CS 455 Computer Graphics

1
CS 455 Computer Graphics

• Fractals in Graphics

2
Texturing with Images

• Texture maps with images are good because images
  are (sometimes) easy to create.
• In some cases, just taking a picture is easier
  than writing a program to generate the picture.
• But
• Images are hard to get for things that arent
  real.
• Images consume space.
• Images have finite precision.

3
Procedural Methods.

• Write a program to generate the texture.
• Save space.
• In some cases, can zoom in forever and retain
  precision
• But
• Hard to write the right procedure.
• However, ideas based on fractals go a long way.
• Especially for objects in nature.

4
Euclidean vs. Fractal

• Standard objects are represented in Euclidean
  geometry
• Described by equations
• This gives smooth, regular objects
• Spheres
• Polygons
• B-spline surfaces
• How do we model natural objects?
• Clouds
• Mountains
• Plants
• These kinds of objects are better represented
  with fractal geometry

5
What is a Fractal?

• Benoit Mandelbrot, 1982,
• clouds are not spheres, mountains are not
  cones, coastlines are not circles, and bark is
  not smooth, nor does lightning travel in a
  straight line.
• Objects are represented as procedures rather than
  equations
• Repeating the fractal procedure produces more and
  more detail

6
Properties of Fractals

• Self-similarity between object parts and the
  overall object features
• Fractional dimension
• No specific size or scaling
• Infinite detail looks good at every resolution
• Generated by a function, so no data to store.
• Some of them look like things you find in nature.
  

7
Fractal Mountain
The closer you get, the more detail you see
8
Fractal Generation

• A fractal object is generated by repeatedly
  applying a specified transformation
• Can apply the transformation to a point set, set
  of primitives (lines, curves, colors ,etc.), or
  anything else
• Theoretically, the procedure is applied an
  infinite number of times
• Practically, we iterate a finite number of times,
  then call it good.

9
Fractal Dimension (Hausdorff variant)
Split into ¼ the size and multiply by
4 Build original by splitting into 4 parts
and multiplying by 4
Split into ¼ the size And multiply by 16
42 Build original by splitting into 4 parts
and multiplying by 42
Build original by splitting into 4 parts and
multiplying by 43
10
Fractal Dimension

• We can talk about the fractalness of an object
  using the fractal dimension
• A measurement of how rough the object is
• The fractal dimension is computed using the scale
  factor we reduce by, and the number of times we
  replicate the scaled down object.

11
Fractal Dimension (Hausdorff variant)
Split into ¼ the size and multiply by
4 Build original by splitting into 4 parts
and multiplying by 4
Split into ¼ the size And multiply by 16
42 Build original by splitting into 4 parts
and multiplying by 42
Build original by splitting into 4 parts and
multiplying by 43
12
Sierpinski Triangle

• Take a triangle.
• Reduce by ½
• Make 3 copies and arrange them into a triangle
• Repeat forever.

13
Sierpinski Triangle
14
A Few Examples
Penrose tiling 1.974 Cauliflower 2.33
Menger sponge 2.726
http//en.wikipedia.org/wiki/List_of_fractals_by_H
ausdorff_dimension
15
c. 13 branches each 1/3 smaller, D
log(13)/log(3) 2.33
16
Fractals in Nature
http//www.treklens.com/gallery/Oceania/New_Zealan
d/photo22113.htm
http//www.funpike.com/images/plant-bushes/2fracta
l_nature_1.jpg
17
Fractal Classification

• Self-similar fractals
• Parts are scaled-down versions of the initial
  object
• Deterministic self-similar
• Non-random
• Statistically self-similar
• Some randomness introduced
• Self-affine
• Different scaling parameters in each coordinate
  direction
• Invariant Fractal Sets
• Formed with nonlinear transformations
• Sets in the complex plane (usually)

18
1. Self-Similar Fractals

• Parts are scaled-down versions of the entire
  object
• Start with the initial shape
• Construct sub-parts by duplicating the initial
  shape and scaling it
• Can apply different scaling factors for different
  parts
• Example
• The von Koch snowflake
• Can also introduce random variations to the
  sub-parts
• These fractals are statistically self-similar
• Typically used to model trees, shrubs, and other
  plants

19
Von Koch Snowflake

• Start with initiator
• And generator
• At each iteration, replace each piece of the
  initiator with the generator

20
Von Koch Snowflake

• Iteration 0

21
Von Koch Snowflake

• Iteration 1

22
Von Koch Snowflake

• Iteration 2

23
Von Koch Snowflake

• Iteration 3

24
Other Self-Similar Fractals
25
Statistically Self-Similar Fractals

• Self-similar fractals in which random variations
  aremade to the sub-parts
• Typically used to model trees, shrubs, and other
  plants

Marco Bubkewww.xfrogdownloads.com
Bernd Lintermann www.xfrogdownloads.com
26
Self-Affine Fractals

• The scale factor differs for each direction

Self Similar
Self Affine
27
Self Affine Fractals
Self Similar
Self Affine
28
Invariant Fractal Sets

• Formed with non-linear transformations
• Includes the self-squaring fractals and the
  self-inversefractals

29
Mandelbrot Set

• Iterate over a complex function
• Color the point in space based on how quickly the
  function diverges at that point
• Points that do not diverge are in the set
• The set is typically colored black, points that
  diverge at different speeds are colored
  accordingly

30
The Mandelbrot Set
31
Zooming into the set
www.chem.vu.nl/feenstra/mandel_gallery.html
32
Zooming into the set
www.chem.vu.nl/feenstra/mandel_gallery.html
33
Zooming into the set
www.chem.vu.nl/feenstra/mandel_gallery.html
34
Zooming into the set
www.chem.vu.nl/feenstra/mandel_gallery.html
35
Zooming into the set
www.chem.vu.nl/feenstra/mandel_gallery.html
36
Zooming into the set
www.chem.vu.nl/feenstra/mandel_gallery.html
37
Zooming into the set
www.chem.vu.nl/feenstra/mandel_gallery.html
38
Zooming into the set
www.chem.vu.nl/feenstra/mandel_gallery.html
39
Zooming into the Mandelbrot Set
Mandelbrot Zoom
Interactive Mandelbrot Zoom
40
Computing the Mandelbrot Set
For each pixel on the screen do x x0 // x
co-ordinate of pixel y y0 // y co-ordinate of
pixel x2 xx y2 yy iteration 0
maxiteration 1000 while ( x2 y2 lt (22)
AND iteration lt maxiteration ) y 2xy
y0 x x2 - y2 x0 x2 xx y2 yy
iteration iteration 1 if ( iteration
maxiteration ) color black else color
iteration
41
Midpoint Displacement Technique

• Start with an initial object
• Find the midpoint of the edges of the object
• Randomly perturb those midpoints
• Repeat recursively
• 2D for coastlines
• 3D for mountains

42
Midpoint Displacement
43
Midpoint Displacement
44
Midpoint Displacement
45
Fractal Mountains

• Start with the basic shape of the mountain
• Subdivide edges of the shape
• Randomly perturb the new vertices
• Repeat recursively

46
Fractal Terrain Results
47
More Fractal Terrain
48
www.cg.tuwien.ac.at
49
www.nextrevision.com
50
djconnel.com/Vue/
51
Fractal Terrain

• Problems?
• Just doesnt look quite right
• Coloration
• Landscape ruggedness
• What else?
• Modifications to the basic technique
• Randomize all points, not just midpoints
• Vary roughness based on height
• User guided terrain generation

52
Other Uses of Fractals

• Sierpinski Triangle

53
Other Uses of Fractals

• Fractal Brociflower

54
Other Uses of Fractals

• Fractal Planets

http//baddoggames.com/planet/gallery.htm
55
Other Uses of Fractals

• Fractal Chandelier