Pappus in optical space

Abstract

Optical space differs from physical space. The structure of optical space has generally been assumed to be metrical. In contradistinction, we do not assume any metric, but only incidence relations (i.e., we assume that optical points and lines exist and that two points define a unique line, and two lines a unique point). (The incidence relations have generally been assumed implicitly by earlier authors). The condition that makes such an incidence structure into a projective space is the Pappus condition. The Pappus condition describes a projective relation between three collinear triples of points, whose validity can -in principle - be verified empirically. The Pappus condition is a necessary condition for optical space to be a homogeneous space (Lobatchevski hyperbolic or Riemann elliptic space) as assumed by for example, the well-known Luneburg theory. We test the Pappus condition in a full-cue situation (open field broad daylight, distances of up to 20 m, visual fields of up to 160° diameter). We found that although optical space is definitely not veridical, even under full-cue conditions, violations of the Pappus condition are the exception. Apparently optical space is not totally different from a homogeneous space although it is in no way close to Euclidean.