1. Fractals and Terrain Synthesis

2. WALL-E, 2008]

3. Proceduralism Philosophy of algorithmic content creation
Frees up artist time to concentrate on most important elements (hero characters, major locations)
Musgrave: "not one concession to the hated user"

4. Simulation and Optimization
models through simulation of underlying process
control through initial settings
may be difficult to adjust rules of simulation
Optimization:
models through energy minimization
control through constraints, energy terms
may be difficult to design energy function

6. [Rusnell, Mould, and Eramian 2009]

11. Height Fields Each point on xy-plane has a unique height value
Convenient for graphics – simplifies representation (can store in 2D array)
Used for terrain, water waves
Drawback: not able to represent full range of possibilities

13. Height Fields and Texture
Can use any texture synthesis process to generate height fields
simply interpret intensity as height, create mesh, render
The most successful processes have used fractals
self-similarity a feature of real terrains
self-similarity defining characteristic of fractals

15. Iterated Function Systems
Show up frequently in graphics
L-systems replacement grammar a celebrated example
Capable of producing commonly cited fractal shapes
Sierpinski gasket
Menger sponge
Koch snowflake

18. Mandelbrot Set Said to “encode the Julia sets”
coloring of the complex plane for connectivity of quadratic Julia sets
say Jc is the set for zn+1 = zn2 + c
Point c is in the Mandelbrot set if Jc is connected, not in the set otherwise
Partitions complex plane
“Mandelbrot separator” – fractal curve

20. Mandelbrot set calculation
Turns out that it is quite straightforward to get the Mandelbrot set computationally:
for each pixel c:
let z0 = c
compute z = z2+c repeatedly, until
(a) |z| > 2 (diverges)
(b) iteration count exceeds constant (say 1000)
if diverged, color it according to the iteration number on which it diverged
if never diverged, color with some special color

22. Fractals
Nonfractal complexity: arises from accretion of different kinds of detail
e.g., people: complex, but not self-similar
Fractal complexity: arises from repeating the same details
What detail to repeat?
Perlin noise a suitable source of detail

23. Multiresolution Noise
Different signals at different scales
Fractals: clouds, mountains, coastlines
sum
1/2
1/4
1/8
1/16

24. Multiresolution Noise
FNoise(x,y,z) = sum((2^-i)*Noise(x*2^i…))
Extremely common formulation – so common that many mistake it for the basic noise primitive

25. Fractional Brownian Motion
aka fBm
requires parameter H (relative size of successive octaves – "roughness")
val = 0;
for (i = 0; i < octaves; i++)
{ val += Noise(point)*pow(2,-H*i);
point *= 2; }

26. Fractional Brownian Motion
aka fBm
requires parameter H (relative size of successive octaves – "roughness")
val = 0;
for (i = 0; i < octaves; i++)
{ val += Noise(point)*pow(2,-H*i);
point *= 2; }
why 2?
"Lacunarity" parameter

27. Lacunarity
"Lacunarity" (from Latin "lacuna", gap) gives the spacing between octaves
Larger values mean fewer octaves needed to cover same range of scales
faster to compute
but individual octaves may be visible
Smaller values mean more densely packed octaves, richer appearance

28. Lacunarity Balance between speed and quality
Value of 2 the "natural" choice
but in genuinely self-similar fractals, may lead to visible artifacts as same features pile up
Transcendental numbers good
genuinely irrational, no piling at any scale
Values slightly over 2 offer good compromise of speed/appearance
e-1/2, π-1

29. Fractal ranges of scale
Real fractals are band-limited: they have detail only at certain scales
Computed fractals also band-limited
practical limitations: don’t write code with infinite loops
Mandelbrot: fractal objects have 3+ scales

30. Midpoint Displacement
Repeated subdivision:
begin with two endpoints; at each step, divide each edge and perturb the midpoint
In 2D: on alternate steps, divide orthogonal and diagonal edges
Among the first fractal terrain systems (Fournier/Fussell/Carpenter 1982)
Problems: seams from early points

31. Midpoint Displacement

32. Midpoint Displacement

33. Characteristics of fBm
Homogeneous: the same everywhere
Isotropic: the same in all directions
Real terrains are neither
mountains differ from plains
direction can matter (e.g., rivers flow downhill)
Require multifractals

34. Multifractals Fractal dimension varies with location
Simple multifractal: multiplicative cascade
val = 1;
for (i = 0; i < oct; i++)
{ val *= (Noise(point)+offset)*pow(2,-H*i)
point *= 2; }

36. Problems Multiplicative formation unstable (can diverge)
Extremely sensitive to value of offset
Control elusive

37. Hybrid multifractals
In real terrains, higher areas are rougher (new mountains) and lower areas smoother (worn down, silted over)
Musgrave: weight of each octave multiplied by current value of function
near value=0 (“sea level”), higher frequencies damped – very smooth
higher values: more jagged
need to clamp value to prevent divergence

38. Ridges Simple trick to get ridges out of noise:
Noise values range from -1 to 1
Take 1-|N(p)|
Absolute value reflects noise about y=0; negative moves reflections to top
Cellular texture (Voronoi regions) naturally has ridges, if distance interpreted as height

41. von Koch snowflake

42. L-Systems "Lindenmeyer systems", after Aristid Lindenmeyer (1960's)
Replacement grammar
set of tokens
rules for transformation of tokens
All rules applied simultaneously across string

43. L-Systems
Very successful for modeling certain classes of structured organic objects
ferns
trees
seashells
Success has impelled others to apply the methods more widely
rust
entire ecosystems

44. L-System example
Tokens: A, B
Rules
A → B
B → AB

45. L-System example Tokens: A, B Rules Initial string: A
B → AB
Initial string: A
Sequence: A, B, AB, BAB, ABBAB…
Lengths are Fibonacci numbers (why?)

46. Geometric Interpretation
Strings are interesting, but application to graphics requires geometric interpretation
Usual method: interpret individual tokens as geometric primitives

47. Turtle Graphics
The language Logo (1967) – once widely used for education
Turtle has heading and position; when it moves, it draws a line behind it
Commands:
F, B: move forward/backward fixed distance
+,- : turn right/left fixed angle
[, ] : push or pop the current state
A : no-op

51. L-Systems and the Turtle
Example replacement rules for the turtle:
F → F-F++F-F
everything else unchanged

52. von Koch snowflake

53. Branching 'Push' and 'pop' operators can produce branching:
A → F[+A][-A]FA
F → FF
A is an 'apex' – the tip of a branch
Each apex sprouts a new branch with buds midway along its length, while existing branches elongate

55. Turtle Graphics in 3D Turtle has orientation and position Commands:
F, B: move forward/backward fixed distance
+,- : turn right/left fixed angle (yaw)
^,& : turn up/down fixed angle (pitch)
\, / : roll right/left fixed angle
[, ] : push or pop the current state
A : no-op

57. Ternary Tree As usual, just one rule: F → F[&F][/&F][\&F]
Each segment has three branches attached to its tip