A universal linear algebraic model for conformal geometries

Abstract

This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown analytically, through a framework comparing the measurement of distances and angles in Cayley-Klein geometries, including Lorentzian geometries, as done by F. Bachmann and later R. Struve. On the other hand, such a relationship may also be expressed in a purely linear algebraic manner, as explained by D. Hestens, H. Li and A. Rockwood. The model described in this article unifies these approaches via a generalization of Lie sphere geometry. Like the work of N. Wildberger, it is a purely algebraic construction, and as such it works over any field of odd characteristic. It is shown that measurement of distances and angles is an inherent property of the model that is easy to identify, and the possible models are classified over the real, complex and finite fields, and partially in characteristic 2, revealing a striking analogy between the real and finite geometries.

MoebInv: C++ libraries for manipulations in non-Euclidean geometry

“…The associated geometries span a wide domain, which includes conformal [8, Ch. 9], hypercomplex [9] and Lie sphere geometries [10,11,12]. For these fields the package facilitates:…”

Constructive curves in non-Euclidean planes

“…On this way we can get nine possibilities which can be realized, these are the so-called Cayley-Klein geometries. The Cayley-Klein geometries have a synthetic description (see in [68]) and also a linear algebraic one (see the paper of Wilderberg [75] and a recent paper of Juhász [44]).…”