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

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

Nov 29, 2023 312 likes •659 views

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

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Introduction

  • Modeling of highly complex and irregular objects
  • Cannot be represented with Euclidian Geometry

Methods

  • Fractional Dimension (D)
  • Self-Similarity
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Generation of Fractals

Basic Principle F : Transformation Function P0(X0,Y0) : Initial Point P1 = F(P0) P2 = F(P1) = F(F(P0)) P2 = F(P2) = F(F(P1)) = F(F(F(P0)))

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Generation of Fractals

Example

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Generation of Fractals

Example

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Generation of Fractals

Example

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Generation of Fractals

Example

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Similarity Ratio

1-D : Line

r = 1/n

2-D : Square

r = 1/n1/2

3-D : Cube

r = 1/n1/3

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D-Dimension : r = 1/n1/D

i.e.

D = log(n)/log(1/r)

Similarity Ratio

Fractal Dimension

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Geometric Fractals

  • Number of Segments ( N)
  • Segment Length – Similarity Ratio r
  • Layout
  • Fractal Dimension

D = log(N) / log(1/r)

L l

  • N = 4
  • r = l / L = 0.5
  • D = 2
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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Herter-Heighway Dragon N = 2 ; r = 0.51/2 ; D = log(2)/log(21/2) = 2

Generator

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Geometric Fractals

Sierpinski’s Gasket N = 3 ; r = 0.5 ; D = log(3)/log(2) = 1.58

Generator

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Geometric Fractals

Effect of Dimension on the Fractal Curve

Generator D = 1.26

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Geometric Fractals

Generator D = 1.89

Effect of Dimension on the Fractal Curve

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Geometric Fractals

Generator D = 2

Effect of Dimension on the Fractal Curve

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Geometric Fractals

Effect of Dimension on the Fractal Curve

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Geometric Fractals

Applications

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

Coastal Lines ( von Koch Curve ) Applications

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Geometric Fractals

Coastal Lines ( von Koch Curve ) Applications

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Geometric Fractals

Trees

Tree Generator

θ b S

Parameterization

  • Branch angle θ
  • Stem branch ratio S/b

Applications

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Geometric Fractals

Trees Applications

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Geometric Fractals

Textured Objects ( 3-D Fractals ) Applications