Hyperbolic Geodesics — Poincaré Disk

Poincaré disk model: all of ℍ² inside the unit disk. Geodesics are circles orthogonal to the boundary (or diameters).

Hyperbolic distance:
d(z,w) = 2 arctanh(|z−w|/|1−z̄w|)

Angle sum of a triangle < π. Gauss-Bonnet: Area = π − (α+β+γ).

Parallel postulate fails: through a point off a line, infinitely many parallels exist.

Click to place 2 points → geodesic. 3 points → triangle. Use Triangle mode to see all three angles and area.