ELLIPTIC GEOMETRY IN TWO DIMENSIONS

Abstract

Spherical and elliptic geometry. As we saw in §1.7, a convenient model for the elliptic plane can be obtained by abstractly identifying every pair of antipodal points on an ordinary sphere. The reflections and rotations which we shall define in §§6.2 and 6.3 are represented on the sphere by reflections in diametral planes and rotations about diameters. In elliptic plane geometry, every reflection is a rotation. This rather startling result is a consequence of the fact that the product of the reflection in any diametral plane and the rotation through π about the perpendicular diameter is the central inversion (or point-reflection in the centre of the sphere), which interchanges antipodal points and so corresponds to the identity in elliptic geometry. In any orientable space (§2.5), a reflection reverses sense, whereas a rotation preserves sense. Thus the above remarks are closely associated with the non-orientability of the real projective plane (in which elliptic geometry operates). On a sphere, corresponding rotations about antipodal points have opposite senses; so the identification of such points abolishes the distinction of sense.