Fractals Ed Angel Professor Emeritus of Computer Science

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Presentation transcript:

Fractals Ed Angel Professor Emeritus of Computer Science University of New Mexico E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Modeling Geometric Procedural Meshes Hierarchical Curves and Surfaces Particle Systems Fractal E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Sierpinski Gasket Rule based: Repeat n times. As n →∞ Area→0 Perimeter →∞ Not a normal geometric object E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Coastline Problem What is the length of the coastline of England? Answer: There is no single answer Depends on length of ruler (units) If we do experiment with maps at various scales we also notice self-similarity each part looks a whole E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Fractal Geometry Created by Mandelbrot Self similarity Dependence on scale Leads to idea of fractional dimension Graftals: graphical fractal objects E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Koch Curve/Snowflake E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Fractal Dimension Start with unit line, square, cube which we should agree are 1, 2, 3 dimensional respectively under any reasonable dimension Consider scaling each one by a h = 1/n E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

How Many New Objects? Line: n Square: n2 Cube: n3 The whole is the sum of its parts implies = 1 d = E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Examples Koch Curve Sierpinski gasket Scale by 3 each time Create 4 new objects d = ln 4 / ln 3 = 1.26186 Sierpinski gasket Scale by Create 3 new objects d = ln 3 / ln 4 = 1.58496 E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Volumetric Examples d = ln 4/ ln 2 = 2 D = ln 20 / ln 3 = 2.72683 E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Midpoint subdivision Randomize displacement using a Gaussian random number generator. Reduce displacement each iteration by reducing variance of generator. E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Fractal Brownian Motion variance ~ length -(2-d) Brownian motion d = 1.5 E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Fractal Mountains E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Iteration in the Complex Plane E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Mandelbrot Set iterate on zk+1=zk2+c with z0 = 0 + j0 Two cases as k →∞ |zk |→∞ |zk | remains finite If for a given c, |zk | remains finite, then c belongs to the Mandelbrot set E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Mandelbrot Set E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

Mandelbrot Set E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012