OpenGL Programming Guide

Appendix GHomogeneous Coordinates and Transformation Matrices

Homogeneous Coordinates

Transforming Vertices

Vertex transformations (such as rotations, translations, scaling, and shearing) and projections (such as perspective and orthographic) can all be represented by applying an appropriate 4 × 4 matrix to the coordinates representing the vertex. If v represents a homogeneous vertex, and M is a 4 × 4 transformation matrix, then Mv is the image of v under the transformation by M. (In computer-graphics applications, the transformations used are usually nonsingular - in other words, the matrix M can be inverted. This isn't required, but some problems arise with nonsingular transformations.)

After transformation, all transformed vertices are clipped so that x, y, and z are in the range [-w, w] (assuming w > 0). Note that this range corresponds in euclidean space to [-1.0, 1.0].

OpenGL Programming Guide