On Real Projective Plane Constructions and Their Isomorphisms

Abstract

The real projective plane has three well know isomorphic constructions: the extended euclidean plane, unit (hemi)sphere, and three dimensional vector space over the reals. In this paper we find the 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

Figures (4)

• Figure 1: Visualization of real projective plane constructions and their isomorphisms. Left: Extended Euclidean plane with three chromatically separated groups of lines each with different, arbitrarily chosen slopes. The point that each line is incident with the line at infinity is notated by the slope of the line. Center: (hemi)sphere construction with two semicircles (lines), the plane containing them, and a normal vector to that plane. Right:$\mathbb{R}^3$ vector space with two 1-dimensional subspaces (points) and the plane (line) incident with them.
• Figure 2: Left: Conceptual design of a direction-sensitive photosensor where red lines represent photons. These photons enter the sensor from above, are refracted by the intermediate lens, and are finally detected on the bottom sensor plane. Figure taken from Ref. Dalmasson_2018. Right: A larger detector outfitted with direction-sensitive photosensors. The sensor planes are red and the lenses are blue. This detector would be filled with a scintillating material which would produce photons when excited by an incident particle.
• Figure 3: Left: Visualization of $s=\mathcal{I}_{PS}(p)$ for all $p=(\alpha, r)\in\mathbb{R}\mathbb{P}^2_P$ with constant $r$ corresponding to constant $\phi$ as shown in the legend. Right: Visualization of $s=\mathcal{I}_{PS}(p)$ for all $p=(\alpha, r)\in\mathbb{R}\mathbb{P}^2_P$ with constant $\alpha$ corresponding to constant $\theta$ as shown in the legend.
• Figure 4: Visualization of $\mathcal{I}_{PS}$ through the relation $\mathbb{R}\mathbb{P}^2_P\cong\lim_{r\to\infty}r\cdot\operatorname{proj}_{\mathbb{R}\mathbb{P}^2_S}{\mathbb{R}^2}$ or $\mathbb{R}\mathbb{P}^2_P\cong\lim_{\rho\to\infty}\operatorname{proj}_{\mathbb{R}\mathbb{P}^2_S(\rho)}{\mathbb{R}^2}$. The top legends contain equations of lines $\ell_1,\ldots,\ell_m\subset\mathbb{R}\mathbb{P}_P^2$; the top plots show $\mathcal{I}_{PS}(\ell)\subset\mathbb{R}\mathbb{P}^2_S$; the bottom plots show $\operatorname{proj}_{\mathbb{R}\mathbb{P}^2_S}{\mathbb{R}^2}$.

Theorems & Definitions (25)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Definition 2.7
• Theorem 2.8
• proof