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Parallel Lines
By Anonymous on Tuesday, September 12, 2000 - 02:11 pm:

From:Anon
On: 3/30/1998

Do parallel lines meet at infinity?

What is infinity and how does it work?

Anon

By Anonymous on Tuesday, September 12, 2000 - 02:18 pm:

From: Craig T. Snydal
On: 4/2/1998 at 12:27

Anon,

Please forgive me if this message is at too low or too high a level for you - since I don't know what maths you have taken. The question you have asked is a very good one. Have you ever looked down a long straight railroad bed? The tracks appear to be getting closer and closer together, seeming to meet at the horizon even though we know that they must stay equally spaced in order for the train to stay on the tracks. I have heard it said that Einstein's idea for Special Relativity came from considering what happens to parallel lines at infinity.

The short answer to your question is - it depends on what kind of geometry you are doing. If you are considering parallel lines in a plane, then there really is no "point at infinity". This is
because when we give the plane an x and y axis, any point in the plane must be able to be given coordinates. The "point at infinity" can't be given coordinates, and so it can not be found
on the plane.

But what if we wanted there to be a point at infinity, where would it be? Remember that this is something we are adding to the plane - it is a special addition that we are constructing, and so using it in an exam might not please your teachers. Of course we'd want it to be at the positive end of the x-axis, so the horizontal line through the origin must intersect this point
at infinity. What about the horizontal line that intersects the y-axis at y=1? Would we want this line to intersect the point at infinity? If we are assuming that there is only one point at
infinity, the of course we'd want them to meet, just like the railroad tracks meet at that point on the horizion. Now we add another horizontal line through the point y=2. This makes a pair
of tracks with the line through y=1, and so it too will intersect that point at infinity (also intersecting the horizontal line through the origin at infinity). We can keep laying down parallel lines (RR tracks) up and down the y-axis at every point, all of which meet at the point of infinity. So, if there actually was a point at infinity - parallel lines _would_ meet there.

Just to make this a bit more interesting (you can stop reading here if you want), where would we want the negative end of the x-axis go? To the point at "negative infinity"? But we decided
early on that we only wanted one point at infinty, so actually the negative end of the x-axis would meet the positive end of the x-axis at the "point at infinity", making a great big circle!
Similarly, all horizontal lines would meet at this point.

So now I'm sure you are getting suspicious that this is all madness. Because I'm sure you are wondering what happens when we go out to the positive end of all the vertical lines in the
plane. But of course we'd want them all to meet at the "point at infinity" too. In fact, we want _both_ ends of ALL lines in the plane to meet at that point at infinity. And what we end up with is a sphere!!! The process by which we project a sphere (minus one "point at infinity") onto a plane is called "stereographic projection".

I hope that this wasn't too confusing. My idea in giving this answer is to explain a way in which we could ima