Chapter 9 - Modeling Geometries

Abstract

The geometric algebra approach considerably extends the classical techniques of homogeneous coordinates. The homogeneous model permits representing offset subspaces as blades and transformations on them as linear transformations and their outermorphisms. The geometric algebra approach exposes some weaknesses in the homogeneous model. It turns out that a useful inner product in the representation space Rn+1 that represents the metric aspects of the original space Rn well cannot be defined but one can only revert to the inner product of Rn. The better model to treat the metric aspects of Euclidean geometry is a representation that can make full use of the power of geometric algebra. That is the conformal model, which requires two extra dimensions. It provides an isometric model of Euclidean geometry. In this representation, all Euclidean transformations become representable as versors and are therefore manifestly structure-preserving. This gives a satisfyingly transparent structure for dealing with objects and operators, far transcending the classical homogeneous coordinate techniques.