Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E...
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Transcript of Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E...
Chapter 9: Recursive Methods and Fractals
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
20121
Mohan SridharanBased on Slides by Edward Angel and Dave Shreiner
Modeling
• Geometric:– Meshes.– Hierarchical.– Curves and Surfaces.
• Procedural:– Particle Systems.– Fractal.
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
20122
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
20123
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
20124
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
20125
Koch Curve/Snowflake
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
20126
Fractal Dimension
• Start with unit line, square, cube which we agree are 1, 2, 3 dimensional respectively.
• Consider scaling each one by a h = 1/n.
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
20127
How Many New Objects?
• Line: n.• Square: n2.• Cube: n3.
• The whole is the sum of its parts, i.e., fractal dimension (d) is given by:
8E. Angel and D. Shreiner: Interactive
Computer Graphics 6E © Addison-Wesley 2012
ndk
= 1n
k
ln
lnd =
Examples
• Koch curve:– Subdivision (i.e., scale) by 3 each time.– Create 4 new objects.– d = ln 4 / ln 3 = 1.26186.
• Sierpinski gasket:– Subdivide (scale) side by 2.– Keep 3 of the 4 new triangles, i.e., create 3 new objects.– d = ln 3 / ln 2 = 1.58496.
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
20129
Volumetric Examples
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201210
d = ln 4/ ln 2 = 2
d = ln 20 / ln 3 = 2.72683
Midpoint subdivision
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201211
Randomize displacement using a Gaussian random number generator.
Reduce displacement in each iteration by reducing variance of generator.
Fractal Brownian Motion
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201212
Variance ~ length -(2-d)
Brownian motion d = 1.5
Fractal Mountains
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201213
Use fractals to generate mountains and natural terrain.
Tetrahedron subdivision + midpoint displacement.
Control variance of randomnumber generator to controlRoughness.
Can apply to mesh surfaces too!
Mandelbrot Set
Based on fractal geometry.Easy to generate but models infinite complexity in the shapes generated.
Based on calculations in the complex plane.
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201214
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
201215
Mandelbrot Set
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201216
Mandelbrot Set
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201217
More Details
• Section 9.8: fractals and recursive methods.
• Section 9.9: procedural noise.
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201218