Parallel Lines Intersecting in the Projective Plane

Question

My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that "two parallel lines intersect." If you type projective space into Wikipedia, or look it up in a textbook, a lot of the time you will see an image of two train tracks. The tracks are parallel, but as they go off into the distance (towards infinity) they appear to be approaching one another.

I believe I understand why we can say that two lines which are parallel in $\mathbb{R}^3$ intersect in $\mathbb{P}^2(\mathbb{R})$ (at least with the equivalence class construnction) but I do not see how this is related to that visual image of the two tracks appearing to approach a common point.

Maybe it is naive, but my question is: why do parallel lines, when looked at as in the image of the train tracks, appear to approach a common point? What is the mathematical reason for this?

Answer 1

Suppose that your eye is at the origin, and the "canvas" on which you draw is in front of your eye, and is the $z = 1$ plane. Then the point $(2, 3, 5)$ in space will project, in your "drawing" or "seeing" of the world, to the point $(2/5, 3/5, 1)$ in the $z = 1$ plane. In general, any point $(x, y, z)$ will project to $(x/z, y/z, 1)$.

Now consider two parallel lines; the first one consisting of all points of the form $(1, t, 4)$ and the second consisting of points $(3, t, 2)$. The projections of these to the drawing plane will consist of points of the form $(1/4, t/4, 1)$ and $(3/2, t/2, 1)$, respectively, i.e., they'll still be parallel.

Now look at the lines $(-1, 0, t)$ and $(1, 0, t)$. These project to $(-1/t, 0, 1)$ and $(1/t, 0, 1)$, which "meet" at the point $(0, 0, 1)$ when $t$ goes to $\infty$.

Does that help at all?

Answer 2

The mathematical reason for this is like this: suppose you look at an infinitely large plane from a point 1 meter above that plane and you have two infinitely long parallel lines drawn on it. Further, suppose the distance between lines is also 1 meter. Then, when you look at these lines right under your point of view the light has to travel 1 meter from each of the lines to your eyes. Then if you start looking at the lines far off, the light has to travel for example 1 km from each line to your eyes. Therefore, in the first case you have an almost equiangular triangle with sides of 1 meter, in the second case you have an isosceles triangle with sides of 1 km and base of 1 meter. In this case points at the base seem to be almost the same, so we perceive the lines to "start intersecting" when we follow them to "infinity". If that is confusing I would try to draw some pictures. Hope that helps

Answer 3

Have a look at this picture

taken from wikipedia

When you look at a camera model where $O$ would be your camera lens and every point is projected on the image plance behind $O$ it would result in the same effect. When the Vanishing point is reached the lines shown in the picture above would collapse to one line.

Compare it to this picture (source)

Now imagine two parallel lines going through $Q'$ and $P'$ and $P$ and $Q$ being your image plane. See what happens when the lines reach out to infinity.

• Some of the mathematical details are shown here and in camera calibration as well
• The first animation is another good approach to visualize this