Today I’d like to take a taste of projective geometry, both because it gives a pleasant twist on a subject on geometry, and also to give an example of a mathematical construction.
Projective Line (Def. 1) Consider all the ordered pairs where are real numbers, not both zero. (Note that this is different from how we make fractions … with fractions the denominator is never allowed to be zero, but the numerator can be; here either can be zero as long as they are not both zero.) We consider two pairs and to be the same if there is a number $\lambda$ so that and . In other words, we think of and as the same object. Likewise and are the same object. So our objects are ordered pairs modulo rescaling. Even though they look like ordered pairs and make you think of points in a plane, these really should be thought of as numbers, or points in a one-dimensional thing.
If you take any line in the plane and find two points on the line and (these are ordinary points in the plane, the kind you’re used to, not projective points). Then the (projective) slope of the line is . This is a bona fide definition, and it doesn’t depend on which points on the line you choose. Furthermore, two parallel lines have the same slope, and nonparallel lines have different slopes. Noice that I do not exclude vertical lines. This way of looking at things lets us view all lines on the same footing. A line which is not vertical which has slope m in the usual sense will have projective slope , and a vertical line has projective slope of (Notice, if you haven’t yet, that (as m varies) and is actually a complete list of projective numbers.)
Lets forget temporarily about the precise numerics and focus on the qualitative. Each projective number corresponds to a “direction” lines can go, which is to that each projective number corresponds to a family of parallel lines. Each family of parallel lines contains a unique line through the origin. So there is a natural correspondence between projective numbers and lines through the origin of the plane. Every line through the origin intersects a circle centered at the origin exactly twice (at opposite ends of a diameter).
Believe it or not, what I just said is most of the reason why the above to definition of the projective line is equivalent to the following.
Projective Line (Def. 2) Consider a circle (centered at the origin, if you like), and treat each point and the antipodal point (directly opposite on the circle) as being the same point.
If you look back at the discussion of projective slopes, you’ll see that we could identify almost all the projective numbers with ordinary numbers, and then there was one extra projective number. The projective line is a sort of “augmented” number line.
Projective Line (Def. 3) The projective line contains all the points of the number line, together with one extra point, called the “point at infinity”, which should be seen as lying “at both ends” of the number line.
This is by no means the only way of extending the real number system in such a way that it encompasses infinity in one way or another. There are many others. One interesting feature of this setup is that if you go up through huge positive numbers, or down through huge negative numbers, its the same infinity on either side. So if you think of “positive infinity” as different from “negative infinity”, which is appropriate in many contexts, then this is not the infinity you’re used to. But that’s another story.
The fact that the infinite point is at the extreme in both directions means that you should imagine going so far to the right (through very very large positive numbers), passing through infinity, and emerging in very very large negative numbers. Notice that this is exactly what happens if you draw a horizontal line and rotate it widdershins. The slope will increase and increase and increase, taking on extremely large positive values, until suddenly the line is vertical, and after that the slope will be extremely large and negative, then become less and less steep as the line returns to horizontal.
Now would be a good time to play around with these ideas a little. Draw yourself some doodles, or draw some diagrams in the air. Look a little odd to passersby, and you’re that much closer to being a mathematician. Try to see why these three things are really just three ways of thinking about the same object.
Now let’s take things just a little further. If that was a projective line, what is a projective plane?
Projective Plane (Def. 1) The points of the projective plane are ordered triples of real numbers, not all zero, modulo rescaling.
Look how simple it is to generalize the concept, once we’ve hit upon the “right” way to look at the one-dimensional case. The circle model for the line has an analogue as well — a sphere model for the projective plane.
Projective Plane (Def. 2) The projective plane is the surface of a sphere, where we consider two points which are directly opposite to be the same point. The lines are the great circles (the largest possible circles on the surface of a sphere, the ones that divide the sphere into hemispheres). (Note that the “lines” within the projective plane all have the structure of a projective line.)
The projective plane is useful for many reasons. Just as the “correct” notion of slope for lines in the plane was a projective number, a point in the projective number line, points in the projective plane are the right kind of object to be the slope of a line in space. That is, points in the projective plane are the most natural (in some sense) way to describe the slope or direction of a line in space. To put it more abstractly, there’s a natural way to label each line in space with a point in the projective plane so that parallel lines get the same label and nonparallel planes don’t. It turns out that points in the projective plane also describe possible directions for planes in space. What a world.
Projective Plane (Def 3.) Just as the projective line can be built by “fortifying” the ordinary number line with a “point at infinity”, we can view the projective plane as an augmented version of the usual plane. The points of the projective plane are points in the ordinary plane, plus an additional “point at infinity” for every direction (so there is the north-south infinite point, the northeast-southwest infinite point, etc.). All these infinite points lie on the “line at infinity”. Augment every usual line by adding the infinite point in that direction. So two parallel lines will end up having a point in common.
This construction has the pleasant side effect of remedying an annoying near miss to symmetry in basic geometry. In the ordinary geometry you learned in high school, every two distinct points determine a line, and any two nonparallel lines intersect in a single point. But in the projective plane, things are simpler. Every two lines intersect in a point, every two points determine a line. No qualifiers, no exceptions, no special cases.
There is something a little strange going on here. In the sphere model, there was a sort of symmetry. Every point played the same role as every other, there were no “special” points or lines. However, in the augmented plane model, this is definitely not true. We designated one line to be the one and only “line at infinity”. Thus the same object can “feel” very different depending on the perspective from which you look at it. Sometimes the ordered triple model is more useful, sometimes the sphere, and sometimes the augmented plane. Each perspective emphasizes certain features of the projective plane and obscures others.
Just in case you didn’t have enough to think about, there’s no need to stop here. What about three-dimensional projective space? It should be clear how to set up the definition — we look at quadruples , not all zero, modulo rescaling. If you can picture a hypersphere, the set of all points in four-dimensional space at a fixed distance from a center, then projective space is the surface of a hypersphere, where we think of two diametrically opposed points as being the same projective point. Projective 3-space can be thought of as ordinary space together with a projective plane. In other words, projective space is an ordinary space, together with an ordinary plane, together with an ordinary line, together with an ordinary point. If you set everything up “correctly”, every projective line intersects every projective plane in a single point, any two projective planes intersect in a projective line. What does “correctly” mean in that sentence? Take this as a geometric koan.