Could someone please relate the notion of parallel lines intersecting to this diagram (at time 4:48 of the following video)?:

https://www.youtube.com/watch?v=q3tu...outu.be&t=4m8s

Idounderstand the algebraic explanation provided here (in the part that's below the text "Proof: Two parallel lines can intersect.").:

Homogeneous Coordinates

I also get that it has to do with perspective, such as railroad tracks, which are parallel in real life, visibly intersecting at a vanishing point.

My only problem is that I'm struggling with visualizing the drawing(s) for homogeneous / projective coordinates that visually describe this phenomenon. Given that the last coordinates needs to approach zero for the phenomenon to occur, I visualize the plane that used to be at z = 1 almost overlapping with the one at z = 0, and it's like the two points cross the z = 0 plane, and though I see how they intersect at (0,0), I don't see how they are parallel.

Could someone please help me demystify this for myself?

Any input would be GREATLY appreciated!

P.S.

If this is in the wrong forum, let me know.