Poincaré Disk:
Hyperbolic
Geometry
Show:
Geodesics
Tiling {7,3}
Circles
Horocycles
Tiling {p,q} — p
7
q
3
Rotation θ
0.00
Translation r
0.00
Click: place point for geodesic through origin
No point selected
Poincaré disk:
Open unit disk with metric ds² = 4(dx²+dy²)/(1−r²)².
Geodesics:
circular arcs perpendicular to the boundary circle.
Curvature:
K = −1.
Angle sum
of triangle < π. Sum decreases with area: π − (α+β+γ) = Area.
Poincaré's theorem:
{p,q} tiling exists in H² iff 1/p + 1/q < 1/2 (more than Euclidean allows).