On Real Projective Plane Constructions and Their Isomorphisms

The real projective plane ($\mathbb{R}^2$) has three well known isomorphic constructions: the extended Euclidean plane, unit (hemi)sphere, and $\mathbb{R}^3$ vector space. In this paper, we find isomorphisms that map between these three constructions. Additionally, we investigate their relationship to direction-sensitive photosensors which use lens(es) to transform light's direction to a position on a local plane. This transformation, done by lenses, is a physical version of an isomorphism between projective plane constructions.