Procedural and Fractal Terrains

Gameplay goals for a terrain engine

• Large enough to travel around for hours

• Detailed when seen at a human scale

• Dynamic modification of terrain data

• Runs at high, stable frame rates

• Need fast rotation of viewpoint

Technical goals

• Level-of-detail management (static or continuous?)

• A lot of polygons (231 in un-reduced terrain, 70,000+ in a given tessellation)

• Efficient graphics primitives

• Near-field detail (geometric and color)

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

Terrain Map

• Connect samples into a mesh

Terrain Map

• Natural looking– 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

Fake Terrain

• Generate the heightfield:– Coarse height map + periodic details– Procedural or function-based– Random process

Perlin Noise

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

Terrain Coloring

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

Terrain Coloring

• Striped 1D texture map.

Terrain Coloring

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

Terrain Coloring

• More advanced coloring is based on altitude and slope.

Rolling Hills

• Scaling in one dimension gives smooth rolling hills.

Random Terrain

• Simple:– Terrain(x,y) = rand( MAX_HEIGHT )– Results in random noise

• Next step:– Smooth the terrain generated above– FIR filter: 1 1 1

1 1 1

1 1 1R

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

Random Fault Lines

• 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

+dHeight

More Realistic Faults

• Apply FIR filter to faulted heightmap

• Smooth out abrupt fault transitions: “Erosion”

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

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.

Particle Deposition

Procedural Modeling With Fractals

• Procedural Modeling– 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

Fractals

“Repetition of form over a variety of scales”

• Mandelbrot set, Julia set

Two Fractal Properties

• Self-similarity

Two Fractal Properties

• Fractal Dimension– Euclidean dimensions : 1, 2, 3, 4, …– Fractal : 1.2342, 2.7656– Measure of detail or roughness of a fractal

D = (ln N)/(ln 1/s)

Midpoint Subdivision

• Midpoint (recursive) subdivision

Midpoint Subdivision

• 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

Fractional Brownian Motion

• 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)

Fractional Brownian Motion

• Fractal dimension = roughness, I.e. H

Fractal Mountains

• 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

Triangle Subdivision

• Subdivide a triangle into 4 triangles at edge midpoint

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

Quadrilateral Subdivision

• Subdivide a quad into 4 quads at edge midpoints and a new center point.

Diamond-Square Subdivision

• Alternate subdivision

Fractal Terrain

Mesh Subdivision

• Square-square Subdivision– Addresses “creasing problem” (slope

discontinuities)

• Subdivide parametric patches

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?

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

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.

Caldera

Multi-Fractal

• This produces a ridged multi-fractal

Add a 1D Texture

More Controls

• To create plateaus, add a min function to flatten out high areas.

More Controls

• To create a plain, add a max function to flatten out low areas.

More Controls

• Multiply by a Guassian to limit the mountain range.

More Controls

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

max (Fractal, Smooth Noise)

max (Fractal, Smooth Noise)

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).

height

New height

Ridges

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.

Cracked Terrains

Crawfis, 2002

Fractal Terrains

• Multifractals

Fractal Terrains

• Color, texture

Fractal Terrains

Ground Clutter

• Grass• Rocks• Trees• Bushes

Placement issues

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.”

Erosion

Erosion by water

Erosion by wind and water