Chapter 13 - The Conformal Model: Operational Euclidean Geometry

Abstract

The conformal model is especially designed for Euclidean geometry, which is the geometry of transformations preserving the Euclidean distances of and within objects. These Euclidean transformations are sometimes called isometries, and they include translations, rotations, reflections, and their compositions. The Euclidean points are represented by null vectors in the representational model. A null vector is a vector with norm zero, and in a Euclidean space such a vector would have to be the zero vector that could not be used to denote a great variety of points. So the (n + 2)-dimensional representational space must be non-Euclidean. The (n + 2)-dimensional representational space can also be constructed by augmenting the regular n Euclidean dimensions with two special dimensions, for which a basis is formed by two vectors that are orthogonal. Euclidean transformations can be representable by versors in the conformal model. Since versors have structure-preserving properties, all constructions that are made using the geometric algebra would transform nicely in that algebra—which implies that they move properly with the Euclidean transformations.