Up to a constant factor, there is just one conformally invariant Riemannian metric on the disk . Indeed, on the tangent space at the metric must be a multiple of the Euclidean one, due to rotational invariance. Normalizing so that at both metrics coincide, we can use the invariance under conformal automorphisms (Möbius transformations)
to find that on the tangent space at the metric differs from Euclidean only by the factor of . This can be written as , indicating what we integrate to find the length of curves with respect to . This is the Poincaré disk model of the hyperbolic plane. The geodesics of are precisely the circles orthogonal to , and diameters.
We could model the hyperbolic plane on any other proper simply-connected domain , just by pulling back under the Riemann map . Explicitly, . But is this really explicit? We have no closed form for except for very special domains . The density can be quite tricky: see this MathOverflow question.
On the disk , the density of is roughly the reciprocal of the distance to the boundary. The Koebe distortion theorem yields the same for all simply connected domains:
This suggests replacing the hyperbolic metric with the quasihyperbolic metric . The density of 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 by , we divide it by , which is the distance measured in the direction of . In other words, this is how long you could walk in the direction before hitting the boundary. It is clear that the length of any curve going to the boundary is infinite due to the divergence of , same as for hyperbolic and quasihyperbolic metrics. Since it is awkward to have a “metric” that is not symmetric (the lengths of and are not the same), we symmetrize:
Now the distances between points are defined in the usual way, as the infimum of lengths of connecting curves ,
This is the Hilbert metric . Despite all its non-Euclideanness, the geodesics of are line segments. This (nontrivial) fact makes it easy to calculate the distance between two points : besides the points themselves, one only needs to consider the pair of points where the line through meets . The integral ends up being (half of) the logarithm of the cross-ratio of these four points. Assuming are situated in the listed order, the relevant cross-ratio is
which is greater than , 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 and .
It turns out that is also a model of the hyperbolic plane, but with geodesics being line segments rather than circular arcs. Along each diameter coincides with , because at Euclidean distance from the center it scales vectors by
Behold the magic of partial fractions!