VIII - Single Elliptic Geometry

Abstract

This chapter discusses single elliptic geometry. Single elliptic geometry resembles double elliptic geometry in that straight lines are finite and there are no parallel lines, but it differs from it in that two straight lines meet in just one point and two points always determine only one straight line. It resembles Euclidean and hyperbolic geometry. It is sometimes called elliptic geometry of the hemispherical type because of its relation to the geometry on a hemisphere. The chapter presents some basic facts of single elliptic geometry, such as:(1) each pair of straight lines meet in exactly one point, (2) through each pair of points there passes exactly one straight line, and (3) through each point there pass infinitely many straight lines, the totality of whose points constitutes the single elliptic plane. The single elliptic plane is a metric space containing at least two points, at least one simple arc of finite length joining each two points, and at least two straight lines.