the Poincaré metric on the disk

The Poincare metric is the specific, named version of the hyperbolic metric living on the unit disk — Poincare's own model of the hyperbolic plane. Its job is to turn the round disk you can draw on a page into a faithful map of an infinite non-Euclidean world, the way a flat map of the Earth distorts the poles to fit a curved surface onto a flat sheet. The Poincare disk is the most famous such map: angles are drawn truthfully, but distances near the edge are radically compressed.

Its defining formula gives, at a point z, the line element ds = 2 |dz| / (1 - |z|^2). Equivalently the metric tensor is 4 (dx^2 + dy^2)/(1 - |z|^2)^2 — a constant multiple of the ordinary Euclidean metric, scaled point-by-point by the density. Because it is just a rescaling of the Euclidean metric (a 'conformal metric'), it preserves angles exactly: the Poincare disk is angle-true even though it is wildly distance-false near the boundary. This is what makes it so vivid for pictures of hyperbolic tilings — the famous Escher 'Circle Limit' prints live in the Poincare disk, with tiles all the same hyperbolic size but drawn shrinking toward the rim.

The Poincare metric is the canonical home for the Schwarz-Pick lemma, hyperbolic geodesics, and the hyperbolic area and distance formulas. It is conformally equivalent to the Poincare upper-half-plane model (where ds = |dz|/y), and a holomorphic universal-cover construction extends a Poincare-type metric to almost any hyperbolic Riemann surface. A careful note: 'the Poincare metric' usually means this disk metric with curvature -1 (the factor 2 is chosen exactly to make the curvature -1), but you will also meet it on the half-plane and, by Schwarz-Pick, as the extremal metric among all conformal metrics under which holomorphic self-maps contract. The name commemorates the model, not a different geometry from 'the hyperbolic metric' — they are two names for the same thing.