Intuitive Understanding of How Parallel Lines Meet in Projective Geometry

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?

• I'm not sure exactly in what sense you mean any sequence diverging to infinity is the same at infinity, but essentially yes, the topology in projective spaces adds "points at infinity" that allow parallel lines to meet. Aug 18, 2020 at 23:29
• No, this meaning is different: In real analysis you have only two points at infinity, $+\infty$ and $-\infty$. In contrast, when you consider the projective plane as obtained by adding a "line at infinity" to the affine plane, you have many "infinities." My suggestion is to learn this material from a textbook. Aug 18, 2020 at 23:57
• It is not an assumption that parallel lines (in ordinary space) meet at infinity. It comes from the definition of a projective space and its representation by ‘homogeneous coordinates’ and a calculation with these homogeneous coordinates. Aug 19, 2020 at 0:23
• @Bernard alright, so we are defining it that way. well then the question is why are we taking so much effort to define something that is not true? Aug 19, 2020 at 12:12
• What's not true? That parallel lines meet? It is not true in affine space, and becomes true in projective space.Furthermore projective geometry is the theoretical basis for perspective as used by painters. Aug 19, 2020 at 13:50

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.

• well now with this railway example, I did get little intuition. But inherintly or in-reality the lines are still parallel right? Also +1 for giving the notion of $l_{\inf}$ to imaginary number. So the reason we add the line $l_{\inf}$ to plane is same reason we add imaginary number to $\mathcal{R}$? Aug 19, 2020 at 18:54
• @GENIVI-LEARNER My answer to your comment was too long to fit in a comment, so I added it to the main answer. Hope it helps. Aug 19, 2020 at 19:22
• this is good insight. thanks Aug 19, 2020 at 20:36

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.

• Alright, but how can we make this definition more intuitive. I am completely lost at "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." Aug 19, 2020 at 12:11
• That is the geometric picture. 1) begin with a Euclidean plane; 2) note that parallelism is an equivalence relation, and call the classes "ideal points"; 3) Declare that the ideal points are a new line; 4) declare that there are new incidences: every old line meets the ideal line at one point: the one corresponding to the equivalence class of parallels that the old line lies in. 5) at this point, it can be verified that the new collection of points and lines satisfies the axioms of a projective plane. Aug 19, 2020 at 12:15
• Another alternative picture is that you call the $1$-dimensional subspaces of $\mathbb R^3$ "points" and you call the $2$-dimensional subspaces "lines". When a $1$-dimensional subspace lies in a given $2$-dimensional subspace, you say the "point" is on the "line." Given this information, you can verify that the "points" and "lines" form the real projective plane. For example. if you have two distinct $2$-d subspaces, they are guaranteed to have a $1$-dimensional subspace in common. Therefore distinct "lines" always intersect at one "point." Aug 19, 2020 at 12:18