Chapter 17 - Euclidean geometry on the two-sided plane

Abstract

This chapter presents Euclidean geometry on the two-sided plane. Euclidean geometry can be emulated in the two-sided space Tv, by treating the two ranges of the straight model as copies of the v-dimensional Euclidean space, and defining perpendicularity, congruence, angular measure, and other Euclidean concepts by reference to the standard formulas of Cartesian geometry. Once this canonical representative exists, the notion of abstract two-sided Euclidean space can be defined, as usual, as any structure that is isomorphic to the canonical one. The fundamental objects that play an important role in Euclidean geometry are used to distinguish horizon hyperplane h, and a polarity-like relation defined on the points of h. In the canonical space, these are by definition and the standard polarity⊥. In classic Euclidean geometry, the normal direction of a hyperplane is usually encoded as a unit vector, and its orientation is not specified. The two-sided definition, interpreted in the straight model, basically agrees with the classical one, except that it fully specifies the orientation of the normal, and encodes it as a point at infinity.