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

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 miles 1 inchScale ~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 similaron 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

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 bys = ½ = 1/N^(1/1)

2D: N=4 parts, scaled down bys = ½ = 1/N^(1/2)

3D: N=8 parts, scaled down bys = 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

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 = 4Scale r = 1/3

D = log(4) / log(3)

D = 1.2619

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

Diamond Square

Again:

Diamond Square

Some steps: taken from http://www.gameprogrammer.com/fractal.html#midpoint

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

Fractals

Fractals

Genres

First Fractals in GamesAnd Movies:

Genres

Rescue on FractalusActivision,1986, Spectrum 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