The line at infinity

I foolishly said this on Twitter about a month ago:

David Butler: My PhD supervisor used to say 2 conics share 4 points - maybe coincident, maybe complex, maybe on the line at infinity.
Tina Cardone: on the line at infinity? Tell me about those!
David Butler: Can of worms! Will need to write something and post later.

At the time I declared this was a bit of a can of worms and I promised to write something and post it later. Well, here it is. But it might take a few blog posts to untangle it.

Tagged in Isnt maths cool