11 - The homogeneous model

Abstract

While the 3-D vector space model can nicely model directions, it is usually considered to be inadequate for use in 3-D computer graphics, primarily because of a desire to treat points and vectors as different elements that are transformed differently by translations. Instead, people commonly use an extension of linear algebra known as homogeneous coordinates. This is often described as augmenting a 3-D vector v with coordinates (v1,v2,v3)T to a 4-vector (v1,v2,v3,1)T. This extension makes nonlinear operations such as translations implementable as linear mappings. For the homogeneous model in geometric algebra, the modeling principle is the same: it embeds the n-dimensional base space Rn in an (n +1)-dimensional representational vector space Rn+1, of which it then uses the inherent algebra. That produces a complete algebraic framework, which is well suited to compute with oriented flats, subspaces offset from the origin in Rn represented as blades in Rn+1. The algebra of Rn+1 provides generally applicable formulas for translation, rotation, and even affine and projective transformations in the base space Rn. The operations of meet and join always return sensible results for incidences of flats. Some of the incidence constructions are cross ratios and can be interpreted as defining motion-invariant measures. Then motions and transformations are studied and show that all direct flats are moved by the same linear transformation and all dual flats by another. This simplifies the code even more.