Why only one $\infty $ point for each parallel class of lines in $\mathbb{R}^2$?

Question

I heard of projective geometry since high school. But I never managed to understand it in a systematic way.

It is said that the projective plane $\mathbb{P^2}(\mathbb{R})$ over the real numbers $\mathbb{R}$ is obtained by adding to the ordinary plane $\mathbb{R}^2$ some infinity points.

The problem is, why just one $\infty $ point for each parallel class while for each line there are two ends?

Answer 1

One way to motivate this is: in the projective plane, any two distinct lines meet at exactly one point. This is a general instance of homogeneity - any two pairs of (distinct) lines in the projective plane "look alike" in an appropriate (specifically, topological) sense. If you added two points at infinity for every family of parallel lines, then parallel lines would behave differently than non-parallel lines.