Fractals and L-Systems

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Fractals and L-Systems

• Brian Cavalier
• 15-462
• November 25, 1997

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Overview

• The notion of a fractal
• Self similarity and Iteration
• Fractal Dimension
• Fractal Uses
• L-Systems

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Whats With Euclid Anyway? (Motivation)

• Arent spheres, cubes, and polygons good enough?
• Well, it depends on your intent, but ...
• Not many interesting real-world objects are
  really traditional geometric objects.
• Coastlines, trees, lightning
• These could be modeled with cylinders, polygons,
  etc, but ...
• Their true level of complexity can only be
  crudely approximated with traditional geometric
  shapes

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First, a Question

• How long is the coast line of Britain?
• How would you measure it?

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A Solution?

• Start by looking at it from a satellite
• Pick equally spaced points and draw lines between
  them. Then measure the lines
• Now you know the length, right?
• Well, sort of ...
• What if you move closer and decrease the distance
  between the points?
• Suddenly what looked like a straight stretch of
  coast becomes bays and peninsulas

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The Notion of a Fractal

• ... a geometrically complex object, the
  complexity of which arises through the repetition
  of form over some range of scales. -- Ken
  Musgrave, Texturing and Modeling
• Say what?

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Self-Similarity and Iteration

• Simple
• Shapes repeat themselves exactly at different
  scales
• Koch curves and snowflakes
• Statistical
• Statistics of a random geometry are similar at
  different scales
• Trees, rivers, mountains, lightning

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Physical Dimension

• A point, line, plane, and cube all have physical
  dimension that we accept as being 0, 1, 2, and 3
  respectively.
• Mandelbrots view
• Dimension is a scale-dependent term
• Consider zooming in on a ball of thread

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Fractal Dimension

• In the Euclidian plane
• we usually speak of topological dimension, which
  is always an integer.
• Fractal dimensions can have non-integer values.
• What does a dimension of, say, 2.3 mean?
• 2 is the underlying Euclidian dimension of the
  fractal
• The fractional value slides from 0 to 0.9999...
  as the fractal goes from occupying its Euclidian
  dimension, e.g. plane, to densely filling some
  part of the next higher dimension.

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Simple Fractals Using Iteration

• Use mathematical iteration and simple logic to
  produce amazing images
• One kind, you have already seen in the Koch and
  Sierpinski fractals
• Another kind is more numerical
• Mandelbrot
• Julia
• fractional Brownian motion

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Mandelbrot Set - How?

• Simple algorithm using points on the complex
  plane
• Pick a point on the complex plane
• The corresponding complex number has the form x
  iy, where i sqrt( - 1 )
• Iterate on the function Zn Zn-1 2 C
• Where Z0 0, and Z1 C2 C
• What does Z do? It either
• Remains near the origin, or
• Eventually goes toward infinity

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Mandelbrot Set - How? (2)

• To produce pretty pictures, do this for every
  point C on the complex plane where -2.5 lt x lt
  1.5, and -1.5 lt y lt 1.5
• See how many iterations it takes for Z to go to
  infinity
• Uh, how do you compute that?
• Just pick some max number of iterations to check
• It turns out if the distance ever grows to a
  value gt 2, the iteration will go to infinity.
• Color the point C based on the number of
  iterations

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Julia Sets

• Just a variant on Mandelbrot Sets
• There is an entire Julia Set corresponding to
  each point in the complex plane.
• An infinite number of Julia Sets
• Most interesting sets tend to be those found by
  using points just outside the Mandlebrot set.
• Similar Iteration formula, but vary Z not C
• Zn Zn-12 C

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Fractional Brownian Motion

• double fBm( Vector point, double H, double
  lacunarity, double octaves )
• if ( firstTimeCalled )
• initialize exponent table for octaves
• for( i0 to octaves-1 )
• value BasisFuction( point )exponentForOctave(
  i )
• point.x lacunarity
• point.y lacunarity
• point.z lacunarity
• if( octaves not and integer )
• value fractionalPartOf( octaves )
• BasisFunction( point )exponentForOctave( i )
• return value

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Cool, But What are They Good For?

• Not just neat pictures in 2D
• Procedural Textures
• Terrain Generation
• Other Realism
• Music?!?

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L-Systems

• Remember the Koch Curve Snowflake?
• Think of it as a grammar

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Just Add Water ...

• L-Systems grow based on their production rules
• Plants
• Cells
• Can control the iteration of production rules and
  actually animate the growth
• Grammars of fine enough granularity could
  simulate growth at the cell level ... talk about
  eating CPU cycles ...