the hyperbolic distance
Once you accept the hyperbolic ruler on the disk, you can ask the most basic question of any geometry: how far apart are two points? The hyperbolic distance is the answer — the length, measured with that special ruler, of the shortest path connecting the two points. It is a genuine distance: it is never negative, it is zero only when the points coincide, it is symmetric, and it satisfies the triangle inequality. But it is wildly different from the straight-ruler distance your eyes report.
There is a clean closed formula. For two points z and w in the unit disk, first form the Mobius-invariant quantity called the pseudo-hyperbolic distance, rho(z, w) = |(z - w)/(1 - w-bar z)|, which lies between 0 and 1. The true hyperbolic distance is then d(z, w) = log((1 + rho)/(1 - rho)), or equivalently 2 artanh(rho) (in the curvature -1 normalization; some texts use a factor of 1/2). The reason this works is automorphism-invariance: the disk automorphisms preserve hyperbolic distance, so you can slide one point to the center 0 by a Blaschke factor, where the distance reduces to the radial formula log((1 + |w'|)/(1 - |w'|)), and the quantity rho is exactly what that Blaschke factor leaves invariant. So measuring distance is: recenter one point to 0, read off the radius of the other, apply the log formula.
Hyperbolic distance is the quantity the Schwarz-Pick lemma controls: holomorphic self-maps of the disk never increase it. It governs the spacing of orbits in complex dynamics, the geometry of Riemann surfaces, the convergence theory of iterated maps, and the modular group's action on the half-plane. An honest caveat: hyperbolic distance is finite between any two interior points but grows to infinity as either point approaches the boundary, so the boundary circle is 'at infinity' — it is not part of the hyperbolic plane at all, only its horizon. And note the pseudo-hyperbolic rho is itself a useful distance (it too is automorphism-invariant), but it is NOT the same as d; rho saturates at 1 while d runs to infinity.
Take z = 0 and w = 1/2. Then rho = |(0 - 1/2)/(1 - 0)| = 1/2, and d(0, 1/2) = log((1 + 1/2)/(1 - 1/2)) = log 3 is about 1.099. Now take z = 1/2 and w = 3/4: rho = |(1/2 - 3/4)/(1 - (3/4)(1/2))| = |(-1/4)/(5/8)| = 2/5, giving d = log((1 + 2/5)/(1 - 2/5)) = log(7/3) is about 0.847 — even though 1/2 and 3/4 look closer on the page than 0 and 1/2, they can sit at a comparable hyperbolic distance because they are nearer the costly rim.
d(z,w) = log((1+rho)/(1-rho)) with rho = |(z-w)/(1 - w-bar z)| — automorphism-invariant.
Do not confuse the pseudo-hyperbolic distance rho (which lives in [0,1)) with the true hyperbolic distance d (which lives in [0, infinity)). They are linked by d = log((1+rho)/(1-rho)) but only d is additive along geodesics; rho is not.