Maths for (Games) Programmers Section 2 - Matrix Methods

Vectors and Coordinate Systems
Coordinate Systems in Two Dimensions
Cartesian Coordinates     Polar Coordinates
Coordinate Systems in Three Dimensions
Cylindrical polar coordinates     Spherical polar coordinates     Scaled Cartesian coordinates     Added square Cartesian Coordinates
Vector Operations
Dot Product     Cross Product     Geometric Product     Other Vector Products     Curvature and Tortion
Polar Function
Derivatives     Euclidean Length     Approximations to Eulidean Length     Calculating the Polar Angle     Grad of distance functions

Matrices: Orientations and Transformations
General Matrices
Identity Matrix     Transpose     Determinants     Inverse Matrices
Discrete Transformations - Orientations
Rotations     Reflections     4x4 Transformation Matrices     Eulerian Angles     Pan Tilt and Cant     Rotations from Reflections     Quaternions
Kopp Rotations
Multiplication-free dot products of unit vectors     Multiplication-free diffuse lit spherical textures     Multiplication-free bump-mapped spheres     Multiplication-free specular bump-mapped spheres
Continuous Transformations - Motion
Discrete Rotation Rates     Example: "Magic Angles"

Representing 2D and 3D Objects
Points Lines and Planes
Points     Lines     Colinearity of Three Points     Planes     Coplanarity of Four Points     Polygons     Convex Hulls
Spheres Circles and Conics
Fitting a Circle to Three 2D Points     Rings and Disks      Conics and Quadrics

3D Clipping
Rectangular View Cone
Checking Spheres     Checking Projected Spheres     Checking and Clipping Lines
Circular View Cone
Checking Spheres     Point Checking Celestial Points
Clipping to a Sphere

Visibility Checks
Facets of Convex Hull
Arbitrary Camera Orientations

3D Projection
Projective Methodologies
Projections of common objects
Projected Lines     Projected Quadrics     Projected Circles     Projected Spheres     Circle fit to projected sphere     Bezier fit to projected sphere     Projected Conic Sections     Projected Bezier Curves
Inverse Projection - Ray Casting
To Plane     Iteration of projected planar boundary in texture space     To Sphere
Texturing
Reconstructed Perspective
Arbitrary Camera Orientations
Rotation and Projection of a Point
Cartesian Coordinates     Spherical Polar Coordiantes     Cylindrical Polar Coordiantes     In summary:     Rotation and Projection of Multiple Points
Image-warp Projections

Illumination
Lit Objects in 3D

Convex Hull Algorithm

Primal Simplex Algorithm
The Basic Algorithm
Application to 3D Convex Hulls
Generating the Vertices of a Convex Hull

Glossary   Contents   Author
Copyright (c) Ian C G Bell 1998
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
18 Nov 2006.