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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
