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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.

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maths in games
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
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
  • 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
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
Momentum in Portal
  • Portal is an action/puzzle game, that makes use of momentum in its puzzles.
lighting
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
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
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
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
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
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)