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Introduction Modeling of highly complex and irregular objects - PowerPoint PPT Presentation

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Introduction Modeling of highly complex and irregular objects Cannot be represented with Euclidian Geometry Methods Fractional Dimension (D) Self-Similarity Generation of Fractals Basic Principle F : Transformation Function P

  2. Introduction • Modeling of highly complex and irregular objects • Cannot be represented with Euclidian Geometry Methods • Fractional Dimension (D) • Self-Similarity

  3. Generation of Fractals Basic Principle F : Transformation Function P 0 (X 0 ,Y 0 ) : Initial Point P 1 = F(P 0 ) P 2 = F(P 1 ) = F(F(P 0 )) P 2 = F(P 2 ) = F(F(P 1 )) = F(F(F(P 0 )))

  4. Generation of Fractals Example

  5. Generation of Fractals Example

  6. Generation of Fractals Example

  7. Generation of Fractals Example

  8. Similarity Ratio 1-D : Line r = 1/n 2-D : Square r = 1/n 1/2 3-D : Cube r = 1/n 1/3

  9. Similarity Ratio D-Dimension : r = 1/n 1/D i.e. D = log(n)/log(1/r) Fractal Dimension

  10. Geometric Fractals • Number of Segments ( N) • Segment Length – Similarity Ratio r • Layout • Fractal Dimension D = log(N) / log(1/r) l • N = 4 • r = l / L = 0.5 • D = 2 L

  11. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  12. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  13. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  14. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  15. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  16. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  17. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  18. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  19. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  20. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  21. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  22. Geometric Fractals Herter-Heighway Dragon N = 2 ; r = 0.5 1/2 ; D = log(2)/log(2 1/2 ) = 2 Generator

  23. Geometric Fractals Sierpinski ’ s Gasket N = 3 ; r = 0.5 ; D = log(3)/log(2) = 1.58 Generator

  24. Geometric Fractals Effect of Dimension on the Fractal Curve Generator D = 1.26

  25. Geometric Fractals Effect of Dimension on the Fractal Curve Generator D = 1.89

  26. Geometric Fractals Effect of Dimension on the Fractal Curve Generator D = 2

  27. Geometric Fractals Effect of Dimension on the Fractal Curve

  28. Geometric Fractals Applications • Coastal Lines ( von Koch curve) • Trees • Textured Objects

  29. Geometric Fractals Applications Coastal Lines ( von Koch Curve )

  30. Geometric Fractals Applications Coastal Lines ( von Koch Curve )

  31. Geometric Fractals Applications Trees Parameterization θ b • Branch angle θ S • Stem branch ratio S/b Tree Generator

  32. Geometric Fractals Applications Trees

  33. Geometric Fractals Applications Textured Objects ( 3-D Fractals )

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