coordinate_transformation.hpp

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/*
 * Copyright (C) 2010-2022 The ESPResSo project
 *
 * 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
 * Convert coordinates from the Cartesian system to the cylindrical system.
 * The transformation functions are provided with three overloads:
 * - one function for the trivial Cartesian <-> cylindrical transformation
 * - one function to transform from/to a cylindrical system with custom axis
 *   (extra @p axis argument, keep in mind the angle phi is under-defined)
 * - one function to transform from/to an oriented cylindrical system with
 *   custom axis (extra @p orientation argument, the angle phi is well-defined)
 */
 
#include "utils/Vector.hpp"
#include "utils/math/vec_rotate.hpp"
#include "utils/matrix.hpp"
 
#include <cassert>
#include <cmath>
 
namespace Utils {
 
/**
 * @brief Basis change.
 */
inline Vector3d basis_change(Vector3d const &b1, Vector3d const &b2,
                             Vector3d const &b3, Vector3d const &v,
                             bool reverse = false) {
  auto const e_x = b1.normalized();
  auto const e_y = b2.normalized();
  auto const e_z = b3.normalized();
  auto const M = Matrix<double, 3, 3>{
      {e_x[0], e_x[1], e_x[2]},
      {e_y[0], e_y[1], e_y[2]},
      {e_z[0], e_z[1],
       e_z[2]}}.transposed();
  if (reverse) {
    return M * v;
  }
  return M.inversed() * v;
}
 
/**
 * @brief Coordinate transformation from Cartesian to cylindrical coordinates.
 * The origins and z-axis of the coordinate systems co-incide.
 * The @f$ \phi = 0 @f$ direction corresponds to the x-axis in the
 * original coordinate system.
 * @param pos    Vector to transform
 */
inline Vector3d
transform_coordinate_cartesian_to_cylinder(Vector3d const &pos) {
  auto const r = std::sqrt(pos[0] * pos[0] + pos[1] * pos[1]);
  auto const phi = std::atan2(pos[1], pos[0]);
  return {r, phi, pos[2]};
}
 
/**
 * @brief Coordinate transformation from Cartesian to cylindrical coordinates
 * with change of basis. The origins of the coordinate systems co-incide.
 *
 * If the parameter @p axis is not equal to <tt>[0, 0, 1]</tt>, the value
 * of the angle @f$ \phi @f$ in cylindrical coordinates is under-defined.
 * To fully define it, it is necessary to provide an orientation vector
 * in Cartesian coordinates that will be used as the reference point
 * (i.e. such that @f$ \phi = 0 @f$), by default it is the x-axis.
 *
 * @param pos    Vector to transform
 * @param axis   Longitudinal axis of the cylindrical coordinates
 * @param orientation   Reference point (in untransformed coordinates) for
 *                      which @f$ \phi = 0 @f$
 */
inline Vector3d transform_coordinate_cartesian_to_cylinder(
    Vector3d const &pos, Vector3d const &axis, Vector3d const &orientation) {
  // check that axis and orientation are orthogonal
  assert(std::abs(axis * orientation) <
         5 * std::numeric_limits<double>::epsilon());
  auto const rotation_axis = vector_product(axis, orientation);
  auto const pos_t = basis_change(orientation, rotation_axis, axis, pos);
  return transform_coordinate_cartesian_to_cylinder(pos_t);
}
 
/**
 * @brief Coordinate transformation from cylindrical to Cartesian coordinates.
 * The origins and z-axis of the coordinate systems co-incide.
 * The @f$ \phi = 0 @f$ direction corresponds to the x-axis in the
 * transformed coordinate system.
 * @param pos    Vector to transform
 */
inline Vector3d
transform_coordinate_cylinder_to_cartesian(Vector3d const &pos) {
  auto const &rho = pos[0];
  auto const &phi = pos[1];
  auto const &z = pos[2];
  return {rho * std::cos(phi), rho * std::sin(phi), z};
}
 
/**
 * @brief Coordinate transformation from cylindrical to Cartesian coordinates
 * with change of basis. The origins of the coordinate systems co-incide.
 *
 * If the parameter @p axis is not equal to <tt>[0, 0, 1]</tt>, the value
 * of the angle @f$ \phi @f$ in cylindrical coordinates is under-defined.
 * To fully define it, it is necessary to provide an orientation vector
 * in Cartesian coordinates that will be used as the reference point
 * (i.e. such that @f$ \phi = 0 @f$).
 *
 * @param pos    Vector to transform
 * @param axis   Longitudinal axis of the cylindrical coordinates
 * @param orientation   Reference point (in Cartesian coordinates) for
 *                      which @f$ \phi = 0 @f$
 */
inline Vector3d transform_coordinate_cylinder_to_cartesian(
    Vector3d const &pos, Vector3d const &axis, Vector3d const &orientation) {
  // check that axis and orientation are orthogonal
  assert(std::abs(axis * orientation) <
         5 * std::numeric_limits<double>::epsilon());
  auto const rotation_axis = vector_product(axis, orientation);
  auto const pos_t = transform_coordinate_cylinder_to_cartesian(pos);
  return basis_change(orientation, rotation_axis, axis, pos_t, true);
}
 
/**
 * @brief Vector transformation from Cartesian to cylindrical coordinates.
 * @param vec    Vector to transform
 * @param axis   Longitudinal axis of the cylindrical coordinates
 * @param pos    Origin of the vector
 */
inline Vector3d transform_vector_cartesian_to_cylinder(Vector3d const &vec,
                                                       Vector3d const &axis,
                                                       Vector3d const &pos) {
  static auto const z_axis = Vector3d{{0, 0, 1}};
  auto const angle = angle_between(axis, z_axis);
  auto const rotation_axis = Utils::vector_product(axis, z_axis).normalize();
  auto const rotated_pos = vec_rotate(rotation_axis, angle, pos);
  auto const rotated_vec = vec_rotate(rotation_axis, angle, vec);
  auto const r = std::sqrt(rotated_pos[0] * rotated_pos[0] +
                           rotated_pos[1] * rotated_pos[1]);
  // v_r = (x * v_x + y * v_y) / sqrt(x^2 + y^2)
  auto const v_r =
      (rotated_pos[0] * rotated_vec[0] + rotated_pos[1] * rotated_vec[1]) / r;
  // v_phi = (x * v_y - y * v_x ) / sqrt(x^2 + y^2)
  auto const v_phi =
      (rotated_pos[0] * rotated_vec[1] - rotated_pos[1] * rotated_vec[0]) / r;
  return Vector3d{v_r, v_phi, rotated_vec[2]};
}
 
} // namespace Utils