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77 lines
2.9 KiB
C++
77 lines
2.9 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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// spherical_geometry.hpp
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//
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// Pure-math building blocks for the spherical discrete conformal map.
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// Ported from de.varylab.discreteconformal.functional.SphericalFunctional
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// (the geometry helpers embedded there).
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//
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// Notation:
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// u_i – vertex conformal factor (DOF)
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// λ°_e – base log-length of edge e (fixed initial value)
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// λ_ij – effective log-length = λ°_ij + u_i + u_j
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// l_ij – spherical arc length = 2·asin(min(exp(λ_ij/2), 1))
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// α_k – interior angle of the spherical triangle at vertex k
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#include "constants.hpp"
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#include <cmath>
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#include <algorithm>
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namespace conformallab {
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/// Backward-compatible alias — prefer conformallab::PI in new code.
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constexpr double PI_SPHER = PI;
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// ── Effective spherical arc length ────────────────────────────────────────────
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/// Spherical arc length `l(λ) = 2·asin(min(exp(λ/2), 1))`.
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/// Clamps `exp(λ/2)` to `[0, 1]` so `asin` stays in domain.
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inline double spherical_l(double lambda)
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{
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double half = std::exp(lambda * 0.5);
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if (half >= 1.0) half = 1.0 - 1e-15;
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if (half <= 0.0) return 0.0;
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return 2.0 * std::asin(half);
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}
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// ── Interior angles of a spherical triangle ──────────────────────────────────
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/// Interior angles of a spherical triangle, plus a `valid` flag.
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struct SphericalFaceAngles {
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double alpha1; ///< Corner angle at vertex v₁.
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double alpha2; ///< Corner angle at vertex v₂.
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double alpha3; ///< Corner angle at vertex v₃.
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bool valid; ///< `false` when the three lengths violate the spherical triangle inequality.
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};
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/// Compute the spherical-triangle corner angles `(α₁, α₂, α₃)` from
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/// the three arc lengths `(l₁₂, l₂₃, l₃₁)` using the half-angle form
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/// of the spherical law of cosines. Returns `valid = false` for
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/// degenerate or out-of-range triangles.
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inline SphericalFaceAngles spherical_angles(double l12, double l23, double l31)
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{
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double s = (l12 + l23 + l31) * 0.5;
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double s12 = s - l12;
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double s23 = s - l23;
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double s31 = s - l31;
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// Spherical triangle inequalities: all s-deficiencies > 0 and s < π.
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if (s12 <= 0.0 || s23 <= 0.0 || s31 <= 0.0 || s >= PI_SPHER)
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return {0.0, 0.0, 0.0, false};
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const double ss = std::sin(s);
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const double ss12 = std::sin(s12);
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const double ss23 = std::sin(s23);
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const double ss31 = std::sin(s31);
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double a1 = 2.0 * std::atan2(std::sqrt(ss12 * ss31), std::sqrt(ss * ss23));
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double a2 = 2.0 * std::atan2(std::sqrt(ss12 * ss23), std::sqrt(ss * ss31));
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double a3 = 2.0 * std::atan2(std::sqrt(ss23 * ss31), std::sqrt(ss * ss12));
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return {a1, a2, a3, true};
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}
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} // namespace conformallab
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