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Fractals and Iterated Affine Transformations

Fractals and Iterated Affine Transformations. Dr. Scott Schaefer. What are Fractals?. What are Fractals?. What are Fractals?. What are Fractals?. What are Fractals?. Recursion made visible A self-similar shape created from a set of contractive transformations. Contractive Transformations.

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Fractals and Iterated Affine Transformations

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  1. Fractals and Iterated Affine Transformations Dr. Scott Schaefer

  2. What are Fractals?

  3. What are Fractals?

  4. What are Fractals?

  5. What are Fractals?

  6. What are Fractals? • Recursion made visible • A self-similar shape created from a set of contractive transformations

  7. Contractive Transformations • A transformation F(X) is contractive if, for all compact sets • Transformations on sets? • Distance between sets?

  8. Transformations on Sets • Given a transformation F(x), • In other words, F(X) means apply the transformation to each point in the set X

  9. Hausdorff Distance

  10. Hausdorff Distance

  11. Hausdorff Distance

  12. Hausdorff Distance

  13. Hausdorff Distance

  14. Hausdorff Distance

  15. Hausdorff Distance

  16. Hausdorff Distance

  17. Hausdorff Distance

  18. Hausdorff Distance

  19. Contractive Transformations • A transformation F(X) is contractive if, for all compact sets

  20. Iterated Affine Transformations • Special class of fractals where each transformation is an affine transformation

  21. Rendering Fractals Given starting set X0 • Attractor is

  22. Rendering Fractals Given starting set X0 • Attractor is

  23. Rendering Fractals Given starting set X0 • Attractor is

  24. Rendering Fractals Given starting set X0 • Attractor is

  25. Rendering Fractals Given starting set X0 • Attractor is

  26. Rendering Fractals

  27. Rendering Fractals

  28. Rendering Fractals

  29. Rendering Fractals

  30. Rendering Fractals

  31. Rendering Fractals

  32. Rendering Fractals

  33. Rendering Fractals

  34. Rendering Fractals

  35. Rendering Fractals

  36. Rendering Fractals

  37. Rendering Fractals

  38. Rendering Fractals

  39. Rendering Fractals

  40. Rendering Fractals

  41. Rendering Fractals

  42. Rendering Fractals

  43. Rendering Fractals

  44. Rendering Fractals

  45. Rendering Fractals

  46. Rendering Fractals

  47. Rendering Fractals

  48. Rendering Fractals

  49. Rendering Fractals

  50. Contractive Transformations • A transformation F(X) is contractive if, for all compact sets • A set of transformations has a unique attractor if all transformations are contractive • That attractor is independent of the starting shape!!!

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