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88 lines
3.3 KiB
C++
88 lines
3.3 KiB
C++
#pragma once
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// Copyright (c) 2024-2026 Tarik Moussa.
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// SPDX-License-Identifier: MIT
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// euclidean_geometry.hpp
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//
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// Corner-angle formula for Euclidean triangles in the discrete conformal
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// (log-length) parametrisation.
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//
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// Ported from de.varylab.discreteconformal.functional.EuclideanCyclicFunctional.
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//
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// In the discrete conformal parametrisation a Euclidean triangle is described by
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// its three effective log-lengths Λ̃_ij = λ°_ij + u_i + u_j (+ edge DOF).
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// The corresponding side lengths are l_ij = exp(Λ̃_ij / 2).
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//
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// Vertex ordering convention (matches EuclideanCyclicFunctional.java):
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// v1 is opposite edge l23, v2 is opposite l31, v3 is opposite l12.
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//
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// t-value trick (Springborn 2008 §3):
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// t12 = −l12 + l23 + l31 = 2(s − l12)
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// t23 = +l12 − l23 + l31 = 2(s − l23)
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// t31 = +l12 + l23 − l31 = 2(s − l31)
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// denom = sqrt(t12 · t23 · t31 · l123) = 4 · Area
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//
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// α_v = 2 · atan2( product of t-values adjacent to v, denom )
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//
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// The centering trick (l_ij ← exp((Λ̃_ij − 2·μ)/2), μ = (Λ̃12+Λ̃23+Λ̃31)/6)
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// rescales all three sides by the same factor, leaving angles unchanged but
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// keeping the arguments of exp in a safe numerical range.
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#include <cmath>
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namespace conformallab {
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/// Interior corner angles of a Euclidean triangle.
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struct EuclideanFaceAngles {
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double alpha1; ///< Corner angle at v₁ (opposite l₂₃).
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double alpha2; ///< Corner angle at v₂ (opposite l₃₁).
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double alpha3; ///< Corner angle at v₃ (opposite l₁₂).
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bool valid; ///< `false` when the triangle is degenerate.
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};
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/// Compute the corner angles of a Euclidean triangle from its three
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/// side lengths. Returns `valid = false` when the triangle inequality
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/// is violated.
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inline EuclideanFaceAngles euclidean_angles_from_lengths(
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double l12, double l23, double l31)
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{
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const double t12 = -l12 + l23 + l31; // 2*(s − l12)
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const double t23 = +l12 - l23 + l31; // 2*(s − l23)
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const double t31 = +l12 + l23 - l31; // 2*(s − l31)
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if (t12 <= 0.0 || t23 <= 0.0 || t31 <= 0.0)
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return {0.0, 0.0, 0.0, false};
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const double l123 = l12 + l23 + l31;
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const double denom2 = t12 * t23 * t31 * l123; // = (4·Area)²
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if (denom2 <= 0.0)
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return {0.0, 0.0, 0.0, false};
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const double denom = std::sqrt(denom2);
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// α at v1 (opposite l23): adjacent t-values are t12 and t31
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// α at v2 (opposite l31): adjacent t-values are t12 and t23
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// α at v3 (opposite l12): adjacent t-values are t23 and t31
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return {
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2.0 * std::atan2(t12 * t31, denom),
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2.0 * std::atan2(t12 * t23, denom),
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2.0 * std::atan2(t23 * t31, denom),
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true
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};
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}
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/// Compute the corner angles of a Euclidean triangle from its three
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/// effective log-lengths `Λ̃ᵢⱼ`. Internally centres lengths so that
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/// `l₁₂·l₂₃·l₃₁ = 1` to avoid float overflow for large `|Λ̃|`.
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inline EuclideanFaceAngles euclidean_angles(
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double lam12, double lam23, double lam31)
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{
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const double mu = (lam12 + lam23 + lam31) / 6.0;
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const double l12 = std::exp((lam12 - 2.0 * mu) * 0.5);
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const double l23 = std::exp((lam23 - 2.0 * mu) * 0.5);
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const double l31 = std::exp((lam31 - 2.0 * mu) * 0.5);
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return euclidean_angles_from_lengths(l12, l23, l31);
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}
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} // namespace conformallab
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