15.3: Space Model
Let us identify the Euclidean plane with a plane \(\Pi\) in the Euclidean space \(\mathbb{R}^3\) that does not pass thru the origin \(O\) . Denote by \(\hat{\Pi}\) the projective completion of \(\Pi\) .
Denote by \(\Phi\) the set of all lines in the space thru \(O\) . Let us define a bijection \(P \leftrightarrow \dot P\) between \(\hat \Pi\) and \(\Phi\) . If \(P\in \Pi\) , then take the line \(\dot P=(OP)\) ; if \(P\) is an ideal point of \(\hat \Pi\) , so it is defined by a parallel pencil of lines, then take the line \(\dot P\) thru \(O\) parallel to the lines in this pencil.
Further denote by \(\Psi\) the set of all planes in the space thru \(O\) . In a similar fashion, we can define a bijection \(\ell\leftrightarrow \dot \ell\) between lines in \(\hat \Pi\) and \(\Psi\) . If a line \(\ell\) is not ideal, then take the plane \(\dot \ell\) that contains \(\ell\) and \(O\) ; if the line \(\ell\) is ideal, then take \(\dot \ell\) to be the plane thru \(O\) that is parallel to \(\Pi\) (that is, \(\dot{\ell} \cap \Pi=\emptyset\) ).
\(P\) and \(\ell\) be a point and a line in the real projective plane. Then \(P \in \ell\) if and only if \(\dot{P} \subset \dot{\ell}\) , where \(\dot{P}\) and \(\dot{\ell}\) denote the line and plane defined by the constructed bijections.