Parallel Lines in Projective Space

Date: 05/18/98 at 21:05:46
From: Mario Hernandez
Subject: Parallels

Is it true that parallel lines meet in the infinite?

I have read that parallel lines intersect in the infinite, but I want 
to know if it's true. If it is, please send me the proof or reason. 

Thank you, Dr. Math.

Date: 05/19/98 at 14:22:59
From: Doctor Rob
Subject: Re: Parallels

Pick a point on one of the lines. Drop a perpendicular from it to the 
other line. Let the length of this perpendicular be d. Then this d 
does not depend on your choice of points on the first line. It is 
called the distance between the lines. No matter how far away from any 
fixed point you go along the first line, the distance to the other 
line is still this fixed value d. That means the lines are not getting 
any closer together, no matter how far you travel along them in either 
direction. One could certainly turn this into an argument that the 
lines never intersect, period.

On the other hand, the answer to this question depends on a bunch of
definitions. First you have to tell me what you mean by "in the
infinite."

There is a mathematical structure called "projective space" which has
our ordinary two-dimensional plane as a part of it. It also contains
other "points" not in the plane. In projective space, you can define
what you mean by a "line." If you take one of these "lines" and look
at the part of it in the plane, it is the same as a line in the plane 
(or is the empty set). Furthermore, if you take a line in the plane, 
it is part of exactly one of the "lines" in projective space. Each 
"line" is gotten from a line by adding a single "point" not in the 
plane. These "lines" in projective space have the property that two 
points determine exactly one of them. This is just like lines in the 
plane. They also share other properties with lines in the plane.

All the "points" not in the plane together form a "line" too.

One property that "lines" in projective space have that lines in the
plane do not is that any two of them intersect in exactly one "point." 
If you start with two parallel lines in the plane, and look at the 
"lines" in projective space containing them, those two intersect in 
some "point" in projective space, which is not a point in the plane.  
Such "points" can be called "points at infinity." In this context, you 
could say that parallel lines in the plane (extended to projective 
space "lines") intersect "at infinity."

I could explain parts of this better if you knew about the Cartesian
coordinate system in the plane.

I hope that this answers your question.

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/