💿🐜 Antkeeper source code
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199 lines
5.1 KiB
199 lines
5.1 KiB
/*
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* Copyright (C) 2023 Christopher J. Howard
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*
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* This file is part of Antkeeper source code.
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*
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* Antkeeper source code is free software: you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation, either version 3 of the License, or
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* (at your option) any later version.
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*
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* Antkeeper source code is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with Antkeeper source code. If not, see <http://www.gnu.org/licenses/>.
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*/
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#ifndef ANTKEEPER_PHYSICS_ORBIT_ANOMALY_HPP
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#define ANTKEEPER_PHYSICS_ORBIT_ANOMALY_HPP
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#include "math/numbers.hpp"
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#include <cmath>
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namespace physics {
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namespace orbit {
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/**
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* Orbital anomaly functions.
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*/
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namespace anomaly {
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/**
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* Derives the eccentric anomaly given eccentricity and true anomaly.
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*
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* @param ec Eccentricity (e).
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* @param ta True anomaly (nu).
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* @return Eccentric anomaly (E).
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*/
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template <class T>
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T true_to_eccentric(T ec, T ta);
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/**
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* Derives the mean anomaly given eccentricity and true anomaly.
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*
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* @param ec Eccentricity (e).
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* @param ta True anomaly (nu).
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* @return Mean anomaly (M).
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*/
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template <class T>
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T true_to_mean(T ec, T ta);
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/**
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* Derives the true anomaly given eccentricity and eccentric anomaly.
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*
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* @param ec Eccentricity (e).
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* @param ea Eccentric anomaly (E).
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* @return True anomaly (nu).
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*/
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template <class T>
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T eccentric_to_true(T ec, T ea);
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/**
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* Derives the mean anomaly given eccentricity and eccentric anomaly.
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*
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* @param ec Eccentricity (e).
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* @param ea Eccentric anomaly (E).
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* @return Mean anomaly (M).
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*/
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template <class T>
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T eccentric_to_mean(T ec, T ea);
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/**
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* Iteratively derives the eccentric anomaly given eccentricity and mean anomaly.
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*
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* @param ec Eccentricity (e).
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* @param ma Mean anomaly (M).
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* @param iterations Maximum number of iterations.
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* @param tolerance Solution error tolerance.
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* @return Eccentric anomaly (E).
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*
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* @see Murison, Marc. (2006). A Practical Method for Solving the Kepler Equation. 10.13140/2.1.5019.6808.
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*/
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template <class T>
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T mean_to_eccentric(T ec, T ma, std::size_t iterations, T tolerance);
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/**
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* Iteratively derives the true anomaly given eccentricity and mean anomaly.
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*
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* @param ec Eccentricity (e).
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* @param ma Mean anomaly (M).
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* @param iterations Maximum number of iterations.
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* @param tolerance Solution error tolerance.
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* @return True anomaly (nu).
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*/
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template <class T>
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T mean_to_true(T ec, T ma, std::size_t iterations, T tolerance);
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template <class T>
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T true_to_eccentric(T ec, T ta)
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{
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// Parabolic orbit
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if (ec == T(1))
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return std::tan(ta * T(0.5));
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// Hyperbolic orbit
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if (ec > T(1))
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return std::acosh((ec + std::cos(ta)) / (T(1) + ec * std::cos(ta))) * ((ta < T(0)) ? T(-1) : T(1));
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// Elliptic orbit
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return std::atan2(std::sqrt(T(1) - ec * ec) * std::sin(ta), std::cos(ta) + ec);
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}
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template <class T>
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T true_to_mean(T ec, T ta)
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{
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return eccentric_to_mean(ec, true_to_eccentric(ec, ta));
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}
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template <class T>
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T eccentric_to_true(T ec, T ea)
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{
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// Parabolic orbit
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if (ec == T(1))
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return std::atan(ea) * T(2);
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// Hyperbolic orbit
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if (ec > T(1))
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return std::atan(std::sqrt((ec + T(1)) / (ec - T(1))) * std::tanh(ea * T(0.5))) * T(2);
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// Elliptic orbit
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return std::atan2(sqrt(T(1) - ec * ec) * std::sin(ea), std::cos(ea) - ec);
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}
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template <class T>
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T eccentric_to_mean(T ec, T ea)
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{
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// Parabolic orbit
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if (ec == T(1))
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return (ea * ea * ea) / T(6) + ea * T(0.5);
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// Hyperbolic orbit
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if (ec > T(1))
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return ec * std::sinh(ea) - ea;
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// Elliptic orbit
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return ea - ec * std::sin(ea);
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}
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template <class T>
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T mean_to_eccentric(T ec, T ma, std::size_t iterations, T tolerance)
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{
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// Wrap mean anomaly to `[-pi, pi]`
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ma = std::remainder(ma, math::two_pi<T>);
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// Third-order approximation of eccentric anomaly starting value, E0
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const T t33 = std::cos(ma);
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const T t34 = ec * ec;
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const T t35 = t34 * ec;
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T ea0 = ma + (T(-0.5) * t35 + ec + (t34 + T(1.5) * t33 * t35) * t33) * std::sin(ma);
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// Iteratively converge E0 and E1
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for (std::size_t i = 0; i < iterations; ++i)
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{
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// Third-order approximation of eccentric anomaly, E1
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const T t1 = std::cos(ea0);
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const T t2 = T(-1) + ec * t1;
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const T t3 = std::sin(ea0);
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const T t4 = ec * t3;
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const T t5 = -ea0 + t4 + ma;
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const T t6 = t5 / (T(0.5) * t5 * t4 / t2 + t2);
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const T ea1 = ea0 - (t5 / ((T(0.5) * t3 - (T(1) / T(6)) * t1 * t6) * ec * t6 + t2));
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// Determine solution error
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const T error = std::abs(ea1 - ea0);
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// Set E0 to E1
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ea0 = ea1;
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// Break if solution is within error tolerance
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if (error < tolerance)
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break;
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}
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return ea0;
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}
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template <class T>
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T mean_to_true(T ec, T ma, std::size_t iterations, T tolerance)
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{
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eccentric_to_true(ec, mean_to_eccentric(ec, ma, iterations, tolerance));
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}
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} // namespace anomaly
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} // namespace orbit
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} // namespace physics
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#endif // ANTKEEPER_PHYSICS_ORBIT_ANOMALY_HPP
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