recapitulation - iit delhi

Recapitulation

•  Exhibit infinite detail •  Show self-similarity •  Generating approach

recursive procedure deterministic

•  Fractal dimension (D) roughness & filling-ness

Recapitulation

Recapitulation

P1 = F(P0) P2 = F(P1)=F(F(P0)) Pn = F(Pn-1)

Introduction

•  Process of Generation: Randomness •  Statistical Self-Similarity •  Different Generating Approaches

Midpoint Subdivision (curve)

A

B

Midpoint Subdivision (curve)

A

B C

M Issues Magnitude of MC Direction of MC

Midpoint Subdivision (curve)

A

B

Example

Example

Subdivision Methods (surface)

Triangle Edge subdivision

Iteration N

Subdivision Methods

v

v

v

Iteration N+1

Triangle Edge subdivision

Subdivision Methods

Triangle Edge subdivision

Subdivision Methods

Triangle Edge subdivision Issues

Amount of displacement Direction of displacement “consistency”

Subdivision Methods

Triangle Edge subdivision

Example

Subdivision Methods

Iteration N

Diamond Square subdivision

Subdivision Methods

Diamond Square subdivision

Iteration N+1

Subdivision Methods

Square Square subdivision

Iteration N

Subdivision Methods

Iteration N+1

Square Square subdivision

9 3

3 1

Bilinear Interpolation

Applications

•  Coast Lines •  Landscapes •  Clouds •  Candies •  Stones •  OTHERS

Applications

•  Coast Lines •  Landscapes •  Clouds •  Candies •  Stones •  OTHERS

Applications

•  Coast Lines •  Landscapes •  Clouds •  Candies •  Stones •  OTHERS

Fractal Plants (L-Systems)

•  Production rules applied to seed axioms. •  Example

•  Axiom B •  Rule B->ABB

Fractal Plants (L-Systems)

Fractal Plants (L-Systems)

http://algorithmicbotany.org Prof. PrzemyslawPrusinkiewicz

Mandelbrot’s Function

z z2 + c z and c are complex Algorithm

c : point in the region (x + iy) z0 : initial value of z zn = zn-1

2 + c Evaluate S(zn): if > Tolerance

assign color (n) to c If n > Limit

assign color (n) to c

Mandelbrot’s Function

General Formulation If the iteration function is z zα + c ; where α  : integer or real, positive or negative

Then the number of complete lobe structures L = floor (abs(α - 1))

Fractional part of α is proportional to the size of an emerging baby Lobe.

General Formulation