1. CIS 781 Fractal Terrains

2. x and y are sampled on a 2D integer grid
Terrain Map
Height Map
z = f(x, y)
x and y are sampled on a 2D integer grid
Real data : Satellite, Elevation maps
Synthetic : Texture map, Noise functions

3. Terrain Map
Connect samples into a mesh

4. Terrain Map Natural looking Use textures to model detail?
Model high detail
Size
Storage (Millions of polygons)
Rendering speed
Zoom-in, zoom-out (does not scale well)
Use textures to model detail?
Not real geometry (bump mapping at silhouettes)
Not scale independent
Repeated textures do not work well

5. Fake Terrain Generate the heightfield Random process
Reflects “realistic” terrain in some way

6. Random Terrain Simple: Next step: Terrain(x,y) = rand( MAX_HEIGHT ) 1
Results in random noise
Next step:
Smooth the terrain generated above
FIR filter: 1 R

7. Filtering FIR (Finite Impulse Response) filter:
Pass a “window” over the data
Multiply data in window by weights
Result = d1*w1 + d2*w2+ … + dN*wN
Window + weights is called “filter kernel”
Larger window grabs more data

8. Random Fault Lines Repeatedly:
+dHeight
Repeatedly:
Create a line that crosses terrain
Add dHeight offset on one side of line
dHeighti = dHeight0 + (i/n)(dHeightn -dHeight0)
Select random endpoints to generate line
Compute “side of line” by plugging X,Y into line equation

9. More Realistic Faults Apply FIR filter to faulted heightmap
Smooth out abrupt fault transitions: “Erosion”

10. Particle Deposition
Drop particles in areas where you want volcanic mountains
Particles flow downhill until they have cause to rest
Move the drop point to create interesting peak

11. Particle Deposition Flow computation
At given point, find lowest neighbor
If local geometry is not flat enough, move new particle on top of lowest neighbor
Roll down hill
“Flat enough” = 27 degrees or so
Check 3x3 pixel window or 5x5 to check neighbors.

12. Particle Deposition

13. Perlin Noise
Using a sampling of 2D perlin Noise provides smooth hills.

14. Terrain Coloring
Using a 1D texture map based on the altitude can provide many useful mapping.

15. Terrain Coloring
Striped 1D texture map.

16. Terrain Coloring
Using a 2D texture map provides richer detail, but is independent of the terrain.

17. Terrain Coloring
More advanced coloring is based on altitude and slope.

18. Rolling Hills
Scaling in one dimension gives smooth rolling hills.

19. Procedural Modeling With Fractals
Compute geometry “on-the-fly”
Fractals
Model Natural Phenomena - Self Similarity
Mountains, fire, clouds, etc.
Scales to infinity
Add or “make up” natural looking details with mathematical tools

20. “Repetition of form over a variety of scales”
Fractals
“Repetition of form over a variety of scales”
Mandelbrot set, Julia set

21. Two Fractal Properties
Self-similarity

22. Two Fractal Properties
Fractal Dimension
Euclidean dimensions : 1, 2, 3, 4, …
Fractal : ,
Measure of detail or roughness of a fractal
D = (ln N)/(ln 1/s)

23. Midpoint Subdivision
Midpoint (recursive) subdivision

24. Midpoint Subdivision Brownian Motion Fractional Brownian Motion
Describes random movement of particles in a gas or fluid
Fractional Brownian Motion
Brownian Motion + Fractal Dimension
A useful model for natural phenomena

25. Fractional Brownian Motion
Equations are compute intensive
Approximate with “A family of 1D Gaussians ”
Zero mean
Standard Deviation : S = k2-iH
H = fractal dimension (roughness)

26. Fractional Brownian Motion
Fractal dimension = roughness, I.e. H

27. Terrain Modeling Criteria
Input
Initial coarse mesh + stochastic parameters
Two criteria
Internal Consistency
Reproducibility : Model is independent of position and orientation
Associate “random numbers” to point index
External Consistency
Continuity between adjacent primitives

28. Fractal Mountains Fournier & Fussell (1980)
Recursively subdivide geometry by random number d:
–dHeight/2 < d < dHeight/2
At each recursion:
dHeight *= 2-r
r=1 : self-similar
r>1 : large features early
r<1 : large features late A B A B A B

29. Triangle Subdivision
Subdivide a triangle into 4 triangles at edge midpoint

30. Quadrilateral Subdivision
Subdivide a quad into 4 quads at edge midpoints and a new center point.

31. Diamond-Square Subdivision
Alternate subdivision

32. Fractal Terrain

33. Mesh Subdivision Square-square Subdivision
Addresses “creasing problem” (slope discontinuities)
Subdivide parametric patches

34. Mesh Subdivision Displacement is scaled by the recursion level.
| b – a | -rn
When do you stop the recursion?
Pixel resolution
Displace upward, or towards the normal to the surface?

35. Mesh Subdivision External Consistency
Avoid tears between neighboring polygons
How do we compute the normals?
Average polygon normals at each vertex.
Propagate normals down the recursion
Cheaper : use the original mesh normals only

36. Ridged Fractal Terrains
To create sharp peaks, add an absolute value and flip the surface upside down.
Or, reflect about a maximum value.
Provides a volcano-like effect.

37. Caldera

38. Multi-Fractal
This produces a ridged multi-fractal

39. Add a 1D Texture

40. More Controls
To create plateaus, add a min function to flatten out high areas.

41. More Controls
To create a plain, add a max function to flatten out low areas.

42. More Controls
Multiply by a Guassian to limit the mountain range.

43. More Controls
To create valleys, create a lower amplitude and rather smooth terrain to use as the max operator.

44. max (Fractal, Smooth Noise)

45. max (Fractal, Smooth Noise)

46. Ridges Quantize a terrain to create ridges
Use directly or as the min function.
Can also be done as a transfer function that maps f(x)->g(x).
New height
height

47. Ridges

48. Other techniques Cracked terrains for dry lakes and riverbeds.
Throw random points onto the plane.
Construct the Voronoi diagram and use the graph to etch into the terrain.

49. Cracked Terrains
Crawfis, 2002

50. Fractal Terrains
Multifractals

51. Fractal Terrains
Color, texture

52. Fractal Terrains

53. Erosion - Higher-level control
Musgrave, et al. “The Synthesis and Rendering of Eroded Fractal Terrains”, Siggraph 1989.
Hydraulic Erosion
Water (rain) depositing to settle in height field
Thermal Weathering
“Any material that knocks material loose, which then falls down to pile up at the bottom of an incline.”

54. Erosion

55. Erosion by water

56. Erosion by wind and water