This week, I thought I’d go a little deeper into the subject of projective geometry (feel free to reread last week’s post for a refresher). Why? I think this is a good opportunity to discuss the idea of an algebraic model of a geometry. Certainly using Cartesian coordinates in the Euclidean plane makes many geometrical problems easier to approach.
So what would a coordinate system in projective geometry look like? The most commonly used system is homogeneous coordinates. Let’s see how they work.
The difficulty with adding the line at infinity is that we need infinitely many points in addition to those already in the plane. Perhaps you might imagine that adding an “infinite” coordinate might do the trick, but there is an alternative — and algebraically simpler — approach.
First, think of the Euclidean plane as being the plane z = 1 in three-dimensional Cartesian coordinates, so that every Euclidean point (x, y) is given homogeneous coordinates (x, y, 1). But we don’t want to make the z-coordinate special — we want all the homogeneous coordinates to have similar interpretations. We accomplish this by giving each point infinitely many names — specifically, for any k ≠ 0, the coordinates (kx, ky, k) describe the same point as (x, y, 1). Geometrically, you might think of this as saying that the line through (x, y, 1) (except the origin) is, in some sense, the point (x, y, 1).
So if z ≠ 0, the point (x, y, z) in homogeneous coordinates is another name for the Euclidean point (x/z, y/z). But why would you want to do this?
The punch line is that we can now use z = 0 to indicate that a point is on the line at infinity! We still keep the convention that if k ≠ 0, the homogeneous coordinates (kx, ky, 0) describe the same point as (x, y, 0), as long as x and y are not both 0.
So our system of homogenous coordinates contains all points (x, y, z) such that not all three coordinates are 0. Any point with z ≠ 0 corresponds to a Euclidean point, and any point with z = 0 corresponds to a point on the line at infinity.
Is it really worth all this trouble? There are many interesting features of such a coordinate system — and I’d like to mention a few of them here, omitting the proofs. There are many resources online that include all the details — one example is the classic Projective Geometry by Veblen and Young available free as an ebook.
Let’s start be looking at equations of lines in projective geometry. In the Cartesian plane, we may represent a line in the form Ax + By + C = 0, where A, B, and C are not all zero. In the projective plane, a line has the form form Ax + By + Cz = 0, where A, B, and C are not all zero. Nice, isn’t it? This form is also consistent with our system of having many names for a point: if Ax + By + Cz = 0 and k ≠ 0, then also A(kx) + B(ky) + C(kz) = 0 as well. So no problems there.
But the really neat aspect of this representation is how you can use linear algebra in three dimensions to solve many problems in projective geometry. This isn’t a post about linear algebra — so I’ll limit myself to just one example. But in case you do know something about linear algebra, I think you should see how it can be used in projective geometry when homogeneous coordinate are used.
We’ll consider the problem of finding the intersection of two lines, say Ax + By + Cz = 0 and Dx + Ey + Fz = 0. The form of these expressions should remind you of taking the dot product, so that we can rewrite these expressions as
Interpreting these coordinates and coefficients as vectors in three-dimensional space, we observe that the common point (x, y, z) is simultaneously perpendicular to both (A, B, C) and (D, E, F), since both dot products are zero. So (x, y, z) can be found using the cross-product
Very nice! Again, there are many such examples, but this is not the place to discuss them….
This algebraic model also suggests another geometric model for the projective plane besides adding a line at infinity to the Euclidean plane. Begin with the surface of a sphere centered at the origin in three-dimensional Cartesian space, and observe that opposite points on the sphere have homogeneous coordinates that are different names for the same point in the projective plane.
So, in some sense, we have exactly twice as many points as we need — so we identify opposite points on this sphere, so that they are in fact the same point. (You might recall a similar discussion about points in the post on spherical geometry.) Thinking about it in another way, we might just consider the top hemisphere of the sphere with only half of the equator, including just one of the endpoints of the semicircle.
And while this model is fairly simple geometrically, it is important to point out that this does not mean that the projective plane lies inside three-dimensional space. Once we have our hemisphere and semicircle, we have to think about it without any of the surrounding space. This is not easy to do, but this type of thinking is necessary all the time when studying differential geometry, a topic for another time….
One last benefit to using homogenous coordinates: it can easily be abstracted to any number of dimensions. Do you want a projective space? Just add a plane at infinity to three-dimensional Euclidean space! Coordinates are easy — all points (x, y, z, w), with not all coordinates equal to 0. When w ≠ 0, the point (x, y, z, w) corresponds to the Euclidean point (x/w, y/w, z/w), and when w = 0, the point is on the plane at infinity.
And clearly, there would be no need to stop at three dimensions — it is just as easy to create a projective space of four or more dimensions.
Finding a workable system of coordinates for a particular geometry is not always a simple matter — but finding a system that allows problems to be solved easily is often a key step to studying any type of geometry. I hope this gives you some insight to yet another aspect of the diverse universe of so many different Geometries….