Poincaré Disk — Hyperbolic Geometry

Click to place points · geodesics are circular arcs orthogonal to boundary

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Hyperbolic distance
Triangle angle sum
Defect (π − sum)
Poincaré disk:
Geodesics = arcs of circles orthogonal to ∂D.

Metric: ds² = 4(dx²+dy²)/(1−r²)²

Curvature K = −1. Infinitely many parallels through a point to a line. Triangle angle sum < π — defect = area.