Chapter 14 - Chain Geometries

Abstract

The focus of this chapter is on chain geometries. In a projective geometry over a ring R, it makes no sense to intrinsically consider a projective line. Nevertheless, if R is a K-algebra, then on the projective line P1(R) there is an interesting incidence structure, the blocks of which are the K-sublines embedded into P1(R) in a canonical way, and this gives the chain geometry over R. The notion of chain geometries arose from efforts to unify the treatment of such different geometries as the geometry of Mobius (lines and circles of the Euclidean plane), of Laguerre–Lie (spears and cycles), and of Minkowski (the pseudo-Euclidean plane with its hyperbolas as circles). The chain geometry Σ(K, R) is called Laguerre geometry provided the parallel relation is an equivalence relation on P (R) and every chain meets every parallel class of points. A weak chain space is an incidence structure Σ = (P, C). The blocks of Σ are called chains and two different points are called distant provided they are incident with a common chain.