s

4. CS5600 3 What is a 2D Linear Transf ? Recall from Linear Algebra:

5. CS5600 4 Example: Scale in x

6. CS5600 5 Example: Scale in x by 2 What is the graphical view?

7. CS5600 6 Scale in x by 2

8. CS5600 7

9. 8

10. 9

11. 10 Summary on Scale “Scale then add,” is same as “Add then scale”

12. CS5600 11 Matrix Representation Scale in x by 2:

13. CS5600 12 Matrix Representation Scale in y by 2:

14. CS5600 13 Matrix Representation Overall Scale by 2:

15. CS5600 14 Matrix Representation Showing Same

16. CS5600 15 What about Rotation? Is it linear?

17. CS5600 16 Rotate by

18. CS5600 17 Rotate by : 1 st Quadrant

19. CS5600 18 Rotate by : 1 st Quadrant

20. CS5600 19 Rotate by : 2 nd Quadrant

21. CS5600 20 Rotate by : 2 nd Quadrant

22. CS5600 21 Rotate by : 2 nd Quadrant

23. CS5600 22 Summary of Rotation by

24. CS5600 23 Summary (Column Form)

25. CS5600 24 Using Matrix Notation (Note that unit vectors simply copy columns)

26. CS5600 25 General Rotation by Matrix

27. CS5600 26 Who had linear algebra? Who understand matrices?

28. CS5600 27 What do the off diagonal elements do?

29. CS5600 28 Off Diagonal Elements

30. CS5600 29 Example 1 S

31. CS5600 30 Example 1 S

32. CS5600 31 Example 1 T(S)

33. CS5600 32 Example 2 S

34. CS5600 33 Example 2 S

35. CS5600 34 Example 2 T(S)

36. CS5600 35 Summary Shear in x: Shear in y:

37. CS5600 36 Double Shear

38. CS5600 37 Sample Points: unit inverses

39. CS5600 38 Geometric View of Shear in x

40. Another Geometric View of Shear in x 39

41. Another Geometric View of Shear in x 40

42. CS5600 41 Geometric View of Shear in y

43. Another Geometric View of Shear in y h h 42

44. Another Geometric View of Shear in y 43

45. CS5600 44 “Lazy 1”

46. CS5600 45 Translation in x

47. CS5600 46 Translation in x

48. CS5600 47 Homogeneous Coordinates

49. CS5600 48 Homogeneous Coordinates

50. CS5600 49 Homogeneous Coordinates Homogeneous term effects overall scaling

51. Homogeneous Coordinates An infinite number of points correspond to (x,y,1). They constitute the whole line (tx,ty,t). w = 1 (tx,ty,t) (x,y,1)

52. CS5600 51 We’ve got Affine Transformations Linear + Translation

53. CS5600 52 Compound Transformations Build up compound transformations by concatenating elementary ones Use for complicated motion Use for complicated modeling

54. CS5600 53 Elementary Transformations Scale Rotate Translate Shear (Reflect)

55. CS5600 54 Refection about y-axis

56. CS5600 55 Reflection about y-axis

57. CS5600 56 Reflection about x-axis

58. CS5600 57 Reflection about x-axis

59. Is Reflection “Elementary?” Can we effect reflection in an elementary way? (More elementary means scale, shear, rotation, translation.) 58

60. CS5600 59 Reflection is Scale (-1)

61. CS5600 60 Example:Move clock hands

62. CS5600 61 Example:Move clock hands

63. CS5600 62 Example:Move clock hands

64. CS5600 63 Example:Move clock hands

65. CS5600 64 Clock Transformations Translate to Origin Move hand with rotation Move hand back to clock Do other hand

66. CS5600 65 Clock Transformations

67. CS5600 66 Clock Transformations

68. CS5600 67 Map [a,b] [0,1] 0 [ ] ab 1

69. CS5600 68 Map [a,b] [0,1] Translate to Origin Map x to translated interval

70. CS5600 69 Map [a,b] [0,1] Normalize the interval Map x to normalized interval

71. CS5600 70 Map [a,b] [0,1]

72. CS5600 71 Just Look at This is a homogeneous form for 1D

73. CS5600 72 Map [a,b] [-1,1] 0 [ ] +1 a b

74. CS5600 73 Map [a,b] [-1,1] Translate center of interval to origin Normalize interval to [-1,1]

75. CS5600 74 Map [a,b] [-1,1] Substitute x =a (analogous for x =b) : x

76. CS5600 75 Now Map [a,b] [c,d] First map [a,b] to [0,1] –(We already did this) Then map [0,1] to [c,d]

77. CS5600 76 Now Map [a,b] [c,d] Scale [0,1] by (d-c) Then translate by c That is, in 1D homogeneous form:

78. CS5600 77 All Together: [a,b] [c,d]

79. CS5600 78 Now Map Rectangles

80. CS5600 79 Transformation in x and y

81. This is the Viewport Transformation Good for mapping objects from one coordinate system to another This is what we do with windows and viewports 80

82. CS5600 81 3D Transformations Scale Rotate Translate Shear

83. CS5600 82 3D Scale in x

84. CS5600 83 3D Scale in x

85. CS5600 84 3D Scale in y

86. CS5600 85 3D Scale in z

87. CS5600 86 Overall 3D Scale

88. CS5600 87 Overall 3D Scale Same in x,y and z:

89. What is a Positive Rotation in 3D ? Sit at end of given axis Look at Origin CC Rotation is in Positive direction 88

90. 3D Positive Rotations

91. CS5600 90 3D Rotation about z-axis by We have already done this:

92. CS5600 91 3D Rotation about x-axis by

93. CS5600 92 3D Rotation about x-axis

94. 3D Rotation about y-axis by 93

95. CS5600 94 3D Rotation about y-axis

96. CS5600 95 Elementary Transformations Scale Rotate Translate Shear (Reflect)

97. CS5600 96 Consider an arbitrary 3D rotation What is its inverse? What is its transpose? Can we constructively elucidate this relationship?

98. Want to rotate by about arbitrary axis a 97

99. CS5600 98 First rotate about by Now in the (y-z)-plane

100. CS5600 99 Then rotate about by Rotate in the (y-z)-plane

101. Now perform rotation about - Now aligned with z-axis 100

102. Now perform rotation about - Now aligned with z-axis 101

103. CS5600 102 Then rotate about by Rotate again in the (y-z)-plane

104. CS5600 103 Then rotate about by Now to original position of a

105. We effected a rotation by about arbitrary axis a 104

106. We effected a rotation by about arbitrary axis a 105

107. CS5600 106 Rotation about an arbitrary axis Rotation about a-axis can be effected by a composition of 5 elementary rotations We show arbitrary rotation as succession of 5 rotations about principal axes

108. CS5600 107 In matrix terms,

109. CS5600 108 Similarly, so,

110. CS5600 109 Recall, Consequently, for because,

111. CS5600 110 It follows directly that,

112. CS5600 111

113. CS5600 112 Constructively, we have shown, This will be useful later

114. CS5600 113 3D Translation in x

115. CS5600 114 3D Translation in y

116. CS5600 115 3D Translation in z

117. CS5600 116 3D Shear in x -direction

118. CS5600 117 3D Shear in x -direction

119. CS5600 118 3D Shears:clamp a principal plane, shear in other 2 DoFs

121. CS5600 120 3D Shear in y -direction

122. CS5600 121 3D Shear in y -direction

124. CS5600 123 3D Shear in z -direction

125. CS5600 124 3D Shear in z

126. CS5600 125 3D Shear in z

127. CS5600 126 What is “Perspective?” A mechanism for portraying 3D in 2D “True Perspective” corresponds to projection onto a plane “True Perspective” corresponds to an ideal camera image

128. Many Kinds of Perspective Used Mechanical Engineering Cartography Art

129. CS5600 128 Perspective in Art Naïve (wrong) Egyptian Cubist (unrealistic) Esher –Impossible (exploits local property) –Hyperpolic (non-planar) –etc

130. CS5600 129 “True” Perspective in 2D (x,y) p h

131. CS5600 130 “True” Perspective in 2D

132. CS5600 131 “True” Perspective in 2D This is right answer for screen projection

133. CS5600 132 “True” Perspective in 2D

134. End Transformations I Lecture Set 5 133