In hyperbolic space, parallel lines diverge and triangle angles sum to less than π
Click two points on the disk to draw a hyperbolic geodesic
About: The Poincaré disk model represents hyperbolic geometry inside the unit disk. Points at the boundary are "at infinity." Geodesics (straight lines in hyperbolic space) appear as circular arcs that meet the boundary at right angles, or as diameters. The hyperbolic distance between points grows exponentially toward the boundary. Triangle angle sum < π, and the deficit equals the area: A = π − (α+β+γ) (Gauss-Bonnet for constant curvature −1). Through any point, infinitely many lines are parallel to a given line (ultra-parallel), in contrast to Euclidean geometry's unique parallel (Playfair's axiom). Poincaré disk is conformal: angles are preserved.