Metrics and Hyperbolic Geometry
One important concept in math is notion of ‘distance,’ called a metric, that we can assign to some set of points. In \(2\)-dimensional Euclidean space, the metric is analogous to the Pythagorean theorem:
\[d s^2 = d x^2 + d y^2.\]
Vaguely, you can think of this expression as measuring how far \(d s\) a point moved, given that it moved a small distance \(d x\) in the \(x\)-direction and a small distance \(d y\) in the \(y\)-direction.
However, there are other types of geometries in \(2\)-dimensional space. For example, on the upper half-plane, defined by the set \(\{(x, y) \in \mathbb{R}^2 : y > 0\}\), we can consider the metric1
\[d s^2 = \frac{d x^2 + d y^2}{y^2},\]
and on the unit disk, defined by the set \(\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1\}\), we can consider the metric2
\[d s^2 = 4\frac{d x^2 + d y^2}{(1 - x^2 - y^2)^2}.\]
Both of these models give rise to what is known as hyperbolic geometry. The former model is known as the Poincaré half-plane, and the latter model is known as the Poincaré disk.
These metrics may seem slightly random, but hyperbolic geometry is very natural from a different perspective. One of the axioms of \(2\)-dimensional Euclidean geometry, called the Parallel Postulate, is the following.
Parallel Postulate: For every line \(\ell\) and every point \(P\) not on \(\ell\), there exists a unique line passing through \(P\) that does not intersect \(\ell\).
However, it turns out that this statement is not a necessary consequence of the other axioms of Euclidean geometry. So, if we replace the Parallel Postulate with a more flexible axiom:
Hyperbolic Version: For every line \(\ell\) and every point \(P\) not on \(\ell\), there exist at least two distinct lines passing through \(P\) that do not intersect \(\ell\).3
We end up with a more flexible4 kind of \(2\)-dimensional geometry: hyperbolic geometry. The metrics we’ve listed above are just models in which we can see how things behave in hyperbolic geometry.
Geodesics and Geodesic Flow
Given two points \(A\) and \(B\) in Euclidean geometry, the shortest path between them (measured using the Euclidean metric) is the line segment \(\overline{AB}\). One feature of Euclidean geometry is that this path is unique.
Similarly, in either the Poincaré half-plane or the Poincaré disk, we can consider the shortest path between two points, measured using the associated metric. Just like Euclidean geometry, this path is unique, and called a geodesic.
Important: A path that minimizes total distance is technically not the definition of a geodesic in general, for other choices of metric.5 However, the general definition happens to be equivalent to what we’ve defined above in Euclidean and hyperbolic geometry.
In fact, in Euclidean and hyperbolic geometry, a point \(P\) and a direction vector \(\vec{v}\) defines a corresponding geodesic: there is a unique geodesic that is tangent to \(\vec{v}\) at \(P\). For instance, in Euclidean geometry, the geodesic corresponding to \(P\) and \(\vec{v}\) is the line passing through \(P\) that is angled in the direction of \(\vec{v}\).
In the Poincaré half-plane model, the following animation depicts several point-vector pairs as they travel along their corresponding geodesics. This is a visualization of what is called the geodesic flow in the hyperbolic plane.
Here’s a similar animation in the Poincaré disk.
The boundary of the hyperbolic plane is represented by the \(x\)-axis in the Poincaré half-plane and the unit circle in the Poincaré disk. In these models of the hyperbolic plane, note that the geodesics are Euclidean circular arcs that intersect the boundary at right angles.
You may also notice that the velocity vectors appear to be changing sizes. While they’re changing sizes if we measure the speed according to the Euclidean metric, Euclidean distances are not the same as hyperbolic distances.
Generally speaking, two points that are the same Euclidean distance apart are further apart (measured using a hyperbolic metric) when they are closer to the boundary.
In terms of the animations, this means that the points appear to move slower as they move closer to the boundary, since we’re looking at these geodesics in the Euclidean plane. However, if we calculate the speed using the hyperbolic metric, all of the points are traveling at a constant speed.