Poincaré disk — geodesic triangle and angle defect
A geodesic triangle in the Poincaré disk model with its interior angles, its Gauss–Bonnet area as the angle defect pi − (alpha+beta+gamma), its hyperbolic side lengths and a numerical check of the hyperbolic law of cosines.
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<text class="ttl" x="28" y="28">Geodesic triangle in the Poincaré disk — angle defect = area</text>
<text class="sub" x="28" y="47">Geodesics of the hyperbolic plane are the circular arcs meeting the boundary at right angles.</text>
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<text x="226.9" y="137.7" text-anchor="middle" font-weight="700">A</text>
<text x="238.9" y="165.7" class="ang">α = 35.32°</text>
<text x="384.1" y="338.3" font-weight="700">B</text>
<text x="356.1" y="310.3" text-anchor="end" class="ang">β = 18.47°</text>
<text x="125.1" y="331.9" text-anchor="end" font-weight="700">C</text>
<text x="148.1" y="303.9" class="ang">γ = 28.08°</text>
<text class="kd" x="250" y="480" text-anchor="middle">boundary circle |z| = 1: the circle at infinity, not part of the plane</text>
<text class="key" x="500" y="96" font-weight="700">Vertices (unit disk, Poincaré metric ds = 2|dz| / (1 − |z|²))</text>
<text class="kd" x="500" y="118">A = (−0.0955, 0.5416)   B = (0.6106, −0.3815)   C = (−0.5597, −0.3497)</text>
<text class="key" x="500" y="152" font-weight="700">Interior angles</text>
<text class="kd" x="500" y="174">α = 35.32°   β = 18.47°   γ = 28.08°</text>
<text class="kd" x="500" y="194">α + β + γ = 81.87° — strictly less than 180°, as it must be.</text>
<text class="key" x="500" y="228" font-weight="700">Gauss–Bonnet (curvature K = −1)</text>
<text class="kd" x="500" y="250">Area = π − (α + β + γ) = 1.7128 = 98.13°</text>
<text class="kd" x="500" y="270">No hyperbolic triangle has area ≥ π; the supremum π is</text>
<text class="kd" x="500" y="288">attained only in the limit, by an ideal triangle.</text>
<text class="key" x="500" y="322" font-weight="700">Side lengths (hyperbolic distance)</text>
<text class="kd" x="500" y="344">d(p,q) = arcosh(1 + 2|p−q|² / ((1−|p|²)(1−|q|²)))</text>
<text class="kd" x="500" y="364">a = d(B,C) = 3.0967   b = d(C,A) = 2.4999   c = d(A,B) = 2.8920</text>
<text class="key" x="500" y="398" font-weight="700">Hyperbolic law of cosines</text>
<text class="kd" x="500" y="420">cosh a = cosh b · cosh c − sinh b · sinh c · cos α</text>
<text class="kd" x="500" y="440">11.085004 = 55.446260 − 44.361256  (residual 5.3e-15)</text>
<text class="kd" x="500" y="472">Grey arcs are the complete geodesics through each pair of</text>
<text class="kd" x="500" y="490">vertices, ideal endpoint to ideal endpoint.</text>
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