# Infinite Parallel Line Paradox?

Imagine two infinitely long lines. They’re perpendicular to each other, so they create an intersection at some arbitrary point. In a completely flat geometry, if we start to rotate one of these lines at a point not at the intersection, until we reach 90º (or perfectly parallel to each other), where and how does the intersection of these two lines disappear? As one line is rotated, I imagine the intersection crawling along the stationary line… but the rotation is a finite motion in time and space. How can this be resolved on an infinite line?

Or can’t it, since there is no such reality, and it’s a purely abstract and contradictory concept?

*It smacks of Zeno’s Paradox to me, but I know that has a resolution

If you are thinking of the rotation of the line as a continuous function, then when you think of the position of the intersection of the two lines it is also a continuous function, except at to point where the two lines are parallel. The name for that point is a “singularity”. When you are dealing with real functions, you can find some that have singularities, e.g., y = 1/x, which has a singularity at x=0. For small positive values of x, y becomes larger and larger, until you can say it “goes off to infinity”. For small negative values of x, y becomes larger and larger in the negative direction, until you might say it “goes off to minus infinity”. Where does it go when x=0? Well, it doesn’t go anywhere: it’s just not defined. Similarly, when two lines are parallel, their intersection is not defined, so the question “where does is go?” has no answer: it doesn’t go anywhere, since it doesn’t exist.

If I’m following your description accurately, the movement of the erstwhile intersection point would continue on out towards infinity as the two lines (previously perpendicular but now moving toward parallel) approach the parallel point. I guess asymptotically? If they ever became truly parallel then I suppose that previous intersection point would become the world famous “vanishing point” that perspective artists assume is there just over the horizon.

If that’s not it, I have no idea.

Yes, both of you have it exactly as I’m picturing it. Thank you Giles, for a mathematical description.

I suppose it’s one of those things that work out on paper, but is impossible to imagine. Stating it that the intersection is simply a singularity and has no definition I suppose is sufficient for my curiosity.

No one post before me! Answer coming…

I’m on pins and needles!

Can we post after you?

Dammit! Too slow. Missed the edit window. Grr.

Let’s define some things:

The vertical line rotates at point A.
The original intersection is piont B.
Point A is at (0,1) while B is at (0,0). That is, they’re 1 unit apart, and this is called mAB.
The rate of rotation is 1 deg/sec, R.
L is the dependent variable. It’s the length from B to the intersection at any given time. In other words, it’s how far the intersection has moved.

On to the calculus:

L/mAB = Tan(theta)
Theta = Rt
mAB=1
L=Tan(R
t)=Tan(t)

Plugging in for 90 seconds, we get that L=1/0. It’s undefined. Hmm. OK, well, what about right before that? Well, we want the limit of L as X->90. Easy enough to do in our heads.

As X gets higher, Cos(x) gets lower. So Cos^2 is even smaller than that. So 1/Cos^2 is very very big. In fact, it approaches infinity.

So how fast is the intersection moving just before the intersection disappears? Infinitely fast.

I intended for that to be a place-holder while I typed out all the math. But I missed the 5-minute mark, so now it’s enshrined in Doperville forever. :mad:

That tends to happen when you start out by trying to imagine things with infinite dimensions.

Well, shit… that’s cool. I never took calculus, but you somehow made it clear.

Of course, very true. And, as always, instills in me a disquiet sensation.

Can we post on him? I’ve always wanted to try that… but only with girl Dopers.

It seems I left out a big portion of the problem. I had it typed out, but didn’t insert it into my post. Here’s the missing section:

As an aside, there’s no paradox if you work on the real projective line, which is often a useful model for these sorts of things.

While this is hard to grasp intuitively, it makes sense in this case. Just before t=90, we already showed that the intersection is going toward infinity at an infinite rate, right? Well if you do the math just after t=90, you’ll see that the intersection comes back, not from infinity, but from negative infinity.

It’s kinda cool to think that for an exact moment in time, the intersection is inifinitely far away for exactly 0.0000000… seconds (which, of course, has infinite digits) and then “appears” at the other end of our line literally instantly.

Very interesting. Thank you.