TITLE: What is the moduli space of lines in $\mathbb R^3$?
QUESTION [8 upvotes]: If we restrict to those lines passing through the origin, we of course get $\mathbb{R}P^2$.  Is there a good topological description of the space that we get when we remove the restriction that they pass through the origin?  Is there a name for this space?

REPLY [2 votes]: Just for kicks here is another "bundle" description of the moduli space.
It is not hard to see that this is the "orthogonal complement" of the canonical bundle over $\mathbb{RP}^2$, that is it is the space
$$ \{ (l,v) \in \mathbb{RP}^2 \times \mathbb{R}^3  | v \perp w \mbox{ for any }w \in l \}$$
after all all you need to describe an arbitrary line is its slope and a point, and you can choose this point to be the one closest to the origin.