Two lines in a plane always intersect in a single point ... unless the lines are parallel. This annoying exception is constantly inserting itself into otherwise simple mathematical statements. Here is an example.
Parallel lines never meet
You may have heard of Desargues' Theorem. Draw any three lines through a point, and draw two triangles with corners on the lines, choosing three colours for corresponding edges, like this:
Now extend the edges of the same colour until they meet in three new (purple) points:
Desargues' Theorem says that, no matter the choices made along the way, the three final purple points will always lie along a single (dotted) line. Click here for an interactive JAVA version of this drawing.
This always works ... unless two of the edges of the same colour are parallel: in that case, no matter how far the lines are extended they will never intersect. Our annoying exception!
However, all is not lost. Start with two well-behaved triangles with non-parallel edges, and imagine transforming the triangles until the extended orange lines are parallel, as illustrated above. In the process, the intersection point of the orange lines will move farther and farther away. It also seems likely that the transformation will result in the dotted line becoming closer and closer to parallel to the orange lines, and will be exactly parallel once the orange lines are parallel. And indeed this is precisely what happens ... unless ... Well, you might be able to guess what else can go wrong. Click here for the answer.
Anyway, as this illustrates, parallel lines tend to give birth to annoying special cases. They take the fun and the elegance out of stating results such as Desargues' Theorem.
The projective plane
So what? The world is not perfect, and presumably we just have to live with this sort of fiddling. Or, is there a way we can handle such issues systematically, without all the special cases?
What do you see when you look along a pair of parallel lines, such as a straight pair of railway tracks? It appears exactly as if the tracks meet at a point infinitely far away. So, perhaps we can just add in those "infinity points"?
Indeed we can. By including such points, we extend the Euclidean plane (a fancy name for the xy-plane) to a slightly larger mathematical world, the so-called projective plane. And, by doing so, we ensure that two lines always intersect, at least somewhere in this larger projective world — no exceptions!
Enlarging the Euclidean world may seem a peculiar thing to do, but such extensions are actually very common in mathematics. For example, the natural numbers are, well, natural, but subtraction quickly becomes problematic: 7 − 3 is fine, but 3 − 7 is not possible within the world of positive numbers. However, enlarging to the world of (positive and negative) integers is a simple and elegant way to guarantee that subtraction between any two numbers is always possible. Of course, whenever a world is extended in this manner, we have to be very clear on how it is being done, on the rules for the new, larger world.
For the projective plane, all lines will include a new point. Moreover, we want any parallel lines to have the same new point. For example, the lines y = 2x + c have a common slope of 2, and so we need just one extra point at infinity, irrespective of c; we'll denote that point at infinity by (2).
This means that our new plane has two different kinds of points: (i) the familiar Euclidean points; and (ii), the slopes of lines in the Euclidean plane. We shall also decree that the projective plane has two different kinds of "lines": (a) the usual Euclidean lines (albeit each with an extra point at infinity); and (b), the line at infinity consisting of all the points at infinity.
The projective plane pays its way
With this set-up, any two distinct lines in the projective plane intersect in exactly one point — no more parallel exceptions! That's simple and elegant, but it is only really helpful if our new creation retains some of the nice properties of the Euclidean plane, and if it is now genuinely easier to state results such as Desargues' Theorem. Luckily (?), this is the case.
For example, just as for the Euclidean plane, two distinct points in the projective plane are always connected by exactly one line. To see this, we just have to check the various possibilities. First note that for two usual Euclidean points, the connecting line is just the same line it always was. Next, for a Euclidean point and a point at infinity, we just have to choose the line of the correct slope through the Euclidean point; for example the Euclidean point (3,7) and the infinite point (2) are connected by the line y = 2x + 1. Finally, two points at infinity are connected by the line at infinity. (This is exactly why we declared the set of points at infinity to also be a line).
And what about our Desarguesian construction? Since lines now always intersect, we are guaranteed to have three purple intersection points to consider. And, though it is not obvious, in the projective plane these three points are always contained in a line — no exceptions!
To illustrate, let's take another look at the Desargues diagram that started all the trouble:
The two orange lines are horizontal and parallel, and so intersect at the infinite point (0) — that's the point at infinity associated to all lines with slope 0. The dotted line is also parallel to the orange lines, and so it also contains (0). Consequently, the dotted line really does contains all three purple points of intersection.
Really no exceptions!
That is well and good, but it may appear that we have merely traded the fiddly special cases of our theorems for fiddly special pieces of our new plane. After all, in the Euclidean plane all points are indistinguishable from each other, and similarly for lines. By contrast, in the projective plane the points at infinity seem different from Euclidean points, and the line at infinity seems very different from the usual Euclidean lines.
All that is true, but it is only true for this particular model of the projective plane. In fact, the projective plane is at least as perfectly uniform as the Euclidean plane, if viewed in just the right way. Let us explain.
In three-dimensional xyz-space, two distinct lines through a point are contained in a unique plane, passing through the same point. Similary, two distinct planes through a point intersect in a unique line through the same point. Given our previous inventiveness this suggests an outrageous idea: let's declare that a line through (0, 0, 0) in xyz-space will be a POINT, and let's declare that a plane through (0, 0, 0) in xyz-space will be a LINE.
Then any any two distinct LINES intersect in exactly one POINT, and any two distinct POINTS are contained in exactly one LINE. That is sounding very much like our projective plane! And, in this new world all the POINTS are indistinguishable, and all the LINES are indistinguishable.
The Euclidean plane can be pictured as part of this POINT-LINE world. To do so, think of the Euclidean plane as hovering at height z = 1: call this hovering plane Euclid-1. Then every point in Euclid-1 gives rise to a POINT: the line connecting the Euclid-1 point to (0, 0, 0). Similarly, every line in Euclid-1 gives rise to a LINE: the plane containing the Euclid-1 line and (0, 0, 0).
This correspondence transforms, and effectively identifies, Euclid-1 as part of the POINT-LINE world. Once this tranformation is made, it's easy to identify the projective extensions of Euclid-1: the points at infinity of Euclid-1 are exactly the POINTS arising as lines in the xy-plane; and, the line at infinity of Euclid-1 is exactly the xy-plane.
When am I ever going to use this in the real world?
The above ideas are very pretty, and they have many applications within mathematics. But does any of it actually help solve real-world problems? Well, the world we live in is 3-dimensional, with computer models of the world based on Euclidean xyz-space. And, just as the Euclidean plane extends to the projective plane, Euclidean space has an extension to so-called projective space.
In projective space, parallel lines always meet, as do parallel planes. In addition, there is a very elegant method of calculating with the objects in projective space, which has important applications in computer graphics. In particular, projective space provides an ideal framework for projecting a three-dimensional virtual world onto the two-dimensional image plane of a computer screen. Projective space also unifies the treatment of the most common transformations and operations in computer graphics.
As to how these projective calculations work, that is perhaps best left for another day. However, the adventuresome among you may wish to follow this link; there, we provide a short introduction to calculating inside the projective plane.
Final thoughts and fiddly bits
- Quite generally, a (non-trivial) mathematical world is called a projective plane if that world consists of "points" and "lines", and any two points determine a unique connecting line, and any two lines intersect in a unique point. Not surprisingly, there are many different projective planes. The particular example that we have been considering is usually referred to as the real projective plane. Similarly, the 3D counterpart of this plane is usually referred to as real projective space. The smallest example of a projective plane has only seven points and seven lines, and the smallest projective space has 15 points, 35 lines and 15 planes. We'll report on these little gems in a sequel to this article.
- All higher-dimensional Euclidean spaces have projective extensions, and a version of Desargues' Theorem holds true in all of them. Desargues' Theorem is also valid in many other projective spaces, and it turns out to be one of the key results in the mathematical discipline of projective geometry.
- There are actually a number of different plausible ways to extend the Euclidean plane, and different extensions are useful in different contexts. However, for eliminating the mess of special cases such as in Desargues' Theorem, the real projective plane is definitely the most successful. For example, we could have extended the xy-plane by including just one single point at infinity. Then all lines would intersect at this extra point, and Desargues' Theorem would in a sense hold true. However, in this world most pairs of lines would be intersecting in two points, with other pairs intersecting in only the one point; we are basically left with the same problems. Similarly, the railway tracks picture suggests we could have extended each line to include two new points, one at either "end". However, this also turns out to not make our lives any easier. We'll leave you to ponder what goes wrong in this world.
About the authors
Burkard Polster and Marty Ross are Australia's tag team of mathematics. They write the Maths Masters column for the The Age newspaper in Melbourne. For many years they have been delivering the mathematics lecture series at the Melbourne Museum, visiting schools and touring the countryside with their Mathematical Mystery Tour. Currently, Burkard is lecturing mathematics at Monash University, and Marty is somewhere, lost in the woods.
Marty Ross (left) and Burkard Polster.
Check out what else Burkard and Marty are up to at www.QEDcat.com.