Geodesics in the Poincaré disk and upper half-plane models of H². Click to place points, watch geodesics (circular arcs perpendicular to boundary), and explore exponential divergence of nearby trajectories.
Hyperbolic plane H² has constant negative Gaussian curvature K<0. Geodesics in Poincaré disk: circular arcs perpendicular to unit circle ∂H². Distance: d(z,w)=2tanh⁻¹|z-w|/|1-z̄w|. Geodesic flow is chaotic — Lyapunov exponent = √|K|. Area of disk of radius r: A=4π/|K|(cosh(r√|K|)-1)∼e^{r√|K|} (exponential growth). Connection to number theory (Riemann surfaces, modular group PSL(2,Z)).