15.3: Space Model
( \newcommand{\kernel}{\mathrm{null}\,}\)
Let us identify the Euclidean plane with a plane Π in the Euclidean space R3 that does not pass thru the origin O. Denote by ˆΠ the projective completion of Π.
Denote by Φ the set of all lines in the space thru O. Let us define a bijection P↔˙P between ˆΠ and Φ. If P∈Π, then take the line ˙P=(OP); if P is an ideal point of ˆΠ, so it is defined by a parallel pencil of lines, then take the line ˙P thru O parallel to the lines in this pencil.
Further denote by Ψ the set of all planes in the space thru O. In a similar fashion, we can define a bijection ℓ↔˙ℓ between lines in ˆΠ and Ψ. If a line ℓ is not ideal, then take the plane ˙ℓ that contains ℓ and O; if the line ℓ is ideal, then take ˙ℓ to be the plane thru O that is parallel to Π (that is, ˙ℓ∩Π=∅).
P and ℓ be a point and a line in the real projective plane. Then P∈ℓ if and only if ˙P⊂˙ℓ, where ˙P and ˙ℓ denote the line and plane defined by the constructed bijections.