1. 1 Geometrical Transformation Tong-Yee Lee

2. 2 Modeling Transform Specify transformation for objects Allow definitions of objects in own coordinate systems Allow use of object definition multiple times in a scene

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

4. 4 2-D Transformations

5. 5

6. 6

7. 7

8. 8

9. 9

10. 10 Basic 2D Transformations

11. 11 Basic 2D Transformations

12. 12 Basic 2D Transformations

13. 13 Rotation around the origin (2-D)

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

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

16. 16 Rotation (3-D)

17. 17 Rotation (3-D)

18. 18 Basic 2D Transformations

19. 19 Basic 2D Transformations

20. 20 Basic 2D Transformations

21. 21 Matrix Representation

22. 22 Matrix Representation

23. 23 2x2 Matrix

24. 24 Scaling

25. 25 Scaling Around A Point

26. 26 2x2 Matrix

27. 27 Shear (2-D)

28. 28 Shear (3-D)

29. 29 2x2 Matrix

30. 30 2x2 Matrix

31. 31 2x2 Matrix

32. 32 2D Translation

33. 33 Basic 2D Transformations

34. 34 Homogeneous Coordinates

35. 35 Matrix Composition

36. 36 Matrix Composition

37. 37 Matrix Composition

38. 38 Matrix Composition

39. 39 3D Transformations

40. 40 Basic 3D Transformations

41. 41 Basic 3D Transformations

42. 42 T1 T2 T3 root P is a local coordinate T3*P T2*(T3*P) T1*(T2(T3*P))  Final global coord. of P Transform # (T1) { Transform { # (T2) Transform { # (T3) }

43. 43 Transformation Hierarchies

44. 44 OpenGL transformation Matrices

45. 45 OpenGL transformation Matrices

46. 46 OpenGL transformation Matrices

47. 47 OpenGL transformation Matrices

48. 48 Transformation Example 1

49. 49 Transformation Example 2

50. 50 Transformation Example 2