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Projective geometry.

The Geometry of the Eye: What you see is what you get.

What is projective geometry? First, it's geometry: It's got points and lines and circles and such. For any two points, there is exactly one line connecting them. It's different from regular Euclidean geometry, though: For any two lines, there is always an intersection point.

But how can parallel lines intersect?

If for example you walk along two parallel lines--say railroad tracks--you can see easily that no matter where you go on the tracks, the two lines do not intersect. Even if you walk a million miles along the tracks, they will still be the same distance apart. (Here I'm assuming for simplicity that the earth is flat. Otherwise only 25,000 miles would take you back where you started!) After such an incredible journey I'm sure you'd be tempted to sit down and vent your frustration. You might wipe your brow and squint off into the distance at the incredible mileage you still have to cover. And this is what you'd see:

You'd see the tracks coming together at a single point on the horizon. You might even be tempted to believe they actually came together at that point. You would indeed believe it if you believed everything you saw. You would indeed believe that parallel lines meet, somewhere, out there...

...at infinity...

Now I'm not trying to tell you that parallel lines intersect. Of course they don't. I'm only trying to convince you that when you look at them, you will seem to see a point on the horizon where they do intersect. If we're developing the geometry of the eye to describe what you see, then these "apparent points" should be regarded as just as real (really seen, that is) as the "real points" on the lines, at least so far as the eye is concerned.

Parallel lines meet at infinity on the horizon, at least according to my eyes.

Question: How many points at infinity are there?

Well, how many?

It's the same to ask how many apparent points on the horizon there are. There are of course plenty (infinitely many) of these, one for each direction the railroad tracks could have gone. Next let's pick two of them, say p and q:

Question: Remember: For any two points, there is exactly one line connecting them. Is there a line connecting these two points (p and q)? Careful! They're infinite points!

If we look on the picture, it might be tempting to simply draw the straight line connecting the two points, and continue it indefinitely on either side. The line we would draw in this manner is of course the horizon line itself. It's the infinite line--a line at infinity made up of all the points at infinity!

(Ponder this: Is the infinite line really a circle?)

Here's a strange fact about our infinite points. Suppose we're standing on some line looking out at the horizon. The line meets a single infinite point (q) on the horizon. But if we turn around, our same line meets a different infinite point (q') on the horizon. What gives?

In fact any other line which hits this "front" point at infinity will be parallel to our line and will also hit the same "back" point at infinity. These two points are Mystically Coupled: any line through one goes through the other! This may be great from a spiritual point of view, but it's not so good from a geometry point of view. After all, we're supposed to have exactly one line connecting any two points, and now we have dozens of different lines connecting these two points.

How do we resolve the "Mystical Coupling," so that we can again say

"Between any two points, there is exactly one line"?

Well, how?

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The page last modified Thursday, 13-May-2004 15:03:01 EDT