1. Game Design, Development, and Technology
CS 382
Game Design, Development, and Technology
Mathematical Concepts
Rotation Interpolation
Fractals
Procedural Texturing with Noise
Part 4.1

2. Mathematical Concepts
Math Topic: Rotation Interpolation
Interpolating between two 3D orientations can be mathematically complex.
Traditional Approach
Define the object’s orientation by means of rotations about the three coordinate axes (i.e., roll, pitch, yaw)
Problem: “Gimbal lock” in which rotation about one axis can override rotation about another
Common Approach
Specify an axis and the rotation about that axis (used in OpenGL)
Problem: Animated interpolation does not appear smooth
Fancy Approach
Essentially, specify a “look-at” vector, supplemented with a roll value (i.e., quaternions)
Problem: Rather complicated mathematics, using an extension of complex analysis
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Mathematical Concepts
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3. Mathematical Concepts
Math Topic: Fractals
Many rendered objects have self-similarity, i.e., when magnified, their details are graphically similar to the objects as a whole.
Note how commonly rendered polyhedral objects don’t have this self-similar quality.
Part 4.1
Mathematical Concepts
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4. Mathematical Concepts
Fractal Generation
Numerous mathematical and statistical techniques have been discovered to yield self-similar images when they’re applied to graphical rendering.
L-Systems (Lindenmayer): Primitive Geometric Models
The Koch Snowflake (above), formed by splitting equilateral triangles.
The Twin Dragon Tile (below), which shifts alternating strips in alternating directions.
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Mathematical Concepts
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5. Mathematical Concepts
Fractal Terrains
In games, fractals are commonly used to generate terrains, often with a 2D grid recursively subdivided with random heights generated at each progressive grid point.
By color-coding different heights, a realistic terrain is produced.
In games, the resulting image is sometimes used as a bitmap for texture-mapping.
Part 4.1
Mathematical Concepts
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6. Mathematical Concepts
Math Topic: Procedural Texturing with Noise
Random number generation can also be used to generate procedural textures, which can add a photorealistic look to object surfaces.
Regular,
concentric-circle wood pattern
Brownian motion turbulence function
Scaled sum texture
Texture mapped to toroidal object
As the turbulence function becomes more complex, the textured image improves, but the processing time increases.
Part 4.1
Mathematical Concepts
Page 120