Hyperbolic Geometry — where parallel lines diverge and angle sums < π
About
The Poincaré disk models the hyperbolic plane. The entire infinite hyperbolic space fits inside the unit disk. Geodesics are arcs orthogonal to the boundary (or diameters).
Click two points to see hyperbolic distance
Triangle mode: place 3 points for angle sum
Key Facts
d(z,w) = 2 tanh⁻¹(|z-w|/|1-z̄w|)
Angle sum of triangle < π
Defect = π − (α+β+γ) = Area
Infinitely many parallels through any external point
{7,3}: 7-gons, 3 per vertex