1. CSCE441: Computer Graphics 2D/3D Transformations
Jinxiang Chai

2. Outline 2D Coordinate transformation Composite transformation
3D transformation
Required readings: HB 5-9 to 5-17

3. Coordinate Transform: 3D Geometry Pipeline
Rotate and translate the camera
Object space
World space
View space
Focal length
Aspect ratio & resolution
Normalized project space
Image space 2

4. Coordinate Transformation: 3D Modeling/Design
Coordinate transformation from one reference frame to another

5. Coordinate Transformation: Animation/Robotics
How to model 2D movement of animated characters or robots?
Click here

6. Coordinate Transformation
Coordinate transformation from one reference frame to another

7. Coordinate Transformation
Coordinate transformation from one reference frame to another

8. Coordinate Transformation
Coordinate transformation from one reference frame to another 7

9. Coordinate Transformation
Coordinate transformation from one reference frame to another ? 8

10. Review – Vector Operations
Dot Product

11. Review – Vector Operations
Dot Product: measuring similarity between two vectors

12. Review – Vector Operations
Dot Product: measuring similarity between two vectors

13. Review – Vector Operations
Dot Product: measuring similarity between two vectors
Unit vector:

14. Review – Vector Operations
Dot Product: measuring similarity between two vectors

15. Review – Vector Operations
Dot Product: measuring similarity between two vectors

16. Review – Vector Operations
Cross Product: measuring the area determined by two vectors 15

17. Review – Vector Operations
Cross Product: measuring the area determined by two vectors 16

18. 2D Coordinates
2D Cartesian coordinate system:
P: (x,y)

19. 2D Coordinate Transformation
2D Cartesian coordinate system:
P: (x,y)

20. 2D Coordinate Transformation
2D Cartesian coordinate system:
P: (x,y) 19

21. 2D Coordinate Transformation
Transform object description from to p

22. 2D Coordinate Transformation
Transform object description from to p
Given the coordinates (x’,y’) in i’j’
how to compute the coordinates (x,y) in ij? 21

23. 2D Coordinate Transformation
Transform object description from to p
Given the coordinates (x’,y’) in i’j’
how to compute the coordinates (x,y) in ij? 22

24. 2D Coordinate Transformation
Transform object description from to p
Given the coordinates (x’,y’) in i’j’
how to compute the coordinates (x,y) in ij? 23

25. 2D Coordinate Transformation
Transform object description from to p

26. 2D Coordinate Transformation
Transform object description from to p

27. 2D Coordinate Transformation
Transform object description from to p

28. 2D Coordinate Transformation
Transform object description from to p

29. 2D Coordinate Transformation
Transform object description from to p

30. 2D Coordinate Transformation
Transform object description from to p

31. 2D Coordinate Transformation
Transform object description from to p

32. 2D Coordinate Transformation
Transform object description from to p

33. 2D Coordinate Transformation
Transform object description from to p

34. 2D Coordinate Transformation
Transform object description from to p

35. 2D Coordinate Transformation
Transform object description from to p

36. 2D Coordinate Transformation
Transform object description from to p

37. 2D Coordinate Transformationp
What does this column vector mean?

38. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean? Vector i’ in the new reference system

39. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean?

40. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean? Vector j’ in the new reference system

41. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean?

42. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean? The old origin in the new reference system

43. 2D Coordinate Transformation
2D translation p

44. 2D Coordinate Transformation
2D translation ? ? p ? ?

45. 2D Coordinate Transformation
2D translation 1 p 1

46. 2D Coordinate Transformation
2D translation&rotation ? p

47. 2D Coordinate Transformation
2D translation&rotation p ?

48. 2D Coordinate Transformation
2D translation&rotation p

49. 2D Coordinate Transformation
2D translation&rotation ? p

50. 2D Coordinate Transformation
2D translation&rotation p

51. 2D Coordinate Transformation
2D translation&rotation p ?

52. 2D Coordinate Transformation
2D translation&rotation p

53. 2D Coordinate Transformation
An alternative way to look at the problem p

54. 2D Coordinate Transformation
An alternative way to look at the problem p

55. 2D Coordinate Transformation
An alternative way to look at the problem p

56. 2D Coordinate Transformation
An alternative way to look at the problem
This transform the point from (x’,y’) to (x,y) p

57. 2D Coordinate Transformation
An alternative way to look at the problem
This transform the point from (x’,y’) to (x,y) p

58. 2D Coordinate Transformation
An alternative way to look at the problem
This transform the point from (x’,y’) to (x,y) p

59. 2D Coordinate Transformation
An alternative way to look at the problem
This transform the point from (x’,y’) to (x,y) p

60. 2D Coordinate Transformation
An alternative way to look at the problem
This transform the point from (x’,y’) to (x,y) p

61. 2D Coordinate Transformation
An alternative way to look at the problem
This transform the point from (x’,y’) to (x,y) p

62. 2D Coordinate Transformation
Same results! p

63. 2D Coordinate Transformation
2D translation&rotation p

64. 2D Coordinate Transformation
2D translation&rotation p

65. 2D Coordinate Transformation
2D translation&rotation p

66. 2D Coordinate Transformation
2D translation&rotation p 65

67. 2D Coordinate Transformation
2D translation&rotation p

68. 2D Coordinate Transformation
2D translation&rotation p

69. Composite 2D Transformation
How to model 2D movement of characters or robots?
Click here

70. Composite 2D Transformation
A 2D lamp character

71. Composite 2D Transformation
A 2D lamp character

72. Composite 2D Transformation
How can we draw the character given the pose ?

73. Articulated Character
A default pose (0,0,0,0,0,0)

74. Composite 2D Transformation
What’s the pose?

75. Composite 2D Transformation
What’s the pose?

76. Composite 2D Transformation
A 2D lamp character ?
Given , , how to compute the global position of a point A?

77. Composite 2D Transformation
What’s local coordinate ? ?

78. Composite 2D Transformation
What’s local coordinate ? ?

79. Composite 2D Transformation
What’s the current coordinate A ? ?

80. Composite 2D Transformation
What’s the current coordinate A ? ?

81. Composite 2D Transformation
What’s the current coordinate A ? ?

82. Composite 2D Transformation
What’s the current coordinate A ? ?

83. Composite 2D Transformation
What’s the current coordinate A ? 82

84. How to Animate the Character?
A 2D lamp character

85. How to Animate the Character?
Keyframe animation
- Manually pose the character by choosing appropriate values for
- Linearly interpolate the inbetween poses.
- Works for any types of articulated characters!

86. 3D Transformation A 3D point (x,y,z) – x,y, and z coordinates
We will still use column vectors to represent points
Homogeneous coordinates of a 3D point
(x,y,z,1)
Transformation will be performed using 4x4 matrix

87. Right-handed Coordinate System
Left hand coordinate system
Not used in this class and
Not in OpenGL

88. Homogenous coordinates
3D Transformation
Very similar to 2D transformation
Translation transformation
Homogenous coordinates

89. Homogenous coordinates
3D Transformation
Very similar to 2D transformation
Scaling transformation
Homogenous coordinates

90. 3D Transformation 3D rotation is done around a rotation axis
Fundamental rotations – rotate about x, y, or z axes
Counter-clockwise rotation is referred to as positive rotation (when you look down negative axis) y + x z

91. 3D Transformation
Rotation about z – similar to 2D rotation y + x z

92. 3D Transformation Rotation about y: z -> y, y -> x, x->z z y

93. 3D Transformation Rotation about x (z -> x, y -> z, x->y) x z
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters, y x z

94. 3D Rotation about Arbitrary Axes
Rotate p about the by the angle
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

95. 3-D Rotation
General rotations in 3-D require rotating about an arbitrary axis of rotation
Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra

96. 3D Rotation about Arbitrary Axes
Rotate p about the by the angle

97. 3-D Rotation
General rotations in 3-D require rotating about an arbitrary axis of rotation
Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra
Standard approach: express general rotation as composition of canonical rotations
Rotations about x, y, z

98. Composing Canonical Rotations
Goal: rotate about arbitrary vector r by θ
Idea: we know how to rotate about x,y,z
So, rotate about z by - until r lies in the xz plane
Then rotate about y by -β until r coincides with +z
Then rotate about z by θ
Then reverse the rotation about y (by β )
Then reverse the rotation about z (by )

99. 3D Rotation about Arbitrary Axes
Rotate p about the by the angle
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

100. 3D Rotation about Arbitrary Axes
Translate so that rotation axis passes through the origin
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

101. 3D Rotation about Arbitrary Axes
Rotation by about z-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

102. 3D Rotation about Arbitrary Axes
Rotation by about y-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

103. 3D Rotation about Arbitrary Axes
Rotation by about z-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

104. 3D Rotation about Arbitrary Axes
Rotation by about y-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

105. 3D Rotation about Arbitrary Axes
Rotation by about z-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

106. 3D Rotation about Arbitrary Axes
Translate the object back to original point

107. 3D Rotation about Arbitrary Axes
Final transformation matrix for rotating about an arbitrary axis

108. 3D Rotation about Arbitrary Axes
Final transformation matrix for rotating about an arbitrary axis
107

109. 3D Rotation about Arbitrary Axes
Final transformation matrix for rotating about an arbitrary axis

110. 3D Rotation about Arbitrary Axes
Final transformation matrix for rotating about an arbitrary axis
A 3 by 3 Rotation matrix

111. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)

112. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its transpose:

113. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its transpose:

114. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its transpose:

115. Rotation Matrices
Why?

116. Rotation Matrices
Why?

117. Rotation Matrices
Why?

118. Rotation Matrices
Why?

119. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its transpose: e.g.,

120. 3D Coordinate Transformation
Transform object description from to p

121. 2D Coordinate Transformation
Transform object description from to p

122. 3D Coordinate Transformation
Transform object description from to p

123. 3D Coordinate Transformation
Transform object description from to p

124. 3D Coordinate Transformation
Transform object description from to p

125. 3D Coordinate Transformation
Transform object description from to p

126. Composite 3D Transformation
Similarly, we can easily extend composite transformation from 2D to 3D

127. Composite 3D Transformation