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Transformations II

Transformations II. CS5600 Computer Graphics Rich Riesenfeld Spring 2005. Lecture Set 7. Arbitrary 3D Rotation. What is its inverse? What is its transpose? Can we constructively elucidate this relationship?. Want to rotate about arbitrary axis a. 3.

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Transformations II

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  1. Transformations II CS5600 Computer Graphics Rich Riesenfeld Spring 2005 Lecture Set 7

  2. Arbitrary 3D Rotation • What is its inverse? • What is its transpose? • Can we constructively elucidate this relationship? Utah School of Computing

  3. Want to rotate about arbitrary axis a Utah School of Computing 3

  4. First rotate about z by Now in the (y-z)-plane Utah School of Computing

  5. Then rotate about x by Rotate in the (y-z)-plane Utah School of Computing

  6. Now perform rotation about Now a-axis aligned with z-axis 6

  7. Then rotate about x by Rotate again in the (y-z)-plane Utah School of Computing

  8. Then rotate about z by Now to original position of a Utah School of Computing

  9. We effected a rotation by about arbitrary axis a Utah School of Computing 9

  10. We effected a rotation by about arbitrary axis a Utah School of Computing 10

  11. Rotation about arbitrary axis a • Rotation about a-axiscan be effected by a composition of 5elementary rotations • We show arbitrary rotation as succession of 5rotations about principal axes Utah School of Computing

  12. In matrix terms, Utah School of Computing

  13. so, Similarly, Utah School of Computing

  14. Recall, Consequently, for because, Utah School of Computing

  15. It follows directly that, Utah School of Computing

  16. Utah School of Computing

  17. Constructively, we have shown, This will be useful later Utah School of Computing

  18. 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 Utah School of Computing

  19. Differert Perspectives Used • Mechanical Engineering • Cartography • Art Utah School of Computing

  20. Perspective in Art • “Naïve” (wrong) • Egyptian • Cubist (unrealistic) • Esher • Impossible (exploits local property) • Hyperpolic (non-planar) • etc Utah School of Computing

  21. “True” Perspective in 2D (x,y) h p Utah School of Computing

  22. “True” Perspective in 2D Utah School of Computing

  23. “True” Perspective in 2D This is right answer for screen projection Utah School of Computing

  24. “True” Perspective in 2D Utah School of Computing

  25. Perspective in Art • Naïve (wrong) • Egyptian • Cubist (unrealistic) • Esher • Miro • Matisse Utah School of Computing

  26. Egyptian Frontalism • Head profile • Body front • Eyes full • Rigid style Utah School of Computing

  27. Uccello's (1392-1475) hand drawing was the first extant complex geometrical form rendered according to the laws of linear perspective Perspective Study of a Chalice, Drawing, Gabinetto dei Disegni, Uffizi, Florence, ca 1430) 65

  28. Perspective in Cubism GeorgesBraque Woman with a Guitar (1913) Utah School of Computing

  29. Perspective in Cubism Utah School of Computing

  30. Pablo Picaso Madre con niño muerto (1937) 68

  31. Pablo Picaso,Cabeza de mujer llorando con pañuelo 69

  32. Perspective (Mural) Games M C Esher, Another World II (1947) Utah School of Computing

  33. Perspective Ascending and Descending (1960) M C Escher Utah School of Computing

  34. M. C. Escher M C Escher, Ascending and Descending (1960) Utah School of Computing

  35. M C Escher • Perspective is “local” • Perspective consistency is not “transitive” • Nonplanar (hyperbolic) projection Utah School of Computing

  36. Nonplanar (Hyperbolic) Projection M C Esher, Heaven and Hell Utah School of Computing

  37. Nonplanar (Hyperbolic) Projection M C Esher, Heaven and Hell Utah School of Computing

  38. David McAllister The March of Progress, (1995) Utah School of Computing

  39. Joan Miro: Flat Perspective The Tilled Field What cues are missing? Utah School of Computing

  40. Flat Perspective: What cues are missing? Henri Matisse, La Lecon de Musique 78

  41. Next 2 Images Contain Nudity ! Utah School of Computing

  42. Henri Matisse, Danse (1909) 80

  43. Henri Matisse, Danse II (1910) 81

  44. Atlas Projection Utah School of Computing

  45. Norway is at High Latitude There is considerable size distortion Utah School of Computing

  46. Isometric View Utah School of Computing

  47. Engineering Drawing: 2 Planes AA AA SectionAA Utah School of Computing

  48. Engineering Drawing: Exploded View Understanding 3D Assembly in a 2D Medium 86

  49. “True” Perspective in 2D (x,y) h p Utah School of Computing

  50. “True” Perspective in 2D Utah School of Computing

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