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Transformations I

Learn how to represent linear transformations using matrices in computer graphics. Explore scaling, rotation, shearing, and translation with practical examples and graphical views. Understand the role of matrices in transforming shapes in a 2D space.

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Transformations I

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  1. Transformations I CS5600Computer Graphics by Rich Riesenfeld 27 February 2002 Lecture Set 5

  2. Transformations and Matrices • Transformations are functions • Matrices are functions representations • Matrices represent linear transf’s CS5600

  3. What is a 2D Linear Transf ? Recall from Linear Algebra: CS5600

  4. Example: Scale in x CS5600

  5. Example: Scale in x by 2 What is the graphical view? CS5600

  6. Scale in x by 2 CS5600

  7. CS5600

  8. CS5600

  9. CS5600

  10. Summary on Scale • “Scale then add,” is same as • “Add then scale” CS5600

  11. Matrix Representation Scale in x by 2: CS5600

  12. Matrix Representation Scale in y by 2: CS5600

  13. Matrix Representation Overall Scale by 2: CS5600

  14. Matrix RepresentationShowing Same CS5600

  15. What about Rotation? Is it linear? CS5600

  16. Rotate by CS5600

  17. Rotate by : 1st Quadrant CS5600

  18. Rotate by : 1st Quadrant CS5600

  19. Rotate by : 2nd Quadrant CS5600

  20. Rotate by : 2nd Quadrant CS5600

  21. Rotate by : 2nd Quadrant CS5600

  22. Summary of Rotation by CS5600

  23. Summary (Column Form) CS5600

  24. Using Matrix Notation (Note that unit vectors simply copy columns) CS5600

  25. General Rotation by Matrix CS5600

  26. Who had linear algebra? Who understand matrices? CS5600

  27. What do the off diagonal elements do? CS5600

  28. Off Diagonal Elements CS5600

  29. Example 1 S CS5600

  30. Example 1 S CS5600

  31. Example 1 T(S) CS5600

  32. Example 2 S CS5600

  33. Example 2 S CS5600

  34. T(S) Example 2 CS5600

  35. Summary Shear in x: Shear in y: CS5600

  36. Double Shear CS5600

  37. Sample Points: unit inverses CS5600

  38. Geometric View of Shear in x CS5600

  39. Another Geometric View of Shear in x 39

  40. Another Geometric View of Shear in x 40

  41. Geometric View of Shear in y CS5600

  42. Another Geometric View of Shear in y h h 42

  43. Another Geometric View of Shear in y 43

  44. “Lazy 1” CS5600

  45. Translation in x CS5600

  46. Translation in x CS5600

  47. Homogeneous Coordinates CS5600

  48. Homogeneous Coordinates CS5600

  49. Homogeneous Coordinates Homogeneous term effects overall scaling CS5600

  50. Homogeneous Coordinates An infinite number of points correspond to (x,y,1). They constitute the whole line (tx,ty,t). (tx,ty,t) w = 1 (x,y,1)

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