2. Transformations II CS5600 Computer Graphics by Rich Riesenfeld 4 March 2002 Lecture Set 6

3. CS56002 What About Elementary Inverses? Scale Shear Rotation Translation

4. CS56003 Scale Inverse

5. CS56004 Shear Inverse

6. CS56005 Shear Inverse

7. CS56006 Rotation Inverse

8. CS56007 Rotation Inverse

9. CS56008 Rotation Inverse

10. CS56009 Translation Inverse

11. CS560010 Translation Inverse

12. CS560011 Shear in x then in y

13. CS560012 Shear in y then in x

14. CS560013 Results Are Different y then x:x then y:

15. CS560014 Want the RHR to Work

16. CS560015 3D Positive Rotations

17. CS560016 Transformations as a Change in Coordinate System Useful in many situations Use most natural coordination system locally Tie things together in a global system

18. CS560017 Example 1 2 3 4

19. CS560018 Example is the transformation that takes a point in coordinate system j and converts it to a point in coordinate system i

20. CS560019 Example

21. CS560020 Example

22. CS560021 Recall the Following

23. CS560022 Since

24. CS560023 Change of Coordinate System Describe the old coordinate system in terms of the new one. x’ y’

25. CS560024 Change of Coordinate System Move to the new coordinate system and describe the one old. x y Old is a negative rotation of the new.

26. CS560025 What is “Perspective?” A mechanism for portraying 3D in 2D “True Perspective” corresponds to projection onto a plane “True Perspective” corresponds to an ideal camera image

27. CS560026 Many Kinds of Perspective Used Mechanical Engineering Cartography Art

28. CS560027 Perspective in Art Naïve (wrong) Egyptian Cubist (unrealistic) Esher Miro Matisse

29. CS560028 Egyptian Frontalism Head profile Body front Eyes full Rigid style

30. Uccello's (1392-1475) handdrawing was the first extant complex geometrical form rendered accor-ding to the laws of linear perspective (Perspective Study of a Chalice, Drawing, Gabinetto dei Disegni, Uffizi, Florence, ca 1430 1440) 29

31. CS560030 Perspective in Cubism Braque, Georges Woman with a Guitar Sorgues, autumn 1913

32. CS560031 Perspective in Cubism

33. Pablo Picaso, Madre con niño muerto (1937) 32

34. Pablo Picaso Cabeza de mujer llorando con pañuelo 33

35. CS560034 Perspective (Mural) Games M C Esher, Another World II (1947)

36. CS560035 Perspective M.C. Escher, Ascending and Descending (1960)

37. CS560036 M. C. Esher M.C. Escher, Ascending and Descending (1960)

38. CS560037 M. C. Esher Perspective is “local” Perspective consistency is not “transitive” Nonplanar (hyperbolic) projection

39. CS560038 Nonplanar Projection M C Esher, Heaven and Hell

40. CS560039 Nonplanar Projection M C Esher, Heaven and Hell

41. CS560040 David McAllister The March of Progress, (1995)

42. CS560041 Joan Miro The Tilled Field Flat Perspective: What cues are missing?

43. Henri Matisse, La Lecon de Musique Flat Perspective: What cues are missing? 42

44. Henri Matisse, Danse II (1910) 43

45. CS560044

46. CS560045 Norway is at High Latitude

47. CS560046 Isometric View

48. CS560047 Engineering Drawing A A Section AA

49. Engineering Drawing: Exploded View Understanding 3D Assembly in a 2D Medium 48

50. CS560049 “True” Perspective in 2D (x,y) p h

51. CS560050 “True” Perspective in 2D

52. CS560051 “True” Perspective in 2D

53. CS560052 “True” Perspective in 2D This is right answer for screen projection

54. CS560053 Geometry is Same for Eye at Origin (x,y) p h Screen Plane

55. CS560054 What Happens to Special Points? What is this point??

56. CS560055 Look at a Limit

57. CS560056 Where does Eye Point Go? It gets sent to on x-axis Where does on x-axis go?

58. CS560057 What happens to? It comes back to virtual eye point!

59. CS560058 What Does This Mean? p

60. CS560059 The “Pencil of Lines” Becomes Parallel

61. CS560060 Parallel Lines Become a “Pencil of Lines” !

62. CS560061 What Does This Mean? p

63. CS560062 “True” Perspective in 2D p

64. CS560063 “True” Perspective in 2D p

65. CS560064 Viewing Frustum

66. CS560065 What happens for large p?”

67. CS560066 Projection Becomes Orthogonal (x,y) p h=y

68. The End of Transformations II 67 Lecture Set 6