Hyperbolic Geometry: Poincaré Disk

Geodesics, parallel lines, and angle defect — click to place points, explore non-Euclidean geometry

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triangle angle sum
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angle defect (180°−Σ)
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hyperbolic area
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hyp. distance A→B
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Poincaré disk model: Hyperbolic plane H² in unit disk. Metric: ds² = 4|dz|²/(1−|z|²)². Geodesics = circular arcs orthogonal to the boundary (or diameters).
Angle defect: For a hyperbolic triangle, Σangles = π − Area(K=−1). Angle sum < 180°. The Gauss-Bonnet theorem: ∫K dA = 2π χ = −Area for K=−1.
Parallel postulate fails: Through a point P not on line ℓ, there are infinitely many parallel geodesics. The two "limiting parallels" bound the "angle of parallelism".
Hyperbolic distance: d(z,w) = 2 arctanh|z−w|/|1−z̄w|. Near the boundary, distances expand — the boundary circle is "at infinity" (the ideal boundary ∂H²).