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Fractals and L-Systems

Fractals and L-Systems. Brian Cavalier 15-462 November 25, 1997. Overview. The notion of a fractal Self similarity and Iteration Fractal Dimension Fractal Uses L-Systems. What’s With Euclid Anyway? (Motivation). Aren’t sphere’s, cubes, and polygons good enough?

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Fractals and L-Systems

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  1. Fractals and L-Systems Brian Cavalier 15-462 November 25, 1997

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

  3. What’s With Euclid Anyway? (Motivation) • Aren’t sphere’s, 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

  4. First, a Question • How long is the coast line of Britain? • How would you measure it?

  5. 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

  6. 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?

  7. 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

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

  9. 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.

  10. 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

  11. 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 + i*y, where i = sqrt( - 1 ) • Iterate on the function Zn = Zn-12 + C • Where Z0 = 0, and Z1 = C2 + C • What does Z do? It either • Remains near the origin, or • Eventually goes toward infinity

  12. Mandelbrot Set - How? (2) • To produce pretty pictures, do this for every point C on the complex plane where -2.5 < x < 1.5, and -1.5 < y < 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 > 2, the iteration will go to infinity. • Color the point C based on the number of iterations

  13. 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

  14. Fractional Brownian Motion double fBm( Vector point, double H, double lacunarity, double octaves ) { if ( firstTimeCalled ) { initialize exponent table for octaves } for( i=0 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; }

  15. Cool, But What are They Good For? • Not just neat pictures in 2D • Procedural Textures • Terrain Generation • Other Realism • Music?!?

  16. L-Systems • Remember the Koch Curve & Snowflake? • Think of it as a grammar:

  17. 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 ...

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