1. CSCE441: Computer Graphics Coordinate & Composite Transformations
Jinxiang Chai

2. Outline 2D/3D Coordinate transformation 2D/3D Composite transformation
Required readings: HB 7-8, 9-6

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
Local reference frame

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

9. Coordinate Transformation
Coordinate transformation from one reference frame to another ?
Local reference frame
Global reference frame 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:

19. 2D Coordinate Transformation
2D Cartesian coordinate system:

20. 2D Coordinate Transformation
2D Cartesian coordinate system: any 2D vector can be represented as 19

21. 2D Coordinate Transformation
2D Cartesian coordinate system:
P: (x,y) 20

22. 2D Coordinate Transformation
2D Cartesian coordinate system:
P: (x,y) 21

23. 2D Coordinate Transformation
Transform object description from to p

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
Given the coordinates (x’,y’) in i’j’
how to compute the coordinates (x,y) in ij? 24

26. 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? 25

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 Transformation
Transform object description from to p

38. 2D Coordinate Transformation
Transform object description from to p

39. 2D Coordinate Transformation
Transform object description from to p 38

40. 2D Coordinate Transformationp
What does this column vector mean?

41. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean? Vector i’ in the new reference system

42. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean?

43. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean? Vector j’ in the new reference system

44. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean?

45. 2D Coordinate Transformation
Transform object description from to p
What does this column vector mean? The old origin in the new reference system

46. 2D Coordinate Transformation
2D translation p

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

48. 2D Coordinate Transformation
2D translation 1 p 1

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
2D translation & rotation p

54. 2D Coordinate Transformation
2D translation & rotation p ?

55. 2D Coordinate Transformation
2D translation & rotation p

56. 2D Coordinate Transformation
An alternative way to look at the problem
set up a transformation that superimposes the x’y’ axes onto the xy axis
transform the point from the new coordinate system to the old one.
P=[x,y]

57. 2D Coordinate Transformation
An alternative way to look at the problem
set up a transformation that superimposes the x’y’ axes onto the xy axis
transform the point from the new coordinate system to the old one.
P=[x,y]

58. 2D Coordinate Transformation
An alternative way to look at the problem
set up a transformation that superimposes the x’y’ axes onto the xy axis
transform the point from the new coordinate system to the old one.
P=[x,y]

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

60. 2D Coordinate Transformation
An alternative way to look at the problem
This transforms the point from (x,y) to (x’,y’)
How to transform the point from (x’,y’) to (x,y)? p 59

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

62. 2D Coordinate Transformation
An alternative way to look at the problem
This transforms the point from (x,y) to (x’,y’)
How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p

63. 2D Coordinate Transformation
An alternative way to look at the problem
This transforms the point from (x,y) to (x’,y’)
How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p

64. 2D Coordinate Transformation
An alternative way to look at the problem
This transforms the point from (x,y) to (x’,y’)
How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p

65. 2D Coordinate Transformation
An alternative way to look at the problem
This transforms the point from (x,y) to (x’,y’)
How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p

66. 2D Coordinate Transformation
Same results! p

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

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

69. 2D Coordinate Transformation
2D translation & rotation p

70. 2D Coordinate Transformation
2D translation & rotation p 69

71. 2D Coordinate Transformation
2D translation & rotation p

72. 2D Coordinate Transformation
2D translation & rotation p

73. 3D Coordinate Transformation
Transform object description from to p

74. 2D Coordinate Transformation
Transform object description from to p

75. 3D Coordinate Transformation
Transform object description from to p

76. 3D Coordinate Transformation
Transform object description from to p

77. 3D Coordinate Transformation
Transform object description from to p

78. 3D Coordinate Transformation
Transform object description from to y x z

79. Composite 2D Transformation
How to model 2D movement of characters or robots?
Click here 78

80. Composite 2D Transformation
A 2D lamp character 79

81. Composite 2D Transformation
A 2D lamp character – skeleton size & model 80

82. Composite 2D Transformation
A 2D lamp character – skeleton size & model 81

83. Composite 2D Transformation
How can we draw the character given the pose ? 82

84. Composite 2D Transformation
How can we draw the character given the pose ?
- This requires computing the global coordinates for any point on the character.
But we only have local coordinates of points.
So how can we map the local coordinates to the global coordinates? 83

85. Articulated Character
Local reference frames with a default pose (0,0,0,0,0,0) 84

86. Composite 2D Transformation
What’s the pose? 85

87. Composite 2D Transformation
What’s the pose? 86

88. Composite 2D Transformation
A 2D lamp character ?
Given , , how to compute the global position of a point (e.g., A) based on its local coordinates? 87

89. Composite 2D Transformation
What’s local coordinate ? ? 88

90. Composite 2D Transformation
What’s local coordinate ? ? 89

91. Composite 2D Transformation
What’s the current coordinate A ? ? 90

92. Composite 2D Transformation
What’s the current coordinate A ? ? 91

93. Composite 2D Transformation
What’s the current coordinate A ? ? 92

94. Composite 2D Transformation
What’s the current coordinate A ? ? 93

95. Composite 2D Transformation
What’s the current coordinate A ? ? 94

96. Composite 2D Transformation
What’s the current coordinate A ? 95

97. How to Animate the Character?
A 2D lamp character 96

98. 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! 97

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

100. Composite 3D Transformation

101. One Remaining Issue
How to draw the object?

102. One Remaining Issue
How to draw the object?