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My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to me?

Thank you.

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3 Answers 3

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If you can think of the point at infinity as a point on a line, then any two lines (it doesn't matter whether they are parallel or not) have the point at infinity as a common point.

For better visualization, use the correspondence between the Euclidean plane and the Riemann sphere (without the north pole). Now you can think of the point at infinity in the plane as the north pole on the Riemann sphere. Each line in the plane corresponds to a circle passing through the north pole. Then clearly any two lines in the plane (or any two circles passing through the north pole on the Riemann sphere) have a point of intersection, namely the point at infinity (the north pole).

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This puts the plane inside a sphere (the one-point compactification), when I think the intention is to put it inside a projective plane, with a whole circle (projective line) of points at infinity. I'm also not entirely convinced that the lines in the plane should still be considered "straight" in the sphere when this point at infinity is added - are they still geodesics? –  Matt Pressland Sep 21 '12 at 10:32
    
@Matt: You'r right. The canonical mapping from the plane to a sphere minus a point is a stereographic projection. It takes lines in the plane to spherical circles through the pole -- but most lines become small circles and thus not geodesics. –  Henning Makholm Sep 21 '12 at 10:44
    
@matt well i am not saying that they are geodesics. I just wanted to let Bek think about the point of infinity as the intersection of two line. My guess is under some certain metric those cirle will be geodesic.At that point I also need help. –  timon Sep 21 '12 at 10:45
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The geodesic problem is not so important - more crucial is that all that differs from the plane picture is that every pair of lines has one more point of intersection than before, so non-parallel lines now intersect twice. Subjectively, this isn't any better than the Euclidean situation, which is why I think the projective picture is the one the OP's teacher was getting at. –  Matt Pressland Sep 21 '12 at 10:53

This is not true in ordinary plane geometry, and so it cannot be proved.

It is true, sort of, in a different form of geometry known as projective geometry, however.

As a quick intuitive introduction to projective geometry, imagine that you're standing at the ordinary Euclidean plane. Your head is about 2 meters above the plane, so when you look down you see whatever is drawn on the plane, stretching out to the horizon. Details on the plane right where you stand look large to you; the same details a long distance away will look small to you and be seen very close to the horizon.

Now it's a common enough experience that if we draw to parallel infinite lines on a plane, when we look at them from a point above the plane, it will look as if they meet at the horizon. We can decide to consider the points on the horizon line "equally real" as points on the plane. The horizon then becomes the "line at infinity" and parallel lines in the plane actually do meet at a point on the line at infinity. Then any two lines always meet. Every line in the plane meets the line at infinity at a point determined by its direction; two lines in the plane with different directions intersect in the plane itself, and two lines in the plane with the same direction both meet the line at infinity at the same point, and therefore meet there.

So far there's a strange thing about points on the horizon: they are at the "edge of the world", with plane below them but nothing above them. This is sort of untidy, and there are two ways to fix that. One is simply to decide to draw stuff on the sky. That leads to spherical geometry, an ancient and venerated area of study that is the foundation of astronomy. But it's not what today's lecture is about.

In projective geometry, we're standing on the Euclidean plane, but we're in a virtual reality constructed by a careless and/or lazy programmer. When we look in any direction, our VR helmet computes the infinite line through our head in the direction we're looking, and figures out where that infinite line intersects the plane below us. Whatever is at the plane there is what we see. So when we look at the sky, what we see is the plane behind our head! Only when we look in a perfectly horizontal direction does this procedure not work, but we posit that there's nevertheless some points to look at there, which form a "line at infintity" as before. Because we're not distinguishing between points in front of us and points behind us, there are only 180° of horizon all in all; if we turn 180° we will be looking at the same points at infinity.

(A different way of looking at it is to imagine that we took a copy of the ordinary Euclidean plane and lifted it up to hover 2 meters above our head, and then turned it by 180° about a vertical axis. Note that since we're looking at the sky plane from below, things in the sky will be the mirror image of the same thing if the turn around and look down at them instead).

In projective geometry any two different lines have exactly one point in common. If they are non-parallel lines in the original plane they will cross once on the ground, and we will also see that crossing in the sky -- but these are just two images of the same point in the projective plane. Two parallel lines will cross exactly once at the line at infinity -- again we see two images of that crossing when we turn around, but they are by definition the same point. And any line in the plane will cross the line at infinity once.

The main thing that makes this cool is that the line at infintity now has no special properties. If we tilt our head and forget which way was up and down, there is no way we can deduce from the geometrical properties of what we see which direction the "true" horizon lies. Projective geometry is about properties of figures that are invariant of rotating our view of the world. It is also about properties of figures that don't change as we walk around on the plane, or move our head closer to (or farther from) the plane. So it has no concept of scale or distance either.

These movements both preserve lines (a line stays a line in our view when we tilt our head and/or walk around), so "line" is a projective concept. They don't preserve circles -- if we stand above the center of a circle, it will look circular, but when we walk away from it it starts to look like an ellipse instead. So "circle" and "ellipse" are not projective concepts. On the other hand, surprisingly, "non-degenerate conic section" is a projective concept, and all non-degenerate conics are equivalent. An ellipse is simply a circle seen from a distance. A parabola is a circle that is tangent to the line at infinity. A hyperbola is a circle or ellipse that intersects the line at infinity twice.

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Thanks Henning for the clear answer, it is comprehensible even for the layman. Though I miss one thing: what exactly was the historical fact/reason that necessitated the extension of the normal Euclidean geometry to comply with other non-Euclidean geometries? I assume that the extension to the projective plane was concieved to make Euclidean geometry consistent with e.g. elliptic geomtery, but why not the other way around? –  István Zachar Oct 18 '13 at 7:23
    
@IstvánZachar: I'm not too clear on the historical development here. It would be better to ask that as a separate question, I think. –  Henning Makholm Oct 18 '13 at 12:50

In elementary geometry, parallel lines do not intersect. A non-metric geometry, called Projective Geometry, was introduced to deal with "points at infinity". It is a very fascinating subject, but I suspect your teacher was just trying to convey a rough idea. Once upon a time, my teacher said that a parabola is "half an ellipse", and the other half is at infinity. It was an attempt to introduce the idea of curved space, but you should not pretend to rigorously prove these sentences in the framework of euclidean geometry.

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Actually I think it is a better description of the situation in projective geometry that a parabola is all of an ellipse except a single point where the ellipse touches the line at infinity. "Half an ellipse" would be one leg of a hyperbola.. –  Henning Makholm Sep 21 '12 at 10:15
    
Well, my teacher was probably confused :-) –  Siminore Sep 21 '12 at 11:08

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