cs4395: computer graphics 1 fractals mohan sridharan based on slides created by edward angel
TRANSCRIPT
![Page 1: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/1.jpg)
CS4395: Computer Graphics 1
Fractals
Mohan SridharanBased on slides created by Edward Angel
![Page 2: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/2.jpg)
Modeling
• Geometric:– Meshes.– Hierarchical.– Curves and Surfaces (coming up soon!).
• Procedural:– Particle Systems.– Fractal.
CS4395: Computer Graphics 2
![Page 3: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/3.jpg)
Sierpinski Gasket
• Rule based:
• Repeat n times. As n →∞:– Area→0– Perimeter →∞
• Not a normal geometric object.
CS4395: Computer Graphics 3
![Page 4: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/4.jpg)
Coastline Problem
• What is the length of the coastline of England?
• There is no single answer:– Depends on length of ruler (units).
• If we experiment with maps at various scales we also notice self-similarity: each part looks like a whole!
CS4395: Computer Graphics 4
![Page 5: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/5.jpg)
Fractal Geometry
• Created by Mandelbrot:– Self similarity.– Dependence on scale.
• Leads to the idea of fractional dimension.
• Graftals: graphical fractal objects.
CS4395: Computer Graphics 5
![Page 6: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/6.jpg)
Koch Curve/Snowflake (Figure 11.12)
CS4395: Computer Graphics 6
• Recursive lengthening:
• In the limit, infinite length and discontinuous first derivative.• Not a 2D object either!
![Page 7: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/7.jpg)
Fractal Dimension• Start with unit line, square, cube which we agree are 1D, 2D,
3D respectively under any reasonable dimension.
• Consider scaling each one by a h = 1/n, the smallest unit we can measure.
• Scale object by h and replicate k times.
CS4395: Computer Graphics 7
![Page 8: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/8.jpg)
How Many New Objects?
• Line: k = n
• Square: k = n2
• Cube: k = n3
• The whole is the sum of its parts implies:
8CS4395: Computer Graphics
ndk
= 1 n
k
ln
lnd =
![Page 9: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/9.jpg)
Examples
• Koch Curve:– Sub-division (scaling) of the original by a factor of 3.– Create 4 new objects.– Fractal dimension: d = ln 4 / ln 3 = 1.26186.
• Sierpinski gasket:– Sub-division (scaling) by a factor of 2.– Keep 3 of the 4 triangles created.– d = ln 3 / ln 2 = 1.58496.
CS4395: Computer Graphics 9
![Page 10: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/10.jpg)
Volumetric Examples
CS4395: Computer Graphics 10
• 3D version of Sierpinski gasket:• d = ln 4/ ln 2 = 2.
• One iteration of the sponge:• d = ln 20 / ln 3 = 2.72683.
![Page 11: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/11.jpg)
Midpoint subdivision
CS4395: Computer Graphics 11
Randomize displacement using a Gaussian random number generator. Reduce displacement each iteration by reducing variance of generator.
![Page 12: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/12.jpg)
Fractal Brownian Motion
CS4395: Computer Graphics 12
variance ~ length -(2-d)
Brownian motion d = 1.5
![Page 13: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/13.jpg)
Fractal Mountains
CS4395: Computer Graphics 13
![Page 14: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/14.jpg)
Iteration in the Complex Plane
CS4395: Computer Graphics 14
![Page 15: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/15.jpg)
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.
CS4395: Computer Graphics 15
![Page 16: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/16.jpg)
Mandelbrot Set (Section 11.8.5)
CS4395: Computer Graphics 16
![Page 17: CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel](https://reader036.vdocuments.net/reader036/viewer/2022072008/56649d755503460f94a5697f/html5/thumbnails/17.jpg)
Mandelbrot Set
CS4395: Computer Graphics 17