Game Programming, Math and the Real World Rolf Lakaemper, CIS, Temple University

The Visual World: Modelling Natural Environments

Modelling Natural Environments Natural environments are needed e.g. in RPG and Strategy Games Gothic 3, Piranha Bytes

Modelling Natural Environments Simulation is needed on different levels: Macro-terrain… Civilization 4, Sid Meier, Firaxis

Modelling Natural Environments …Mid level (mountains, rocks, clouds)… Gothic 3, Piranha Bytes

Modelling Natural Environments …trees, leaves… Gothic 3, Piranha Bytes

Modelling Natural Environments Q.: How can we describe a ‘natural’ environment ? A.: a mathematical description would be helpful

Modelling Natural Environments ‘Visual Math’ = Geometry Geometry = Euclidean Geometry (really?)

Modelling Natural Environments An attempt to model nature using Euclidean geometry Age of Kingdoms (shareware)

Modelling Natural Environments Let’s have a look at nature to see why Euclidean geometry fails. Q.: What makes the appearance of objects in nature ‘natural’ ?

Modelling Natural Environments New Jersey

Modelling Natural Environments New Jersey 1.2 miles1 inch Scale ~1:100000

Modelling Natural Environments Broccoli

Modelling Natural Environments Coastline 1 (computer generated)

Modelling Natural Environments Coastline 2 (computer generated)

Modelling Natural Environments Coastline 3 !

Modelling Natural Environments Observation 1: Nature seems to be self similar on different scales

Modelling Natural Environments Is self similarity sufficient to describe nature ? M.C. Escher: Circle Limits IV

In a certain sense, Eucledian geometry sometimes is self similar, too. self similar not self similar What’s missing is some ‘roughness’ Modelling Natural Environments

Two waterways

Modelling Natural Environments Natural or not ?

Modelling Natural Environments A measure to describe ‘roughness’: Fractal Dimensions

Modelling Natural Environments Motivation: Defining Dimensionality 1D: N=2 parts, scaled down by s = ½ = 1/N^(1/1) 2D: N=4 parts, scaled down by s = ½ = 1/N^(1/2) 3D: N=8 parts, scaled down by s = ½ = 1/N^(1/3)

Fractals We can also state: N= (1/s)^D D results from s, N : D = log(N) / log(1/s)

Fractals D doesn’t have to be integer…

Fractals Fractals are self similar geometric objects, which have not necessarily an integer dimension (though their topological dimension is still integer)

Fractals The simplest: von Koch Snowflake N=4, r=1/3

What ? Von Koch Snowflake

Fractals Von Koch Snowflake Iterating the snowflake algorithm to infinity, the boundary of the 1d snowflake becomes part of the 2d AREA of the plane it is constructed in (take it intuitively !)

Fractals Von Koch Snowflake It therefore makes sense to define its dimensionality BETWEEN one and two !

Fractals N = 4 Scale r = 1/3 D = log(4) / log(3) D = Intuitive ?

Fractals This definition of the dimensionality gives us a direct measure for the roughness of self similar objects.

Fractals Interestingly, studying nature shows that a fractal roughness of ~x.25 (x=1,2,3,…) seems to be found everywhere, and perceived by humans a ‘natural’ Coastlines, clouds, trees, the distribution of craters on the moon, microscopic ‘landscapes’ of molecules, …

Fractals So let’s build fractals with a dimensionality of x.25 !

Fractals Algorithms for Random Fractals

Fractals Random fractals: In contrast to exact self similar fractals (e.g. the Koch snowflake), also termed as deterministic fractals, an additional element of randomness is added to simulate natural phenomena. An exact computation of fractals is impossible, since their level of detail is infinite ! Hence we approximate (i.e we stop the iteration on a sufficient level of detail)

Fractals We will use MIDPOINT DISPLACEMENT

Fractals A 1D example to draw a mountain : Start with a single horizontal line segment. Repeat for a sufficiently large number of times { Repeat over each line segment in the scene { Find the midpoint of the line segment. Displace the midpoint in Y by a random amount. Reduce the range for random numbers. }

Fractals Result:

Fractals Result:

Fractals Extension to 2 dimensions: The Diamond – Square Algorithm (by Fournier, Fussel, Carpenter)

Fractals Data Structure: Square Grid  Store data (efficiently) in 2D Array.  Modification is very trivial.  Not possible to define all terrain features.  Good for Collision detection

Fractals Data Structure: (Square) Grid (“Heightfield”)

Diamond Square The basic idea: Start with an empty 2D array of points. To make it easy, it should be square, and the dimension should be a power of two, plus one (e.g. 33x33). Set the four corner points to the same height value. You've got a square.

Diamond Square This is the starting-point for the iterative subdivision routine, which is in two steps: The diamond step: Take the square of four points, generate a random value at the square midpoint, where the two diagonals meet. The midpoint value is calculated by averaging the four corner values, plus a random amount. This gives you diamonds when you have multiple squares arranged in a grid.

Diamond Square Step 2: The square step: Taking each diamond of four points, generate a random value at the center of the diamond. Calculate the midpoint value by averaging the corner values, plus a random amount generated in the same range as used for the diamond step. This gives you squares again.

This is done repeatedly, but the next pass is different from the previous one in two ways. First, there are now four squares instead of one. Second, and this is main point: the range for generating random numbers has been reduced by a scaling factor r, e.g. r = 1/4 (remember the fractal dimension ?) Diamond Square

Again:

Diamond Square Some steps: taken from

Diamond Square The scaling factor r, determining the range of random displacement R, defines the roughness ( => fractal dimension !) of the landscape. Some examples for diff. r and R R(n+1) = R(n) * 1 / (2^H), 0 < H < 1

Diamond Square Dimension: 2.8

Diamond Square 2.6

Diamond Square 2.5

Diamond Square 2.4

Diamond Square 2.3

Diamond Square 2.2

Diamond Square 2.15

Diamond Square 2.5

Diamond Square 2.8

Diamond Square Some tricks: power law

Diamond Square Some tricks: power law

Fractals Remember ? Simulation is needed on different levels. There are different approaches and algorithms to model nature, all of them fractal, all of them taking care of the correct dimensionality.

Fractals

Genres First Fractals in Games And Movies:

Genres Rescue on Fractalus Activision,1986, Spectrum VersionSpectrum Version

Genres Star Trek II: The wrath of Khan (1982) (movie)

Conclusion: Fractal Geometry helps to analyze and model the visual properties of nature. Breakout