Mathematics in Computer Games

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# Mathematics in Computer Games - PowerPoint PPT Presentation

Mathematics in Computer Games. Nicolas Redfern. Maths in Games. There 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.

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### Mathematics in Computer Games

Nicolas Redfern

Maths in Games

There 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=?

u=0

v=

a=9.8

t=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 2

Team 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
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)