Math 3329-Uniform Geometries — Lecture 12 1. The Hyperbolic Plane and the Pseudosphere The xy-plane (a.k.a. the Euclidean plane) is a concrete model for Euclidean geometry. That is, the lines, points, angles, and distances satisfy Hilbert’s Axioms. The analogous object for hyperbolic geometry is called the hyperbolic plane. There is another non-Euclidean geometry, called elliptic geometry, and associated with it is the elliptic plane. Both the hyperbolic plane and elliptic plane are difficult for us to visualize within our (at least approximately) Euclidean space, but surfaces with the same local geometries do exist in Euclidean space. Recall that we made a cylinder by rolling up a piece of paper. The cylinder is certainly different from the Euclidean plane, but the local geometry is the same as the local geometry of the Euclidean plane, because the angles and distances were not deformed. The unit sphere, it turns out, has the same local geometry as the elliptic plane. If you were to look at a small piece of the sphere and a small piece of the elliptic plane, they would look the same, and sit in Euclidean space with the same curvature (the surfaces would be “bowl shaped”). Geometers describe the curvature of surfaces in terms of something called Gaussian curvature. We won’t get into that, but for the unit sphere, the Gaussian curvature is K = 1. In general, a sphere with radius r has Gaussian curvature K =1/r2. Note that really big spheres would have really small curvatures, and an infinite radius sphere would have zero curvature. This is the curvature of the Euclidean plane. If our universe were elliptic, a piece of a really big sphere would actually lay flat in our universe. A piece of the hyperbolic plane has Gaussian curvature K = 1, and a piece of it would have to be saddle shaped to fit into our Euclidean space. One surface that has− this same curvature is the pseudosphere. You can find pictures of it on the internet (e.g. at http://en.wikipedia.org/wiki/Pseudosphere). The pseudosphere is related to the hyperbolic plane kind of the way the cylinder and Euclidean plane are related. The pseudosphere and hyperbolic plane have the same local geometry, but the pseudosphere is somewhat tube shaped. Note that a really big pseudosphere would have a much smaller curvature, and so it’s conceivable that our universe is hyperbolic, and a piece of a really big pseudosphere would actually lay flat. 2. Another model for hyperbolic geometry So far, we have seen that Euclid came up with a set of definitions and postulates for what we call Euclidean geometry. Much later, Hilbert formalized Euclid’s system into a somewhat modern axiom system using Euclid’s approach as much as possible. For convenience, let’s pretend that Euclid’s system of postulates and Hilbert’s axiom system are the same, since Hilbert’s work can be thought of as patching up Euclid’s original attempt. Until the time of Gauss, Lobachevski, and Bolyai, the general assumption was that Euclid’s axiom system was not independent. In particular, it was conjectured that Euclid’s Fifth Postulate/Euclid’s Axiom could be proven from the other axioms. Gauss, Lobachevski, and Bolyai realized that this was impossible. In fact, there was another geometry, hyperbolic geometry, that satisfied all of the “other” axioms, but not Euclid’s Fifth Postulate. This marked the beginning of what is generally called non-Euclidean geometry. Hyperbolic geometry, after all, is a geometry that is not Euclidean. Somewhat oddly, non-Euclidean geometry doesn’t refer to all geometries that are not Euclidean, but for the most part, only refers to hyperbolic geometry and something called elliptic geometry. We’ll talk about elliptic geometry later, but it is very similar to, or exactly the same as (depending on who you’re talking to), spherical geometry. As I’ve said before, hyperbolic geometry is sometimes referred to as the geometry of Lobachevski and/or Bolyai, although Gauss, Cayley, and Klein are important in its development. The other non-Euclidean geometry, elliptic geometry, is sometimes referred to as the geometry of Riemann. The names hyperbolic and elliptic are due to Felix Klein. The study of geometry through axiom systems like Hilbert’s is called synthetic geometry, and this line of geometric reasoning pretty much comes to a dead end at the geometries 1 2 of Euclid, Lobachevski/Bolyai, and Riemann. The study of geometry through coordinates, equations, and functions is called analytic geometry, and we will start to head in that direction today. 3. The Half-Plane Model for Hyperbolic Geometry As much as you can, try to think of the hyperbolic plane as an infinite plane that is no more or less real than the Euclidean plane (i.e., the xy-plane). Our brains have a very strong Euclidean bias. Try to suppress this geometricism. Because of this geometrism, it is difficult to visualize the hyperbolic plane, but by deforming the hyperbolic plane, we can squeeze it into our familiar Euclidean plane as an open disk, the Poincar´edisk. This is only one way of visualizing the hyperbolic plane, and I want to explore another way here. The set of points in the half-plane model is the set (1) HP = (x, y) R2 y > 0 . ∈ | These are all the points above the x-axis, but not including the x -axis itself. In other words, we have an open upper-half-plane. As we deform the hyperbolic plane into the upper-half of the Euclidean plane this way, the lines in the hyperbolic plane become half-circles with endpoints on the x-axis (but without the endpoints) or vertical rays with endpoint on the x-axis (but not the endpoint). Note that the lines intersect the boundary at right angles. In Figure 1, a few of the lines are shown. Be clear on the fact that the endpoints on the x-axis are not points on the lines. This allows infinite lines to correspond point for point to these circles and rays. Figure 1. Some lines in the Half-Plane Model. Compared to the Poincar´edisk, I don’t think this model gives as good a general feeling for the hyperbolic plane, but in many ways, computations are easier to do here. In this model, the angles between lines in the hyperbolic plane have been preserved under the deformation. That is, the angles we see are the same as the actual angles. The distances have been warped, however. They have been changed by a factor of y, where y is the height above the x-axis. When we transform back to the hyperbolic plane, therefore, we have to divide distances by y. As an example, consider the half-circle of radius 2 in Figure 2. In the Euclidean plane, the circumference of the entire circle is 2π(2) = 4π. Since there are 2π radians around the circle, the length of the arc s divided ϕ by C must be equal to 2π . Therefore, (2) s =2ϕ. π π If ϕ = 6 radians, for example, then s = 3 . This is only the length of the segment as we see it in the Euclidean plane. We want to know the hyperbolic length. The highest point on the arc is at height y = 2, and the rest of the arc is not too much lower. We can say, therefore, that the height is approximately y =2 for the entire arc. To get an approximation of the length of the corresponding hyperbolic segment, we need π to divide the length of the arc by y = 2, so the hyperbolic length is about 6 . This is only an estimate, however, since the height of the points on the arc are not all exactly y = 2. 3 2 ϕ Figure 2. An arc of the circle of radius 2. This would correspond to a hyperbolic line segment in the hyperbolic plane. 3.1. Computation of hyperbolic length. As we move along a circle, the conversion factor to hyperbolic length keeps changing. In order to do the conversion, therefore, we need to do an integration. File this away in your brain: In general, the conversion between geometries is a calculus problem. ds r dθ r sin θ θ Figure 3. A small arc of a circle of radius r. This would correspond to a small hyperbolic line segment in the hyperbolic plane. OK. Let’s take a circle of radius r in the half-plane model. We’ll talk about points on the circle in terms of the angle θ shown in Figure 3. Measuring from the positive x-axis, we come to a point on the circle corresponding to the angle θ. From this point, we move a small distance ds along the circle. The distance ds corresponds to a small angle dθ, so in radians, (3) ds = r dθ. Since the height is r sin(θ), the hyperbolic length of the arc is ds r dθ dθ (4) = = . y r sin(θ) sin(θ) These quantities are good approximations for small dθ and ds, but exact in the limit, or in terms of in- finitesimals, which I think is a good way to think about things like this. Therefore, for an arc determined by running the angle θ from a to b, the length of the corresponding hyperbolic line segment is b dθ (5) . Za sin θ 4 Note that the radius of the circle is irrelevant. This integral is simple looking, but probably not one of the ones you’ve got memorized.