Three hyperbolic metrics

Up to a constant factor, there is just one conformally invariant Riemannian metric {\rho} on the disk {\mathbb{D}=\{z\in {\mathbb C}\colon |z|<1\}}. Indeed, on the tangent space at {0} the metric must be a multiple of the Euclidean one, due to rotational invariance. Normalizing so that at {0} both metrics coincide, we can use the invariance under conformal automorphisms (Möbius transformations)

\displaystyle \psi_a(z) = \frac{z+a}{1+\bar a z},\quad |a|<1

to find that on the tangent space at {a} the metric {\rho} differs from Euclidean only by the factor of {|\psi'_a(a)| (1-|a|^2)^{-1}}. This can be written as {d\rho(z) = (1-|z|^2)^{-1}|dz|}, indicating what we integrate to find the length of curves with respect to {\rho}. This is the Poincaré disk model of the hyperbolic plane. The geodesics of {\rho} are precisely the circles orthogonal to {\partial\mathbb{D}}, and diameters.

Hyperbolic geodesics
Hyperbolic geodesics

We could model the hyperbolic plane on any other proper simply-connected domain {\Omega\subset {\mathbb C}}, just by pulling back {\rho} under the Riemann map {\phi\colon\Omega\rightarrow\mathbb{D}}. Explicitly, {d\rho_{\Omega}(z)=(1-|\phi(z)|^{-2})\,|\phi'(z)|\,|dz|}. But is this really explicit? We have no closed form for {\phi} except for very special domains {\Omega}. The density {\rho_\Omega} can be quite tricky: see this MathOverflow question.

On the disk {\mathbb{D}}, the density of {\rho} is roughly the reciprocal of the distance to the boundary. The Koebe distortion theorem yields the same for all simply connected domains:

\displaystyle \mathrm{dist}\,(z,\partial \Omega)^{-1} \le \frac{|d\rho_\Omega(z)|}{|dz|}\le 4\, \mathrm{dist}\,(z,\partial \Omega)^{-1}

Disk and slit plane realize 1 and 4, respectively.
Disk and slit plane realize 1 and 4, respectively.

This suggests replacing the hyperbolic metric {\rho_\Omega} with the quasihyperbolic metric {d\delta_\Omega(z)=\mathrm{dist}\,(z,\partial \Omega)^{-1}\,|dz|}. The density of {\delta_\Omega} is about as simple as we could wish for, but the shape of its geodesics is not as obvious; indeed, a number of papers were written on this subject in the last 35 years.

Can we have a hyperbolic-type metric with explicit geodesics and explicit density? The Hilbert metric delivers both, at least in bounded convex domains. The idea is simple: instead of dividing the length of each tangent vector {v\in T_a} by {\mathrm{dist}\,(a,\partial \Omega)}, we divide it by {\mathrm{dist}_v\,(a,\partial \Omega)}, which is the distance measured in the direction of {v}. In other words, this is how long you could walk in the direction {v} before hitting the boundary. It is clear that the length of any curve going to the boundary is infinite due to the divergence of {\int_{0^+} \frac{dx}{x}}, same as for hyperbolic and quasihyperbolic metrics. Since it is awkward to have a “metric” that is not symmetric (the lengths of {v} and {-v} are not the same), we symmetrize:

\displaystyle \|v\|_{\Omega,a} =\frac12 \left(\frac{1}{\mathrm{dist}_v\,(a,\partial \Omega)} +    \frac{1}{\mathrm{dist}_{-v}\,(a,\partial \Omega)} \right) |v|_{{\mathbb R}^2}

Directional distances to the boundary
Directional distances to the boundary

Now the distances between points are defined in the usual way, as the infimum of lengths of connecting curves {\gamma \colon [a,b]\rightarrow\Omega},

\displaystyle L(\gamma) = \int_a^b \|\gamma'(t)\|_{\Omega,\gamma(t)}\,dt

This is the Hilbert metric {d_\Omega}. Despite all its non-Euclideanness, the geodesics of {d_\Omega} are line segments. This (nontrivial) fact makes it easy to calculate the distance between two points {a,b\in\Omega}: besides the points themselves, one only needs to consider the pair of points {a',b'} where the line through {a,b} meets {\partial \Omega}. The integral ends up being (half of) the logarithm of the cross-ratio of these four points. Assuming {a',a,b,b'} are situated in the listed order, the relevant cross-ratio is

\displaystyle \frac{|a'-b|\, |a-b'|}{|a'-a|\,|b-b'|}

which is greater than {1}, making the logarithm positive.

I personally find the integral of reciprocals of distances more intuitively accessible than the logarithm of cross-ratio. Either version of the definition makes it clear that the Hilbert metric is invariant under projective transformations, a property not shared by the metrics {\rho} and {\delta}.

It turns out that {d_{\mathbb D}} is also a model of the hyperbolic plane, but with geodesics being line segments rather than circular arcs. Along each diameter {d_{\mathbb D}} coincides with {\rho}, because at Euclidean distance {r} from the center it scales vectors by

\displaystyle    \frac{1}{2}\left(\frac{1}{1-r}+\frac{1}{1+r}\right) = \frac{1}{1-r^2}

Behold the magic of partial fractions!