Intuitive Understanding of How Parallel Lines Meet in Projective Geometry

Question

I am watching a lecture on Topology, where it is mentioned that in Projective Geometry Parallel Lines Meet. I am interested in intuitive idea of how is that even possible. Is in projective geometry do we make an assumption that at infinity the parallel lines meet just like we make an assumption that any sequence diverging to infinity is same at the infinity?

Answer 1

Since you asked for an intuitive idea of how it is possible for parallel lines to meet, consider the common observation that railroad tracks (which are parallel) meet at the horizon. You know, of course, that the earth is not a plane, and that a powerful telescope would show that they don't really meet. But pretend that the earth is a flat infinite plane. Do the tracks meet on the horizon or not?

In projective geometry the allowable transformations are called projective transformations. They are bijections of the plane that map lines to lines. Four non-collinear points that map to another four non-collinear points uniquely determine a projective transformation. If you play with projective transformations you'll see that they feel like changes in perspective.

Getting back to railroad tracks on an infinite plane, consider perspective A, which looks at them from above, and perspective B, which sees them converging at the horizon (line $h$). There is a projective transformation $T$ that takes perspective A to perspective B. But consider $T^{-1}$, which takes $B$ to $A$. Since lines go to lines, what is $T^{-1}(h)$? Since the horizon is "at infinity",$T^{-1}(h)$ can't be a finite line. It is the "line at infinity" $l_{\infty}$, which is a line consisting of "points at infinity", which in turn can be thought of as directions (suppose you have two railways going in different directions. They will meet at different points on the horizon). Furthermore, $T(l_{\infty})=h$, so $T$ is way of viewing $l_{\infty}$ as a visible line.

Adding the line $l_{\infty}$ to the plane is a little like adding $i=\sqrt{-1}$ to $\mathbb R$ to get the complex numbers. In both cases we add something that strikes us a imaginary and intangible, but in return we get a more consistent and complete mathematical framework.

So yes, in projective geometry the railway tracks (as seen from above as parallel lines) meet at a point on $l_{\infty}$. And that's why in projective geometry there is no concept of "parallel".

Answer to question in a comment (But inherently or in-reality the lines are still parallel right?): The mindset of projective geometry is that it is just lines and points. There is no metric info such as distance and angle. On the other hand, we tend to use the Euclidean plane as a starter model to help us visualize things. That's useful, but we have to drop our metric notions, and the statement "parallel lines never meet" is no longer true because it has been replaced by the axiom "two lines meet in a point". So the Euclidean plane is sort of training wheels for picturing what's going on. The analogy with imaginary numbers is only suggestive here, because "i" expands R to C, but with projective geometry "parallel lines don't meet" is replaced with "two distinct lines meet". You can go the other way and start with the projective plane and by tweaking things get the euclidean plane. The parallel axiom is also replaced in hyperbolic geometry but in a different way, and people like Gauss famously wondered whether the parallel axiom was "true in reality" (like, in the real world) but kept his thoughts to himself because they were too controversial. And in spherical geometry two lines (defined as great circles) always meet.

But, to your question, if you want to play by the rules of the game, you don't say that two lines are parallel, you say they meet at $l_{\infty}$. And there's nothing special about $l_{\infty}$. In fact if you have a theorem about parallel lines you can get often get a new theorem for free by applying a projective transformation and replace "parallel lines" with "lines that meet on a particular line (like $h$)". You can still insist that the lines are parallel, but at that point you're stepping out of bounds and saying something about a specific model of projective geometry.

Answer 2

in projective geometry, parallel lines meet

Is an oxymoronic statement.

It is more accurate to say

in projective geometry, no two distinct lines are parallel

The way the oxymoronic statement arose is this way: from any affine plane (like the Euclidean plane, where a single line had uncountably many parallel compatriots) you can add points, which form one new line, and extend incidence relations to create a projective plane containing that affine plane.

For each equivalence class, you declare a new point, called an ideal point, corresponding to that class. All the lines in the class are “extended” by one point, and they all share the point in common.