EulerAngles.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2023 Juraj Oršulić, University of Zagreb <juraj.orsulic@fer.hr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
#ifndef EIGEN_EULERANGLES_H
#define EIGEN_EULERANGLES_H
 
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
template<typename Derived>
EIGEN_DEVICE_FUNC inline Matrix<typename MatrixBase<Derived>::Scalar,3,1>
MatrixBase<Derived>::canonicalEulerAngles(Index a0, Index a1, Index a2) const
{
  /* Implemented from Graphics Gems IV */
  EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Derived, 3, 3)
 
  Matrix<Scalar, 3, 1> res;
 
  const Index odd = ((a0 + 1) % 3 == a1) ? 0 : 1;
  const Index i = a0;
  const Index j = (a0 + 1 + odd) % 3;
  const Index k = (a0 + 2 - odd) % 3;
 
  if (a0 == a2)
  {
    // Proper Euler angles (same first and last axis).
    // The i, j, k indices enable addressing the input matrix as the XYX archetype matrix (see Graphics Gems IV),
    // where e.g. coeff(k, i) means third column, first row in the XYX archetype matrix:
    //  c2      s2s1              s2c1
    //  s2s3   -c2s1s3 + c1c3    -c2c1s3 - s1c3
    // -s2c3    c2s1c3 + c1s3     c2c1c3 - s1s3
 
    // Note: s2 is always positive.
    Scalar s2 = numext::hypot(coeff(j, i), coeff(k, i));
    if (odd)
    {
      res[0] = numext::atan2(coeff(j, i), coeff(k, i));
      // s2 is always positive, so res[1] will be within the canonical [0, pi] range
      res[1] = numext::atan2(s2, coeff(i, i));
    }
    else
    {
      // In the !odd case, signs of all three angles are flipped at the very end. To keep the solution within the canonical range,
      // we flip the solution and make res[1] always negative here (since s2 is always positive, -atan2(s2, c2) will always be negative).
      // The final flip at the end due to !odd will thus make res[1] positive and canonical.
      // NB: in the general case, there are two correct solutions, but only one is canonical. For proper Euler angles,
      // flipping from one solution to the other involves flipping the sign of the second angle res[1] and adding/subtracting pi
      // to the first and third angles. The addition/subtraction of pi to the first angle res[0] is handled here by flipping
      // the signs of arguments to atan2, while the calculation of the third angle does not need special adjustment since
      // it uses the adjusted res[0] as the input and produces a correct result.
      res[0] = numext::atan2(-coeff(j, i), -coeff(k, i));
      res[1] = -numext::atan2(s2, coeff(i, i));
    }
 
    // With a=(0,1,0), we have i=0; j=1; k=2, and after computing the first two angles,
    // we can compute their respective rotation, and apply its inverse to M. Since the result must
    // be a rotation around x, we have:
    //
    //  c2  s1.s2 c1.s2                   1  0   0
    //  0   c1    -s1       *    M    =   0  c3  s3
    //  -s2 s1.c2 c1.c2                   0 -s3  c3
    //
    //  Thus:  m11.c1 - m21.s1 = c3  &   m12.c1 - m22.s1 = s3
 
    Scalar s1 = numext::sin(res[0]);
    Scalar c1 = numext::cos(res[0]);
    res[2] = numext::atan2(c1 * coeff(j, k) - s1 * coeff(k, k), c1 * coeff(j, j) - s1 * coeff(k, j));
  }
  else
  {
    // Tait-Bryan angles (all three axes are different; typically used for yaw-pitch-roll calculations).
    // The i, j, k indices enable addressing the input matrix as the XYZ archetype matrix (see Graphics Gems IV),
    // where e.g. coeff(k, i) means third column, first row in the XYZ archetype matrix:
    //  c2c3    s2s1c3 - c1s3     s2c1c3 + s1s3
    //  c2s3    s2s1s3 + c1c3     s2c1s3 - s1c3
    // -s2      c2s1              c2c1
 
    res[0] = numext::atan2(coeff(j, k), coeff(k, k));
 
    Scalar c2 = numext::hypot(coeff(i, i), coeff(i, j));
    // c2 is always positive, so the following atan2 will always return a result in the correct canonical middle angle range [-pi/2, pi/2]
    res[1] = numext::atan2(-coeff(i, k), c2);
 
    Scalar s1 = numext::sin(res[0]);
    Scalar c1 = numext::cos(res[0]);
    res[2] = numext::atan2(s1 * coeff(k, i) - c1 * coeff(j, i), c1 * coeff(j, j) - s1 * coeff(k, j));
  }
  if (!odd)
  {
    res = -res;
  }
 
  return res;
}
 
template<typename Derived>
EIGEN_DEPRECATED EIGEN_DEVICE_FUNC inline Matrix<typename MatrixBase<Derived>::Scalar,3,1>
MatrixBase<Derived>::eulerAngles(Index a0, Index a1, Index a2) const
{
  /* Implemented from Graphics Gems IV */
  EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Derived, 3, 3)
 
  Matrix<Scalar, 3, 1> res;
 
  const Index odd = ((a0 + 1) % 3 == a1) ? 0 : 1;
  const Index i = a0;
  const Index j = (a0 + 1 + odd) % 3;
  const Index k = (a0 + 2 - odd) % 3;
 
  if (a0 == a2)
  {
    res[0] = numext::atan2(coeff(j, i), coeff(k, i));
    if ((odd && res[0] < Scalar(0)) || ((!odd) && res[0] > Scalar(0)))
    {
      if (res[0] > Scalar(0))
      {
        res[0] -= Scalar(EIGEN_PI);
      }
      else
      {
        res[0] += Scalar(EIGEN_PI);
      }
 
      Scalar s2 = numext::hypot(coeff(j, i), coeff(k, i));
      res[1] = -numext::atan2(s2, coeff(i, i));
    }
    else
    {
      Scalar s2 = numext::hypot(coeff(j, i), coeff(k, i));
      res[1] = numext::atan2(s2, coeff(i, i));
    }
 
    // With a=(0,1,0), we have i=0; j=1; k=2, and after computing the first two angles,
    // we can compute their respective rotation, and apply its inverse to M. Since the result must
    // be a rotation around x, we have:
    //
    //  c2  s1.s2 c1.s2                   1  0   0
    //  0   c1    -s1       *    M    =   0  c3  s3
    //  -s2 s1.c2 c1.c2                   0 -s3  c3
    //
    //  Thus:  m11.c1 - m21.s1 = c3  &   m12.c1 - m22.s1 = s3
 
    Scalar s1 = numext::sin(res[0]);
    Scalar c1 = numext::cos(res[0]);
    res[2] = numext::atan2(c1 * coeff(j, k) - s1 * coeff(k, k), c1 * coeff(j, j) - s1 * coeff(k, j));
  }
  else
  {
    res[0] = numext::atan2(coeff(j, k), coeff(k, k));
    Scalar c2 = numext::hypot(coeff(i, i), coeff(i, j));
    if ((odd && res[0] < Scalar(0)) || ((!odd) && res[0] > Scalar(0)))
    {
      if (res[0] > Scalar(0))
      {
        res[0] -= Scalar(EIGEN_PI);
      }
      else
      {
        res[0] += Scalar(EIGEN_PI);
      }
      res[1] = numext::atan2(-coeff(i, k), -c2);
    }
    else
    {
      res[1] = numext::atan2(-coeff(i, k), c2);
    }
    Scalar s1 = numext::sin(res[0]);
    Scalar c1 = numext::cos(res[0]);
    res[2] = numext::atan2(s1 * coeff(k, i) - c1 * coeff(j, i), c1 * coeff(j, j) - s1 * coeff(k, j));
  }
  if (!odd)
  {
    res = -res;
  }
 
  return res;
}
 
} // end namespace Eigen
 
#endif // EIGEN_EULERANGLES_H