The Poincaré disk model is one of the most useful ways to visualize the hyperbolic plane. In this model,
the “whole” hyperbolic plane is represented inside the open unit disk
\(\{(x,y):x^2+y^2<1\}\). The boundary circle \((x^2+y^2=1)\) is not part of the space—it behaves like an
“infinite horizon.” Points closer to the boundary are still inside the disk in the Euclidean sense, but they are
very far away in the hyperbolic metric.
Distances in the Poincaré disk are defined so that the geometry has constant negative curvature. One important
consequence is that Euclidean-looking straight segments are usually not the shortest paths. The shortest
paths are called geodesics. In the disk model, geodesics are exactly:
(1) diameters of the unit circle, and (2) circular arcs that meet the boundary circle at right angles
(orthogonally). This is why the visualization often shows a curved arc between two points \(A\) and \(B\):
that arc is the hyperbolic “straight line.”
A clean way to compute hyperbolic distance uses complex notation. Let \(z=x_1+i y_1\) and \(w=x_2+i y_2\).
The hyperbolic distance is
\[
d_{\mathbb H}(z,w)=2\,\operatorname{artanh}\!\left(\left|\frac{z-w}{1-\overline{z}w}\right|\right).
\]
The term inside \(\operatorname{artanh}\) is always between 0 and 1 for points strictly inside the disk.
As either point approaches the boundary, the denominator \(1-\overline{z}w\) can become small in magnitude,
pushing the ratio toward 1, and \(\operatorname{artanh}(s)\) grows without bound. That is the “infinite distance
to the edge” effect.
A particularly common special case is distance from the origin. If \(|z|=r\), then
\[
d_{\mathbb H}(0,z)=2\,\operatorname{artanh}(r).
\]
This also explains hyperbolic circles centered at the origin. A hyperbolic circle of radius \(\rho\) corresponds
to a Euclidean circle of radius
\[
r=\tanh(\rho/2).
\]
So when you increase \(\rho\), the Euclidean circle expands rapidly toward the boundary but never reaches it,
because \(\tanh(\rho/2)<1\) for every finite \(\rho\).
Comparing Euclidean and hyperbolic distances reveals the non-Euclidean nature of the space: near the center, the
two metrics can look similar, but near the boundary the hyperbolic distance becomes much larger. This is one reason
hyperbolic geometry is useful in areas like geometric group theory, network embeddings, and models of negatively
curved spaces—distances “stretch” near the boundary in a systematic way.
