2. Transformations II CS5600 Computer Graphics Rich Riesenfeld Spring 2005 Lecture Set 7

3. Student Name Server Spring 2005Utah School of Computing2 Arbitrary 3D Rotation What is its inverse? What is its transpose? Can we constructively elucidate this relationship?

4. Student Name Server Spring 2005Utah School of Computing3 Want to rotate about arbitrary axis a 3

5. Student Name Server Spring 2005Utah School of Computing4 First rotate about z by Now in the (y-z)-plane

6. Student Name Server Spring 2005Utah School of Computing5 Then rotate about x by Rotate in the (y-z)-plane

7. Now perform rotation about Now a-axis aligned with z-axis 6

8. Student Name Server Spring 2005Utah School of Computing7 Then rotate about x by Rotate again in the (y-z)-plane

9. Student Name Server Spring 2005Utah School of Computing8 Then rotate about z by Now to original position of a

10. Student Name Server Spring 2005Utah School of Computing9 We effected a rotation by about arbitrary axis a 9

11. Student Name Server Spring 2005Utah School of Computing10 We effected a rotation by about arbitrary axis a 10

12. Student Name Server Spring 2005Utah School of Computing11 Rotation about arbitrary axis a 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

13. Student Name Server Spring 2005Utah School of Computing12 In matrix terms,

14. Student Name Server Spring 2005Utah School of Computing13 Similarly, so,

15. Student Name Server Spring 2005Utah School of Computing14 Recall, Consequently, for because,

16. Student Name Server Spring 2005Utah School of Computing15 It follows directly that,

17. Student Name Server Spring 2005Utah School of Computing16

18. Student Name Server Spring 2005Utah School of Computing17 Constructively, we have shown, This will be useful later

19. Student Name Server Spring 2005Utah School of Computing18 3D Translation in x

20. Student Name Server Spring 2005Utah School of Computing19 3D Translation in y

21. Student Name Server Spring 2005Utah School of Computing20 3D Translation in z

22. Student Name Server Spring 2005Utah School of Computing21 3D Shear in x -direction

23. Student Name Server Spring 2005Utah School of Computing22 3D Shear in x -direction

24. Student Name Server Spring 2005Utah School of Computing23 3D Shears :clamp a principal plane, shear in other 2 DoFs

26. Student Name Server Spring 2005Utah School of Computing25 3D Shear in y-direction

27. Student Name Server Spring 2005Utah School of Computing26 3D Shear in y-direction

28. Student Name Server Spring 2005Utah School of Computing27

29. Student Name Server Spring 2005Utah School of Computing28 3D Shear in z-direction

30. Student Name Server Spring 2005Utah School of Computing29 3D Shear in z-direction

31. Student Name Server Spring 2005Utah School of Computing30 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

32. Student Name Server Spring 2005Utah School of Computing31 Differert Perspectives Used Mechanical Engineering Cartography Art

33. Student Name Server Spring 2005Utah School of Computing32 Perspective in Art “Naïve” (wrong) Egyptian Cubist (unrealistic) Esher –Impossible (exploits local property) –Hyperpolic (non-planar) –etc

34. Student Name Server Spring 2005Utah School of Computing33 “ True ” Perspective in 2D (x,y) p h

35. Student Name Server Spring 2005Utah School of Computing34 “ True ” Perspective in 2D

36. Student Name Server Spring 2005Utah School of Computing35 “ True ” Perspective in 2D This is right answer for screen projection

37. Student Name Server Spring 2005Utah School of Computing36 “ True ” Perspective in 2D

38. Student Name Server Spring 2005Utah School of Computing37 What Are Elementary Inverses? Scale Shear Rotation Translation

39. Student Name Server Spring 2005Utah School of Computing38 Scale Inverse

40. Student Name Server Spring 2005Utah School of Computing39 Shear Inverse

41. Student Name Server Spring 2005Utah School of Computing40 Shear Inverse

42. Student Name Server Spring 2005Utah School of Computing41 Rotation Inverse

43. Student Name Server Spring 2005Utah School of Computing42 Rotation Inverse

44. Student Name Server Spring 2005Utah School of Computing43 Rotation Inverse

45. Student Name Server Spring 2005Utah School of Computing44 Translation Inverse

46. Student Name Server Spring 2005Utah School of Computing45 Translation Inverse

47. Student Name Server Spring 2005Utah School of Computing46 Double Shear

48. Student Name Server Spring 2005Utah School of Computing47 Shear in x then in y

49. Student Name Server Spring 2005Utah School of Computing48 Shear in y then in x

50. Student Name Server Spring 2005Utah School of Computing49 Results Are Different y, then x:x, then y:

51. Student Name Server Spring 2005Utah School of Computing50 Want the RHR to Work

52. Student Name Server Spring 2005Utah School of Computing51 3D Positive Rotations

53. Student Name Server Spring 2005Utah School of Computing52 Transformations as Change in Coordinate System Useful in many situations Use most natural coordination system locally Tie things together in a global system

54. Student Name Server Spring 2005Utah School of Computing53 Example 1 2 3 4

55. Student Name Server Spring 2005Utah School of Computing54 Example is the transformation that takes a point in coordinate system j and converts it to a point in coordinate system i

56. Student Name Server Spring 2005Utah School of Computing55 Example

57. Student Name Server Spring 2005Utah School of Computing56 Example

58. Student Name Server Spring 2005Utah School of Computing57 Recall, from Matice Algebra

59. Student Name Server Spring 2005Utah School of Computing58 Since

60. Student Name Server Spring 2005Utah School of Computing59 Change of Coordinate System Describe the old coordinate system in terms of the new one. x’ y’

61. Student Name Server Spring 2005Utah School of Computing60 Move to the new coordinate system and describe the one old. x y Old is a negative rotation of the new Change of Coordinate System

62. Student Name Server Spring 2005Utah School of Computing61 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

63. Student Name Server Spring 2005Utah School of Computing62 Many Kinds of Perspective Used Mechanical Engineering Cartography Art

64. Student Name Server Spring 2005Utah School of Computing63 Perspective in Art Naïve (wrong) Egyptian Cubist (unrealistic) Esher Miro Matisse

65. Student Name Server Spring 2005Utah School of Computing64 Egyptian Frontalism Head profile Body front Eyes full Rigid style

66. Uccello's (1392-1475) hand drawing was the first extant complex geometrical form rendered according to the laws of linear perspective Perspective Study of a Chalice, Drawing, Gabinetto dei Disegni, Uffizi, Florence, ca 1430) 65

67. Student Name Server Spring 2005Utah School of Computing66 Perspective in Cubism Woman with a Guitar (1913) Georges Braque

68. Student Name Server Spring 2005Utah School of Computing67 Perspective in Cubism

69. Madre con niño muerto (1937) 68 Pablo Picaso

70. Pablo Picaso, Cabeza de mujer llorando con pañuelo 69

71. Student Name Server Spring 2005Utah School of Computing70 Perspective (Mural) Games M C Esher, Another World II (1947)

72. Student Name Server Spring 2005Utah School of Computing71 Perspective Ascending and Descending (1960) M C Escher

73. Student Name Server Spring 2005Utah School of Computing72 M. C. Escher M C Escher, Ascending and Descending (1960)

74. Student Name Server Spring 2005Utah School of Computing73 M C Escher Perspective is “ local ” Perspective consistency is not “ transitive ” Nonplanar ( hyperbolic ) projection

75. Student Name Server Spring 2005Utah School of Computing74 Nonplanar ( Hyperbolic ) Projection M C Esher, Heaven and Hell

76. Student Name Server Spring 2005Utah School of Computing75 Nonplanar ( Hyperbolic ) Projection M C Esher, Heaven and Hell

77. Student Name Server Spring 2005Utah School of Computing76 David McAllister The March of Progress, (1995)

78. Student Name Server Spring 2005Utah School of Computing77 Joan Miro: Flat Perspective The Tilled Field What cues are missing?

79. Henri Matisse, La Lecon de Musique Flat Perspective: What cues are missing? 78

80. Student Name Server Spring 2005Utah School of Computing79 Next 2 Images Contain Nudity !

81. Henri Matisse, Danse (1909) 80

82. Henri Matisse, Danse II (1910) 81

83. Student Name Server Spring 2005Utah School of Computing82 Atlas Projection

84. Student Name Server Spring 2005Utah School of Computing83 Norway is at High Latitude There is considerable size distortion

85. Student Name Server Spring 2005Utah School of Computing84 Isometric View

86. Student Name Server Spring 2005Utah School of Computing85 Engineering Drawing: 2 Planes AA Section AA

87. Engineering Drawing: Exploded View Understanding 3D Assembly in a 2D Medium 86

88. Student Name Server Spring 2005Utah School of Computing87 “ True” Perspective in 2D (x,y) p h

89. Student Name Server Spring 2005Utah School of Computing88 “ True” Perspective in 2D

90. Student Name Server Spring 2005Utah School of Computing89 “ True” Perspective in 2D

91. Student Name Server Spring 2005Utah School of Computing90 Geometry is Same for Eye at Origin (x,y) h Screen Plane

92. Student Name Server Spring 2005Utah School of Computing91 What Happens to Special Points? What is this point??

93. Student Name Server Spring 2005Utah School of Computing92 Let’s Look at Limit We see that Observe,

94. Student Name Server Spring 2005Utah School of Computing93 Where does Eye Point Go? It gets sent to on x-axis Where does on x-axis go?

95. Student Name Server Spring 2005Utah School of Computing94 What happens to ? It comes back to virtual eye point!

96. Student Name Server Spring 2005Utah School of Computing95 What Does This Mean?

97. Student Name Server Spring 2005Utah School of Computing96 What Does This Mean?

98. Student Name Server Spring 2005Utah School of Computing97 The “Pencil of Lines” Becomes Parallel

99. Student Name Server Spring 2005Utah School of Computing98 Parallel Lines Become “ Pencil of Lines ” !

100. Student Name Server Spring 2005Utah School of Computing99 Parallel Lines Become “ Pencil of Lines ” !

101. Student Name Server Spring 2005Utah School of Computing100 What Does This Mean?

102. Student Name Server Spring 2005Utah School of Computing101 “ True ” Perspective in 2D

103. Student Name Server Spring 2005Utah School of Computing102 “ True ” Perspective in 2D

104. Student Name Server Spring 2005Utah School of Computing103 Viewing Frustum

105. Student Name Server Spring 2005Utah School of Computing104 What happens for large p ?”

106. Student Name Server Spring 2005Utah School of Computing105 Projection Becomes Orthogonal: “Right Thing Happens” (x,y) h=y

107. The End Transformations II Lecture Set 7