the Poincare disk model
/ pwan-kah-RAY /
Hyperbolic geometry can feel like a rumour until you can actually draw it. A model is a concrete picture, living inside ordinary Euclidean space, in which all the hyperbolic axioms come true — and the Poincare disk is the most beloved one, the world of Escher's 'Circle Limit' woodcuts. The entire infinite hyperbolic plane is squeezed inside one open disk. The points are the points strictly inside the boundary circle; the boundary itself is 'infinity' and is NOT part of the plane. The catch is that the ruler is warped: distances stretch enormously as you approach the rim, so an object drifting toward the edge must shrink endlessly on the page yet never actually arrives.
The model comes with a dictionary you must learn to read it. A 'straight line' (a geodesic) is NOT a Euclidean straight line; it is either a diameter of the disk, or an arc of a circle that meets the boundary circle at right angles. Two such arcs are 'parallel' if they do not cross inside the disk; since you can fit infinitely many non-crossing arcs through a point off a given arc, the parallel postulate fails exactly as hyperbolic geometry demands. The model's signature virtue is that it is CONFORMAL: it preserves ANGLES faithfully, so the hyperbolic angle between two arcs equals the ordinary Euclidean angle you measure between them on paper.
What the model trades away for that fidelity is straightness and uniform scale: hyperbolic lines look bent, and equal hyperbolic lengths look wildly unequal near the edge. This is the crucial honesty about ALL these models — none is 'the real' hyperbolic plane; each is a faithful but distorted map of it, like a flat atlas of a round Earth. The disk keeps angles right and sacrifices shape; the Klein model will do the reverse. They are different charts of one and the same geometry.
Draw a unit disk. A diameter is one hyperbolic line. For another, draw a circular arc from one boundary point to another that crosses the rim at 90 degrees at both ends — that is a second hyperbolic line. Pick a point not on the diameter: you can draw infinitely many such arcs through it that never touch the diameter inside the disk, so it has infinitely many parallels — yet every angle you read with a protractor is the true hyperbolic angle.
Lines are diameters or arcs meeting the rim at right angles; the model preserves angles but badly distorts distance near the edge.
A model is not the geometry itself — the disk is a distorted but faithful MAP of the hyperbolic plane. Its lines only look curved because of the chosen chart; intrinsically they are perfectly straight, and the rim is unreachable infinity, not a wall.