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Transformations II

Transformations II. CS5600 Computer Graphics by Rich Riesenfeld 4 March 2002. Lecture Set 6. What About Elementary Inverses?. Scale Shear Rotation Translation. Scale Inverse. Shear Inverse. Shear Inverse. Rotation Inverse. Rotation Inverse. Rotation Inverse. Translation Inverse.

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Transformations II

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

  2. What About Elementary Inverses? • Scale • Shear • Rotation • Translation CS5600

  3. Scale Inverse CS5600

  4. Shear Inverse CS5600

  5. Shear Inverse CS5600

  6. Rotation Inverse CS5600

  7. Rotation Inverse CS5600

  8. Rotation Inverse CS5600

  9. Translation Inverse CS5600

  10. Translation Inverse CS5600

  11. Shear in x then in y CS5600

  12. Shear in y then in x CS5600

  13. Results Are Different y then x: x then y: CS5600

  14. Want the RHR to Work CS5600

  15. 3D Positive Rotations CS5600

  16. Transformations as a Change in Coordinate System • Useful in many situations • Use most natural coordination system locally • Tie things together in a global system CS5600

  17. Example 4 3 2 1 CS5600

  18. Example is the transformation that takes a point in coordinate system j and converts it to a point in coordinate system i CS5600

  19. Example CS5600

  20. Example CS5600

  21. Recall the Following CS5600

  22. Since CS5600

  23. Change of Coordinate System • Describe the old coordinate system in terms of the new one. x’ y’ CS5600

  24. Change of Coordinate System Move to the new coordinate system and describe the one old. Old is a negative rotation of the new. y x CS5600

  25. 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 CS5600

  26. Many Kinds of Perspective Used • Mechanical Engineering • Cartography • Art CS5600

  27. Perspective in Art • Naïve (wrong) • Egyptian • Cubist (unrealistic) • Esher • Miro • Matisse CS5600

  28. Egyptian Frontalism • Head profile • Body front • Eyes full • Rigid style CS5600

  29. 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

  30. Perspective in Cubism Braque, Georges Woman with a Guitar Sorgues, autumn 1913 CS5600

  31. Perspective in Cubism CS5600

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

  33. Pablo Picaso Cabeza de mujer llorando con pañuelo 33

  34. Perspective (Mural) Games M C Esher, Another World II (1947) CS5600

  35. Perspective M.C. Escher, Ascending and Descending (1960) CS5600

  36. M. C. Esher M.C. Escher, Ascending and Descending (1960) CS5600

  37. M. C. Esher • Perspective is “local” • Perspective consistency is not “transitive” • Nonplanar (hyperbolic) projection CS5600

  38. Nonplanar Projection M C Esher, Heaven and Hell CS5600

  39. Nonplanar Projection M C Esher, Heaven and Hell CS5600

  40. David McAllister The March of Progress, (1995) CS5600

  41. Joan Miro The Tilled Field Flat Perspective:What cues are missing? CS5600

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

  43. Henri Matisse, Danse II (1910) 43

  44. CS5600

  45. Norway is at High Latitude CS5600

  46. Isometric View CS5600

  47. Engineering Drawing A A Section AA CS5600

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

  49. “True” Perspective in 2D (x,y) h p CS5600

  50. “True” Perspective in 2D CS5600

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