1. Jinxiang Chai CSCE441: Computer Graphics Coordinate & Composite Transformations 0

2. Outline 2D/3D Coordinate transformation 2D/3D Composite transformation Required readings: HB 7-8, 9-6 1

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

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

5. Coordinate Transformation: Animation/Robotics How to model 2D movement of animated characters or robots? 4 Click herehere

6. Coordinate Transformation Coordinate transformation from one reference frame to another 5

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

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

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

10. Review – Vector Operations Dot Product 9

11. Review – Vector Operations Dot Product: measuring similarity between two vectors 10

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

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

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

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

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: 17

19. 2D Coordinate Transformation 2D Cartesian coordinate system: 18

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 22

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

25. 2D Coordinate Transformation Transform object description from to p 24 Given the coordinates (x’,y’) in i’j’ - how to compute the coordinates (x,y) in ij?

26. 2D Coordinate Transformation Transform object description from to p 25 Given the coordinates (x’,y’) in i’j’ - how to compute the coordinates (x,y) in ij?

27. 2D Coordinate Transformation Transform object description from to p 26

28. 2D Coordinate Transformation Transform object description from to p 27

29. 2D Coordinate Transformation Transform object description from to p 28

30. 2D Coordinate Transformation Transform object description from to p 29

31. 2D Coordinate Transformation Transform object description from to p 30

32. 2D Coordinate Transformation Transform object description from to p 31

33. 2D Coordinate Transformation Transform object description from to p 32

34. 2D Coordinate Transformation Transform object description from to p 33

35. 2D Coordinate Transformation Transform object description from to p 34

36. 2D Coordinate Transformation Transform object description from to p 35

37. 2D Coordinate Transformation Transform object description from to p 36

38. 2D Coordinate Transformation Transform object description from to p 37

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

40. 2D Coordinate Transformation p 39 What does this column vector mean?

41. 2D Coordinate Transformation Transform object description from to p 40 What does this column vector mean? Vector i’ in the new reference system

42. 2D Coordinate Transformation Transform object description from to p 41 What does this column vector mean?

43. 2D Coordinate Transformation Transform object description from to p 42 What does this column vector mean? Vector j’ in the new reference system

44. 2D Coordinate Transformation Transform object description from to p 43 What does this column vector mean?

45. 2D Coordinate Transformation Transform object description from to p 44 What does this column vector mean? The old origin in the new reference system

46. 2D Coordinate Transformation 2D translation p 45

47. 2D Coordinate Transformation 2D translation p ? ? ? ? 46

48. 2D Coordinate Transformation 2D translation p 1 0 0 1 47

49. 2D Coordinate Transformation 2D translation&rotation p ? 48

50. 2D Coordinate Transformation 2D translation&rotation p ? 49

51. 2D Coordinate Transformation 2D translation&rotation p 50

52. 2D Coordinate Transformation 2D translation&rotation p ? 51

53. 2D Coordinate Transformation 2D translation&rotation p 52

54. 2D Coordinate Transformation 2D translation&rotation p ? 53

55. 2D Coordinate Transformation 2D translation&rotation p 54

56. 2D Coordinate Transformation An alternative way to look at the problem P=[x,y] 55 - set up a transformation that superimposes the x’y’ axes onto the xy axis

57. 2D Coordinate Transformation An alternative way to look at the problem P=[x,y] 56 - set up a transformation that superimposes the x’y’ axes onto the xy axis

58. 2D Coordinate Transformation An alternative way to look at the problem P=[x,y] 57 - set up a transformation that superimposes the x’y’ axes onto the xy axis

59. 2D Coordinate Transformation An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) p 58

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 60

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 61

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 62

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 63

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 64

66. 2D Coordinate Transformation Same results! p 65

67. 2D Coordinate Transformation 2D translation&rotation p 66

68. 2D Coordinate Transformation 2D translation&rotation p 67

69. 2D Coordinate Transformation 2D translation&rotation p 68

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

71. 2D Coordinate Transformation 2D translation&rotation p 70

72. 2D Coordinate Transformation 2D translation&rotation p 71

73. 3D Coordinate Transformation Transform object description from to p 72

74. 2D Coordinate Transformation Transform object description from to p 73

75. 3D Coordinate Transformation Transform object description from to p 74

76. 3D Coordinate Transformation Transform object description from to p 75

77. 3D Coordinate Transformation Transform object description from to 76 p

78. 3D Coordinate Transformation Transform object description from to 77 x y z

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

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

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

82. Composite 2D Transformation How can we draw the character given the pose ? 81

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

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

85. Composite 2D Transformation What’s the pose? 84

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

87. 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? ? 86

88. Composite 2D Transformation What’s local coordinate ? ? 87

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

90. Composite 2D Transformation What’s the current coordinate A ? ? 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. How to Animate the Character? A 2D lamp character 95

97. 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! 96

98. Composite 3D Transformation Similarly, we can easily extend composite transformation from 2D to 3D 97

99. Composite 3D Transformation 98