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I posited this question to my geometry teacher in highschool many years ago, and it stumped her. I've recently brought it up again in conversations with friends and have not gotten any answer that's satisfactory. I have an intuitive feeling of what I expect, but I don't know enough to really back it up with solid reasoning.

Basically, consider this theoretical situation: Suppose you have two infinitely long lines (you could say a couple laser beams which never diffuse, but any perfectly straight line would work), and they are mounted so that they intersect at some point along an infinite plane in space. Then, you gradually rotate one of them so that the angle of intersection gets gradually closer and closer to 0, moving the lines closer and closer to parallel. At some point, intuition tells us that there is a position at which these lines will be perfectly parallel and never intersect. However, math tells us that they will always intersect, and the angle at which they intersect will simply approach 0 forever but never reach it.

Theoretically, parallel exists. If these lines were angled away from each other, they will not intersect, because at each increment, the distance between the two lines increases. If one of the lines in this case were rotated closer and closer to parallel, eventually they will angle towards each other and intersect at some distance away.

Math, again, tells us that this intersection point is found by a limit calculation, but it evaluates to infinity.

Considering that theoretical parallel does intuitively exist, when the lines become perfectly parallel, what happens at the point of intersection? There couldn't be a final point of intersection, could there?

EDIT I am de-emphasizing the following paragraph as it confuses the question, more than anything. It addresses a physical world representation where this question is largely theoretical, and concerns the mathematics behind the concept.

The only way I can wrap my brain around this situation, in my relatively limited understanding of physics and mathematics, is if these lines physically manifested as infinitely large loops, and then behave like intersecting rings. At any frame of reference, there is a point which appears parallel (the tangent lines) but where elsewhere, on the other side of the ring, they intersect. But this then assumes that the universe is curved along itself. Is there a better theory to explain this? I'm sure I can't be the first one to imagine this kind of situation.

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migrated from Aug 11 '11 at 19:53

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Your geometry teacher should have said that in terms of projective geometry, parallel lines intersect at an ideal point at infinity. Or intuitively, non-parallel lines intersect on the plane itself, while parallel lines intersect on the geometrical horizon of that plane, i.e., infinitely far away. There really isn't any physics here, though. – Stan Liou Aug 9 '11 at 2:36
Have a look at Homogeneous Coordinates. Quote from Wikipedia: "They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates". This model might give you more insight of how you can think of points in infinity. – whoplisp Aug 9 '11 at 23:18
@sidran32: as I understand your question, what you are wondering about is a version of Zeno's Paradox; the resolution is that the closer you get to the two beams being parallel, the more sensitive the "point of intersection" is to small changes of the angle, so it is possible to get to "intersection at infinity" in a finite turn of angle. – Willie Wong Aug 11 '11 at 18:31
As it appears this is basically a recreation of Zeno's Paradox, it would be useful if someone who suggested this posted that as an answer so I can mark it as such, and they can be recognized. – Ben Richards Aug 11 '11 at 21:05
up vote 6 down vote accepted

If you start with two infinitely long lines, which intersect at a point that is a finite distance in front of you, and straighten them so that they are parallel, then the point of intersection will shoot off to infinity in finite time.

This may seem counterintuitive, but stuff like this happens when you have an infinitely long line and you move it around as a rigid body. For example, if you just think about one line, when you rotate it, a point on the line that is distance "x" away from you sweeps out an arc at a certain speed. Another point at distance "2x" will sweep out an arc twice as long in the same time, so it moves twice as fast. And since the line, by definition of the thought experiment, is infinite, there are points on the line arbitrarily far away from you, which sweep out arbitrarily huge distances at arbitrarily fast speeds when the line is rotated.

This has nothing to do with physical reality because physical reality doesn't contain infinitely long, infinitely rigid physical bodies that respond instantly, all along their infinite length, when you rotate them at the origin. If you point a laser in one direction, and then change the direction, it takes time for the redirected beam to spread out into space, the beam was never infinitely long (because the laser would have had to be switched on forever in order to have created an infinite beam), etc. Anything you actually do will only involve finite distances and changes that take time to travel; you don't need to bring up the curvature of space in order to explain why your thought experiment can't happen in the real world.

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This is all true. However, the meat of the question is whether the lines actually do become parallel or not (physical reality or no). I can understand if the use of curvature of space may be a cop-out, but then if once is speaking of a truly infinite space with infinite lines, where (I suppose I should have clarified) transfer of energy is infinite fast, then does the concept of "parallel" truly exist? I can understand if this would be then better suited for the Math site, so it can be moved if necessary. – Ben Richards Aug 9 '11 at 2:25
Yes, the concept would exist... In your post you said "math tells us that they will always intersect, and the angle at which they intersect will simply approach 0 forever but never reach it". This is false. You are reinventing Zeno's paradox of the tortoise and Achilles: Achilles can never overtake the tortoise because it has always moved on by the time he reaches its last position. The paradox is resolved by convergence of infinite sums. 1+1/2+1/4+...=2, not infinity. When you straighten the lines, the angle approaches zero, and the limit case of parallelism is reached in finite time. – Mitchell Porter Aug 9 '11 at 3:42

If you have to circles intersecting in two points and you move them apart then there is a catastrophe happening at some moment: They will have only one point in common and in the next millisecond none. There is nothing mystical about this. In your case you have two lines, and they intersect in a point. Now you turn one of the lines, and at some particular angle a catastrophe is happening: They will have no point in common, and a millisecond later they will intersect again.

Now perform the same experiment (turning one line around a point $P$ keeping the other line fixed) on (a) the sphere and (b) the hyperbolic plane. On the sphere you don't have catastrophes, but all the time two intersection points which move continuously along the line which is kept fixed. On the hyperbolic plane however from a certain turning angle $\alpha$ onwards there is no intersection point, and at some later angle $\beta$ again an intersection point appears.

This has nothing to do with physics. It is just a fact of life that the morphology of a geometrical situation can change drastically under continuous changes of the defining parameters.

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One unmentioned point is that there is a way to make precise your notion of the parallel lines intersecting at an angle of zero, but this does not imply that they intersect at a finite distance. That is the crux of the error in reasoning.

Stan Liou mentioned the projective plane in a comment. That is one possible way to view parallel lines as intersecting. In the projective plane every two distinct lines intersect at a unique point. But we can't really define a meaningful notion of angle in the projective plane.

There is another possible view, that can capture the notion of parallel lines intersecting at an angle of $0$, namely the extended complex plane. One can prove that every (generalized) circle (including lines) maps to a circle on the Riemann sphere. Every line passes through $\infty$, which is defined as the point that maps to the north pole, and thus every two distinct lines intersect at two points, counting tangency twice. Parallel lines map to circles tangent at the north pole, while non-parallel lines map to circles that intersect at a positive angle at the north pole. It turns out that we can define angle in a natural way (angle on the sphere surface), and it amazingly agrees with the angle in the plane when the intersection is not at $\infty$! [See below if you're interested in a proof.]

Now we can consider what happens on the Riemann sphere when you have two lines as in your question with one gradually changed to become parallel to the other. For simplicity, let us assume that one of them is a line through the origin, so that it maps to a great circle $C$ through $\infty$. Now where does the other line map to? I made a Google search for an image and was pleasantly surprised to find that Ali Caglayan has already made one here.

enter image description here

Suppose the black line is the fixed one and the blue line is the one we are changing. Note that the red circle intersects $C$ at the north pole and at some other point $P$. Now imagine what happens as you rotate the blue line to make it parallel to the black line. The red circle sort of rotates too, and $P$ moves along $C$ closer and closer to the north pole. Finally when the blue line is parallel to the black line, $P$ has reached the north pole (corresponding to $\infty$) and the angle between the lines is $0$, but their two intersections are now both at $\infty$. You can see that the crucial error in your reasoning is that the intersection between the lines are at a finite point. Of course, technically your reasoning is flawed to begin with, since you cannot just claim that parallel lines ought to intersect (not to say at zero angle) just because they are a result of some limiting process. Just to make that clear, consider that $\frac1n \to 0$ as $n \to \infty$ and $\frac1n > 0$ for every positive $n$. There is no reason to expect $\lim_{n\to\infty} \frac1n > 0$.

Here is a simple proof sketch that angles are preserved by the projection mapping $p$ from the plane to the Riemann sphere. First prove (fact 1) that inversion at origin on the extended complex plane, namely $f = ( z \mapsto {\large\frac1{z^*}} )$, corresponds to a reflection $F$ on the Riemann sphere about the equator plane. Next prove (fact 2) that $f$ preserves angles between lines that do not pass through the origin. These proofs can be done using similar triangles alone. And note (fact 3) that $p$ preserves angles between lines through the origin. Now take any two lines $l,m$ that intersect at $z \ne 0$ in the plane, and let $l',m',z'$ be their images under $p$. $l',m'$ intersect at $z'$ on the Riemann sphere at the same angle as they intersect at the north pole (since they are circles on the sphere), which is the angle that $F(l'),F(m')$ intersect at the south pole (by fact 1), which is the angle that $p(f(l)),p(f(m))$ intersect at the south pole (by definition of $F$), which is the angle that $f(l),f(m)$ intersect at $0$ in the plane (by fact 3 since the tangents at the intersection are lines through the origin), which is the angle that $f(l),f(m)$ intersect at $f(z)$ (since they are circles in the plane), which is the angle that $l,m$ intersect at $z$ (by fact 2).

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Also see for a proof that $p$ maps generalized circles in the plane to circles on the sphere. – user21820 Jul 25 at 12:36

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