math/quaternion.hpp

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/*
 * Copyright (C) 2010-2022 The ESPResSo project
 * Copyright (C) 2002,2003,2004,2005,2006,2007,2008,2009,2010
 *   Max-Planck-Institute for Polymer Research, Theory Group
 *
 * This file is part of ESPResSo.
 *
 * ESPResSo is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * ESPResSo is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program.  If not, see <http://www.gnu.org/licenses/>.
 */
 
#pragma once
 
/** \file
 *  Quaternion algebra.
 */
 
#include "utils/Vector.hpp"
#include "utils/quaternion.hpp"
 
#include <cmath>
#include <limits>
#include <numbers>
 
namespace Utils {
 
/** Convert quaternion to director
 *  @return A (non-normalized) director.
 */
template <class T>
Vector<T, 3> convert_quaternion_to_director(Quaternion<T> const &quat) {
  return {2 * (quat[1] * quat[3] + quat[0] * quat[2]),
          2 * (quat[2] * quat[3] - quat[0] * quat[1]),
          quat[0] * quat[0] - quat[1] * quat[1] - quat[2] * quat[2] +
              quat[3] * quat[3]};
}
 
/** Convert director to quaternion
 *  @param d  Director
 *  @return A (non-normalized) quaternion from the director, or {1, 0, 0, 0}
 *  if the director is the null vector.
 */
template <class T>
Quaternion<T> convert_director_to_quaternion(Vector<T, 3> const &d) {
 
  auto const dm = d.norm();
 
  // null vectors cannot be converted to quaternions
  if (dm < std::numeric_limits<T>::epsilon()) {
    return {{{{1, 0, 0, 0}}}};
  }
 
  // Calculate angles
  auto const d_xy = std::sqrt(d[0] * d[0] + d[1] * d[1]);
  T theta2, phi2;
  if (d_xy == 0.) {
    // Here the director is co-linear with the z-axis
    // We need to distinguish between (0, 0, +d_z) and (0, 0, -d_z)
    theta2 = (d[2] > 0.) ? 0. : std::numbers::pi_v<T> / 2.;
    phi2 = 0.;
  } else {
    // Here we take care of all other directions
    // We suppose that theta2 = theta/2 and phi2 = (phi - pi/2)/2,
    // where angles theta and phi are in spherical coordinates
    theta2 = std::acos(d[2] / dm) / 2.;
    // here we do not use the signum function due to the edge case d[1] = 0
    auto const phi = ((d[1] > 0.) ? 1. : -1.) * std::acos(d[0] / d_xy);
    phi2 = phi / 2. - std::numbers::pi_v<T> / 4.;
  }
 
  // Calculate the quaternion from the angles
  auto const cos_theta2 = std::cos(theta2);
  auto const sin_theta2 = std::sin(theta2);
  auto const cos_phi2 = std::cos(phi2);
  auto const sin_phi2 = std::sin(phi2);
  return {{{{cos_theta2 * cos_phi2, -sin_theta2 * cos_phi2,
             -sin_theta2 * sin_phi2, cos_theta2 * sin_phi2}}}};
}
 
} // namespace Utils