Mathematics in Computer Games

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Mathematics in Computer Games

• Nicolas Redfern

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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

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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.

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Gravity

• Gravity is simulated in games to ensure that
  objects fall realistically.
• On earth, the acceleration due to gravity is 9.8
  m/s2.

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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 sut½at2
• s09.8 (2)2/2
• s19.6 m

s? u0 v a9.8 t2
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Momentum in Portal

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

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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
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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.

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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.

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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.

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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)
• Garrys Mod (Garry Newman)
• Crysis (Crytek)