2. 1 Geometrical Transformation

3. 2 Outline General Transform 3D Objects Quaternion & 3D Track Ball

4. 3 Modeling Transform Specify transformation for objects –Allow definitions of objects in own coordinate systems –Allow use of object definition multiple times in a scene

5. 4Overview 2D transformations –Basic 2-D transformations –Matrix representation –Matrix composition 3D transformations –Basic 3-D transformation –Same as 2-D Transformation Hierarchies –Scene graphs

6. 5 2-D Transformations

7. 6

8. 7

9. 8

10. 9

11. 10 2-D Transformations

12. 11 Basic 2D Transformations

13. 12 Basic 2D Transformations

14. 13 Basic 2D Transformations

15. 14 Rotation around the origin (2-D)

16. 15 Rotation around the origin (2-D)

17. 16 Rotation around the origin (2-D)

18. 17 Rotation (3-D)

19. 18 Rotation (3-D)

20. 19 Basic 2D Transformations

21. 20 Basic 2D Transformations

22. 21 Basic 2D Transformations

23. 22 Matrix Representation

24. 23 Matrix Representation

25. 24 2x2 Matrix

26. 25Scaling

27. 26 Scaling Around A Point

28. 27 2x2 Matrix

29. 28 Shear (2-D)

30. 29 Shear (3-D)

31. 30 2x2 Matrix

32. 31 2x2 Matrix

33. 32 2D Translation

34. 33 Basic 2D Transformations

35. 34 Homogeneous Coordinates

36. 35 Linear Transformations

37. 36 Affine Transformations

38. 37 Projective Transformations

39. 38 Matrix Composition

40. 39 Matrix Composition

41. 40 Matrix Composition

42. 41 Matrix Composition

43. 42 3D Transformations

44. 43 Basic 3D Transformations

45. 44 Basic 3D Transformations

46. 45 GENERAL ROTATION ABOUT ANAXIS An axis in space is specified by a point P and a vector direction. Suppose that we wish to rotate an object about this arbitrary axis.

47. 46 Developing the General Rotation Matrix

48. 47 Developing the General Rotation Matrix

49. 48 Developing the General Rotation Matrix

50. 49 Developing the General Rotation Matrix

51. 50 Developing the General Rotation Matrix

52. 51 Developing the General Rotation Matrix Be careful ………… Z X (+,+) (-,-) In both cases, tan(y/x) are positive. So, we need to carefully choose it by checking the signs of x and y

53. 52 Developing the General Rotation Matrix Another problem is: rotation interpolation is not easy and not good reported in many papers.

54. 53

55. 54

56. 55 Angular displacement glRotate( , Ax,Ay,Az) (,n) defines an angular displacement of  about an axis u or n for rotating a vector v

57. 56 The above formula is a matrix form, so we can use Matrix to compute rotation In above equation, v=(x,y,z) T and n=(a x,a y,a z ) T

58. 57

59. 58

60. 59

61. 60

62. 61 Inverse Transformation

63. 62 Inverse Transformation

64. 63 Transform points, lines, planes etc.

65. 64 Transforming Normals

66. 65 Transformation Hierarchies

67. 66 OpenGL transformation Matrices

68. 67 OpenGL transformation Matrices

69. 68 OpenGL transformation Matrices

70. 69 OpenGL transformation Matrices

71. 70 Transformation Example 1

72. 71 Transformation Example 2

73. 72 Transformation Example 2

74. 73 Hierarchical Scene Graph This topics will be taught in future or the next semester!!

75. 74

76. 75

77. 76

78. 77

79. 78

80. 79

81. 80

82. 81

83. 82

84. 83

85. 84

86. 85

87. 86Applications

88. 87 Applications

89. 88 Applications

90. 89

91. 90

92. 91

93. 92

94. 93