In the Poincaré half-plane model of hyperbolic space, the upper half-plane is the set of complex numbers with positive imaginary part:
The boundary of , , is the real axis plus the point . Formally,
is often referred to as the circle at infinity because it is topologically equivalent to a circle. This is the two-dimensional analogue of the topological equivalence of the 2-sphere and the extended complex plane (`the Riemann sphere’) which I wrote about in a previous note. As in that note, the argument is based on stereographic projection. We represent the unit circle in the complex plane as the set
and we construct a homeomorphism
by imagining a line passing through at the `north pole’ of the circle and another point .
This line meets the real axis at a unique point . We define . The map is a homeomorphism from to , which is a topological way of saying that and are `the same’.
What I want to focus on in this note is really the `at infinity’ part of the circle at infinity. It is said to be `at infinity’ because the upper half-plane model of hyperbolic space has the curious property that points on are at an infinite distance from any point in . This arises from the fact that the hyperbolic length of a path in the upper half-plane is obtained by integrating the functon along that path.
To clarify, let be a (piecewise differentiable) path in with parametrisation
As usual in complex analysis, if is a continuous function, then the integral of along the path is defined to be
where denotes the usual modulus of a complex number, i.e.,
(Note that no matter what parametrisation is chosen for the path, the value of this integral will be the same so one usually tries to find a parametrisation that is as easy to integrate as possible).
As stated earlier, the hyperbolic length of the path is obtained by integrating the function along , so
I want to highlight two curious results that follow from this setup. First, consider the points and in where and .
A parametrisation of the `vertical’ path between them in the diagram could be
We then have
Since the length of the path is we see that as the lower point gets closer and closer to the real axis (i.e., as ), the length of the path goes to . This is the motivation for the `at infinity’ nomenclature for the boundary .
The second curious result involves the lengths of paths between two points at the `same height’ in , say and .
A parametrisation of the `horizontal’ path between them in the leftmost diagram above could be
We then have
so the length of the horizontal path between the two points is . Now consider the situation shown in the remaining two pictures above. These involve piecewise linear paths between the two points that go diagonally from to a point and then diagonally from to . I am leaving unspecified so that we can study the lengths of the piecewise linear paths as a function of .
A parametrisation of the piecewise linear paths is given by
where denotes the indicator function for set . We then have
This final expression gives the lengths of the piecewise linear paths as a function of . If we plot this function of we see something completely at odds with Euclidean geometry:
We see that for values of between and the lengths of the piecewise linear paths are actually less than the length of the straight horizontal line between the two points and . The minimum possible path length between the two points is achieved by one of the piecewise linear paths (with a value of somewhere between and ) rather than by the straight horizontal path between them. This shows that hyperbolic lengths behave very differently from the lengths we are used to in Euclidean geometry.
Clearly, this approach to calculating distances between points is at odds with the triangle inequality, so it cannot be used directly as the metric for measuring distances between points in hyperbolic space. Instead, given , the hyperbolic distance between and is defined as
So what we do is consider all piecewise differentiable paths between and , calculate the hyperbolic length of each such path, then take the shortest. It turns out that this infimum is always actually achieved by some path (a geodesic) and this path is unique. The graph I plotted above is an illustration of this.
The metric clearly does satisfy the triangle inequality, i.e.,
for all . The distance between two points according to this metric must be increased if one goes via a third point, because it is by definition the infimum of all hyperbolic lengths between the points, so those involving any third point must have already been discounted as being longer.