Skip to main content
$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 15.3: Space Model

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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, $$\dot{\ell} \cap \Pi=\emptyset$$).

Observation $$\PageIndex{1}$$

$$P$$ 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.