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

2. What About Elementary Inverses?
Scale
Shear
Rotation
Translation
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3. Scale Inverse
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4. Shear Inverse
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5. Shear Inverse
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6. Rotation Inverse
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7. Rotation Inverse
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8. Rotation Inverse
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9. Translation Inverse
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10. Translation Inverse
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11. Shear in x then in y
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12. Shear in y then in x
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13. Results Are Different
y then x:
x then y:
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14. Want the RHR to Work
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15. 3D Positive Rotations
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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
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17. Example 4 3 2 1
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18. Example
is the transformation that takes a point in coordinate system j and converts it to a point in coordinate system i
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19. Example
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20. Example
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21. Recall the Following
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22. Since
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23. Change of Coordinate System
Describe the old coordinate system in terms of the new one. x’ y’
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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
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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
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26. Many Kinds of Perspective Used
Mechanical Engineering
Cartography
Art
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27. Perspective in Art Naïve (wrong) Egyptian Cubist (unrealistic) Esher
Miro
Matisse
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28. Egyptian Frontalism Head profile Body front Eyes full Rigid style
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29. Uccello's ( ) 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 ) 29

30. Perspective in Cubism Woman with a Guitar Sorgues, autumn 1913
Braque, Georges
Woman with a Guitar
Sorgues, autumn 1913
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31. Perspective in Cubism
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32. Pablo Picaso, Madre con niño muerto (1937)32

33. Pablo Picaso Cabeza de mujer llorando con pañuelo33

34. Perspective (Mural) Games
M C Esher, Another World II (1947)
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35. M.C. Escher, Ascending and Descending (1960)
Perspective
M.C. Escher, Ascending and Descending (1960)
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36. M.C. Escher, Ascending and Descending (1960)
M. C. Esher
M.C. Escher, Ascending and Descending (1960)
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37. M. C. Esher Perspective is “local”
Perspective consistency is not “transitive”
Nonplanar (hyperbolic) projection
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38. M C Esher, Heaven and Hell
Nonplanar Projection
M C Esher, Heaven and Hell
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39. M C Esher, Heaven and Hell
Nonplanar Projection
M C Esher, Heaven and Hell
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40. The March of Progress, (1995)
David McAllister
The March of Progress, (1995)
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41. Flat Perspective:What cues are missing?
Joan Miro
The Tilled Field
Flat Perspective:What cues are missing?
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42. Flat Perspective:What cues are missing?
Henri Matisse, La Lecon de Musique 42

43. Henri Matisse, Danse II (1910)
Sample speaker note (rfr)
Henri Matisse, Danse II (1910) 43
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44. CS5600

45. Norway is at High Latitude
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46. Isometric View
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47. Engineering Drawing A A Section AA CS5600 CS5600 Section AA shows:
1. What object looks like when viewed orthogonally
to cut-away looking in direction of arrows.
2. Bottom part shows solid material, no holes
encountered.
3. Top part of Section AA shows orthogonal view in
looking in direction of arrows, BUT as though the
cut-away were pivoted to vertical.
4. Depth of "counter bore hole" which cannot be
determined from main drawing. Concentric holes
only indicate that something like this is going on.
5. This is a interesting way in which engineering
drawing communicate 3D info on 2D media. A
Section AA
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48. Engineering Drawing: Exploded View
Understanding 3D Assembly in a 2D Medium 48

49. “True” Perspective in 2D
(x,y) h p
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50. “True” Perspective in 2D
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51. “True” Perspective in 2D
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52. “True” Perspective in 2D
This is right answer for screen projection
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53. Geometry is Same for Eye at Origin
Screen Plane
(x,y) h p
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54. What Happens to Special Points?
What is this point??
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55. Look at a Limit
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56. Where does Eye Point Go? It gets sent to on x-axis
Where does on x-axis go?
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57. It comes back to virtual eye point!
What happens to ?
It comes back to virtual eye point!
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58. What Does This Mean? p
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59. The “Pencil of Lines” Becomes Parallel
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60. Parallel Lines Become a “Pencil of Lines” !
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61. What Does This Mean? p
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62. “True” Perspective in 2D
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63. “True” Perspective in 2D
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64. Viewing Frustum
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65. What happens for large p?”
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66. Projection Becomes Orthogonal
(x,y)
h=y p
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67. The End of Transformations II
Lecture Set 6 67