17 - Operational models for geometries

Abstract

Now that the conformal model of Euclidean geometry has been described and its effectiveness has been seen, it is useful to take a step back and look in a more abstract manner at what has actually been done and why it works so well. Doing so will provide a tentative glimpse of future developments in this way of encoding geometries. Geometry (affine, Euclidean, conformal, projective, or any other kind) is characterized by certain operators that act on the objects of the geometry. These objects can range from geometrical entities such as a triangle to properties such as length, so they may have various dimensionalities (which we call “grades”). In order to use this structural capability for a given geometry, a good representational space has to be found which then uses the geometric algebra. In this representational space, the symmetries should become isometries. Isometries are distance-preserving transformations; they are the orthogonal transformations in the representational space that can be represented by versors in the corresponding geometric algebra. Such a model of geometry is called an operational model, since it is fully designed around the operational symmetry that defines the geometry.