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...
![Page 1: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/1.jpg)
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
![Page 2: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/2.jpg)
Modeling
• Geometric:– Meshes.– Hierarchical.– Curves and Surfaces.
• Procedural:– Particle Systems.– Fractal.
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
20122
![Page 3: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/3.jpg)
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
![Page 4: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/4.jpg)
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
![Page 5: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/5.jpg)
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
![Page 6: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/6.jpg)
Koch Curve/Snowflake
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
20126
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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
![Page 8: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/8.jpg)
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 =
![Page 9: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/9.jpg)
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
![Page 10: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/10.jpg)
Volumetric Examples
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
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d = ln 4/ ln 2 = 2
d = ln 20 / ln 3 = 2.72683
![Page 11: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/11.jpg)
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.
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Fractal Brownian Motion
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201212
Variance ~ length -(2-d)
Brownian motion d = 1.5
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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!
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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
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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
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Mandelbrot Set
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201216
![Page 17: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/17.jpg)
Mandelbrot Set
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley
201217
![Page 18: Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Mohan Sridharan Based on Slides.](https://reader036.fdocuments.us/reader036/viewer/2022072007/56649d345503460f94a0b811/html5/thumbnails/18.jpg)
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
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