Fractals

5.

Iterated Function Systems

5.1.

Mathematical Framework

5.1.1.

Affine Transformations

5.1.1.1.

Scaling Transformations

5.1.1.2.

Rotation Matrices

5.1.1.3.

Translation Vectors

5.1.1.4.

Shear Transformations

5.1.2.

Contraction Mappings

5.1.2.1.

Contraction Ratio

5.1.2.2.

Fixed Point Theorem

5.1.3.

The Collage Theorem

5.1.3.1.

Statement and Proof Outline

5.1.3.2.

Applications in Fractal Construction

5.2.

IFS Construction Methods

5.2.1.

Deterministic IFS

5.2.2.

5.2.3.

Weighted Probability Systems

5.3.

Classic IFS Examples

5.3.1.

Barnsley's Fern

5.3.1.1.

Transformation Parameters

5.3.1.2.

Probabilistic Generation

5.3.2.

5.3.3.

Spiral Fractals

5.4.

Applications and Extensions

5.4.1.

Image Compression

5.4.2.

Natural Form Modeling

5.4.3.

Fractal Interpolation

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4. Classical Fractals

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6. Escape-Time Fractals