What players see as jumping up to smash bricks or zooming in to snipe soldiers is, behind the curtain, a vast array of variables and calculations. Even with large teams of individuals all working on different aspects of the game, advanced mathematics fall into aspects of game design from complex physics to modeling to lighting -- and that's just what the players can see.
Basic dynamics of the game fall into the category of linear algebra. Vectors are lines extending from a single point and are used in calculations regarding directions, camera angles and distances between objects. This also includes information on matrices, arrays of related data boxed into columns and rows. Linear algebra also couples with trigonometry to calculate velocity and user control in the game. This same pair functions for axis rotations, not only for cameras, but any dynamic movement in the game.
Geometry is an integral part of all game design that has any element of shape or position. From creating models to wrapping textures, very little visualization can exist in a game without geometry. Beyond this are the vital calculations of positioning, collision detection properties of rigid bodies, and minimum and maximum outputs. Where this really kicks into gear is in the design of hefty 3D terrains and ensuring that everything locks neatly into place with no overlapping.
Calculus is one of those math subjects that may never appear, depending on the hats worn during development. Commonly aligned with physics development, most calculus falls into the hands of the programmers responsible for creating the game engine. With many of these being packaged and sold to production studios, there's a strong possibility that it can be avoided altogether. Other uses for calculus revolve around dynamic variances in difficulty based on gameplay and programming AI to prioritize goals and targets.
Statistics and probability are the driving force of understanding success, challenges, artificial intelligence and event-driven gameplay. In any game, these are essential for calculating random events and the reactions surrounding them. But beyond the code, statistics are essential to pinpointing target audiences. Not only will this allow producers to keep a tab on what the current hot-button items are, but proper study enables them to predict trends towards where the industry will be when the game is finished and delivered.
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