Parallel lines diverge behind the observer in projective geometry

Question

Of course parallel lines converge at a point at infinity in projective geometry, but visually, they appear to diverge as one gets closer and closer to the start of one's vision, i.e. they diverge behind the observer. I can't find much about this fact. How is this dealt with in the different projective spaces? Are there any interesting hyperbolic 'perspectives' on this fact?

Answer 1

The usual notion of parallelism is foreign to projective geometry. We need to go back to definitions, and the best way to do this is to define a projective space $P(E)$ associated with a vector space $E$. A projective line $D_i$ is then by definition a projective linear manifold associated with a vector plane $P_i$ of $E$. And $$D_1\cap D_2=P(P_1)\cap P(P_2)=P(P_1\cap P_2)$$ If $P_1\neq P_2 $ then $P_1\cap P_2 $ is a vector line $D$ and $P(D)$ is a point of $P(E)$.

All these statements are easily illustrated in $E:=\mathbb R^3$: