Poincaré Disk — Geodesics & Reflections

Hyperbolic geometry — click to place points, observe non-Euclidean distance
ds² = 4(dx²+dy²)/(1-|z|²)²
Click inside the disk to place points. Geodesics are arcs of circles orthogonal to the boundary. The angle sum of any hyperbolic triangle is < π.
Poincaré Disk Model (Beltrami 1868, Poincaré 1882): The open unit disk with metric ds² = 4|dz|²/(1−|z|²)² models the hyperbolic plane H² with curvature K=−1. Geodesics are arcs of Euclidean circles meeting the boundary at right angles (or diameters). The disk is a conformal model — angles are preserved but distances are distorted (boundary is "at infinity"). Isometries are Möbius transformations preserving the disk: z → (az+b)/(b̄z+ā) with |a|²−|b|²=1. The {p,q} tessellation (meeting q polygons at each vertex with pq>2(p+q)) tiles the disk if 1/p+1/q<1/2 — the condition for hyperbolic geometry.