Mathematics in Computer GamesNicolas Redfern

Maths in GamesThere are many uses for maths in games:•Object interaction/collisions•Gravity and other forces•Breakable objects•Realistic vehicle movement•Realistic positional sound•Particle systems•Lighting

Object Interaction/Collisions Through use of mechanics, it is possible to

allow objects in a game to collide realistically with each other.

Collisions are calculated using the following formula: v2-v1=-e(u2-u1)

u2 and u1 are the velocities before collision, and v2 and v1 are the velocities after collision.

e is the coefficient of restitution. In a game this would be set by the developers.

Gravity

•Gravity is simulated in games to ensure that objects fall realistically.

•On earth, the acceleration due to gravity is 9.8 m/s2.

Calculating the Height of the Tower•The items all took 2 seconds to fall (t)•Acceleration (a) (due to gravity) is 9.8

m/s2

•The initial velocity was 0 m/s2 (u)•Distance travelled is worked out using:

s=ut+½at2

•s=0+9.8 (2)2/2•s=19.6 m

s=?s=?u=0u=0v=v=a=9.8a=9.8t=2t=2

Momentum in Portal

•Portal is an action/puzzle game, that makes use of momentum in its puzzles.

Lighting• Lighting is of immense importance to games.

A game can be much more immersive, if the lighting is realistic.

• Often, dynamic lights are used, for moving light sources.

+ =

colour luminance(colour)*colour

Illustrative Lighting in Team Fortress 2Team Fortress 2 uses a cartoon-like look to

achieve a unique visual style and comedic effect.

It uses complex equations to produce lighting effects.

Character Lighting Equation• This is the equation that determines the

lighting for characters in Team Fortress 2:

•Spatially-varying directional ambient

•Modified Lambertian terms•Unclamped Lambertian term•Scale, bias and exponent•Warping function

•Albedo

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Realistic positional sound

• Many games use positional sound: this is where sound varies depending upon where it is coming from.

• Another way in which sound is made more realistic is varying it upon the surrounding, environment.

• The Doppler effect can also be simulated in game: as an object moves towards you, the sound becomes lower in pitch, and continues to do so as it moves away.

• In Day Of Defeat: Source, another effect is applied: sounds far from the player lack higher frequencies and thus sound more like they naturally would, for example, abrupt pops for gunshots.

Breakable objects

• Having objects that can be broken up, adds another level of realism to a game.

• Usually, only certain objects can be broken, such as crates and fragile items.

Bibliography

•Jason Mitchell, Moby Francke and Dhabih Eng, "Illustrative Rendering in Team Fortress 2," International Symposium on Non-Photorealistic Animation and Rendering, 2007

•Games:▫Half-Life 2, Day Of Defeat: Source, Portal

and Team Fortress 2 (Valve)▫Garry’s Mod (Garry Newman)▫Crysis (Crytek)