Fractals Introduction to Fractals •Term Fractal coined by Benoit Mandelbrot •Properties of fractal: –Self-similarity (small portion looks like the whole object )

$Page 1: Fractals Introduction to Fractals •Term Fractal coined by Benoit Mandelbrot •Properties of fractal: –Self-similarity (small portion looks like the whole object )$
1

CS 430/536Computer Graphics I

FractalsWeek 10, Lecture 19

David Breen, William Regli and Maxim PeysakhovGeometric and Intelligent Computing Laboratory

Department of Computer ScienceDrexel University

http://gicl.cs.drexel.edu

2

Overview

• Fractal representations• Algorithms for drawing fractals• Mandelbrot Set• Brownian Motion

1994 Foley/VanDam/Finer/Huges/Phillips ICG

3

Introduction to Fractals• Term Fractal coined by Benoit Mandelbrot• Properties of fractal:

– Self-similarity (small portion looks like the whole object )– Have fractional dimensions– Non-differentiable– Infinite length

Construction of Koch Curve

Compiled from Gary W. Flake “The Computational Beauty of Nature”

4

Length of the Fractals

• Lewis Richardson(1961) measuredlength of coastlines

• Total length increaseas measurementstick reduced

• Does this meancoastlines are infinitein length?


5

Koch Curve

http://www.arcytech.org/java/fractals/

6

• Has no tangent at any point• Length of the curve at iteration n is (4/3)n

• Koch curve is 1.26186 …dimensional object

Properties of Koch Curve


1.77777780.11111111621.33333330.333333341

3.11798*10121.94033*10-481.04858*1060100

1110Total lengthLength of segmentNum of segmentsStep

7

Fractal Dimensions• 1D line of unit length divided into n segments

looks like the whole line scaled by 1/n• 2D square of unit area divided into n segments

looks like the whole square scaled by 1/n1/2– i.e. 4 segments, each one 1/2 the size of the original

• Divide an object into n segments where eachsegment is scaled by 1/s

• s = n1/d d = log(n)/log(s) 2 = log(4)/log(2)• The square is divided into 4 parts and each part is

1/2 the size of the original• d = log(4)/log(2) = 2

8

Fractal Dimensions• s = n1/d d = log(n)/log(s)• Koch curve divides line into 4 parts, each part is

1/3 the size of the original• Therefore the dimension of the Koch curve is:

!

41d = 3, n1/ d

= s

d =log(4)

log(3)=

log(n)

log(s)=1.26186 ...

9

Space Filling Fractals• Peano curve (1890)• Fills a 2D region• Fractal dimension - 2• s = n1/d 9 segments 1/3 original size

2 = log(9)/log(3)

Construction of Peano Curve


10

Sierpinski Triangle

http://www.arcytech.org/java/fractals/

11

• Fractals often occur in nature• All natural fractals are grown• We can model natural objects with fractals

Naturally Occurring Fractals

Brain Lungs Kidney


13

Production Systems

• In 1968 A. Lindenmayer developeda formalism describing plant growth.

• Called L-system• L-system consists of:

– Seed cell (axiom)– Production rules

Axiom: B Rules: B=>F[-B][+B] F=>FF


14

PlantlikeFractals 1


15

PlantlikeFractals 2


16

PlantlikeFractals 3


17

• Fractal often described to contain miniatureversions of itself.

• We can use affine transformations to describewhere these miniatures should be placed– Translation– Rotation– Scaling

• Self-affine fractals may have different scalingfactors in different dimensions

Affine Transformation Fractals


18

• Can be simulated with real copy machine• Copy machine takes a seed image• Creates several transformed copies of it as an output.• Procedure is recursive

Multiple Reduction CopyMachine Algorithm


19

MRCM Examples


20

Problems with MRCM

• To get good image wemust compute MRCMto large depth

• Computing time growsexponentially with depth

• Problem amplified whenthe reduction in sizeduring one iteration issmall


Fern Leaf

21

• Observations:– Fractal consist entirely of points– Randomly and recursively apply affine

transformation Li(p) to point p– Converges to a fractal

Iterated Functional System


500 points 5000 points

22

• Pick random point of the seed image• Randomly pick one of the affine

transformations• Transform the point• Continue recursively

IFS Algorithm

Fern LeafCompiled from Gary W. Flake “The Computational Beauty of Nature”

23

IFS Generated Tree


24

• xn=xn-12+c for some complex number c

– For some c, x→ 0 as n→ ∞c ∈ Mandelbrot Set

– For some c, x→∞ as n→ ∞– For others neither

Mandelbrot Set

For each c in the complex planex0=0

for (n=1 to nmax) {xn= xn

2+cif (|xn|>2) break

}If (n < nmax) color c = white

else color c = blackfrom http://www.astro.su.se/~alexis/fractals/

Mandelbrot Set

Magnified region

25

Mandelbrot Set


26

Mandelbrot Set

Recorded with XaoS

27

Newton Set f(x) = x3 - 1 xn+1 = xn - f(x) / f'(x)

Recorded with XaoS

28

Phoenix Setzn + 1 = zn

2 + Re(c) + Im(c) * zn - 1

Recorded with XaoS

29

• The dance of pollen grains in a water drop(Robert Brown in 1827)

• Explained by AlbertEinstein in 1905

• Self-similar motion• Infinite length• Used to simulate

rivers mountainsand other randomnatural phenomena

Brownian Motion

30

• Fractional BM (fBM) is a generalization of BM toinclude memory– Integral on progress of random walk

• fBM characterized by it’s power spectrum– BM has 1/f2 power spectrum– fBM had 1/fβ power spectrum

with 1.0 ≤ β ≤ 3.0• Think of β as controlling

terrain roughness

Fractional Brownian Motion

Voss & Saupe in “The Science of Fractal images”

31

• Landscape generated by the XMOUNTAINS programby Steven Booth

• Realistic, self-similar image

XMOUNTAINS


32

XMOUNTAINS: Changing Fractal Dimension

34

Random MidpointDisplacement

• fBM computations are time consuming• RMD are faster to compute but less realistic• Basic idea:

– Start with 2 (2D) 4 (3D)random points.

– Compute the midpoint foreach interval

– Displace the midpoint atrandom but based on thedifference between endpoints

Michael Shantz @ University of Waterloo

35

RMD Algorithm

x

y

y(b)

y(a)

a b

x

y

y(b)

y(a)

a b

ymid

a+b2

x

y a b

d c

x

y a b

d c

h f

e

g

m

• Generate points a and b at randomymid = (1/2)(y(a)+y(b))+rr = s·(|b-a|)·rg()s - is a surface roughness

parameterrg()- returns a Gaussian random value

(mean = 0, variance = 1)

In 3D version:ym = (1/4)(y(a)+y(b)+y(c)+y(d))+r

36

Controlling the Topography

• To model real environments we want tocontrol the placement of peaks and valleys

• We can do it by setting up Control Surfaces.• Calculate random elevations based on:

– Elevation of controlsurface

– Average elevation ofthe midpoint

37

Fractal MountainsKen Musgrave

38

Cellular Automata

• Invented by Johnvon Neumann in1940s

• Thoroughly studiedby Wolfram

• Mechanism to studyreproduction

• Dynamic system– Discrete in space– Discrete in time


39


Cellular Automata in Nature

• Process which creates seashells has beenlinked to a one-dimensional CA

40

Conus Textile Seashell

Source: wikipedia.org

Documents

Fractals Introduction to Fractals •Term Fractal coined by Benoit Mandelbrot •Properties of fractal: –Self-similarity (small portion looks like the whole object )