Eigen/src/Geometry/EulerAngles.h - eigen - Git at Google

 // 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

 // IWYU pragma: private
 #include "./InternalHeaderCheck.h"

 namespace Eigen {

 /** \geometry_module \ingroup Geometry_Module
  *
  *
  * \returns the canonical Euler-angles of the rotation matrix \c *this using the convention defined by the triplet (\a
  * a0,\a a1,\a a2)
  *
  * Each of the three parameters \a a0,\a a1,\a a2 represents the respective rotation axis as an integer in {0,1,2}.
  * For instance, in:
  * \code Vector3f ea = mat.eulerAngles(2, 0, 2); \endcode
  * "2" represents the z axis and "0" the x axis, etc. The returned angles are such that
  * we have the following equality:
  * \code
  * mat == AngleAxisf(ea[0], Vector3f::UnitZ())
  *      * AngleAxisf(ea[1], Vector3f::UnitX())
  *      * AngleAxisf(ea[2], Vector3f::UnitZ()); \endcode
  * This corresponds to the right-multiply conventions (with right hand side frames).
  *
  * For Tait-Bryan angle configurations (a0 != a2), the returned angles are in the ranges [-pi:pi]x[-pi/2:pi/2]x[-pi:pi].
  * For proper Euler angle configurations (a0 == a2), the returned angles are in the ranges [-pi:pi]x[0:pi]x[-pi:pi].
  *
  * The approach used is also described here:
  * https://d3cw3dd2w32x2b.cloudfront.net/wp-content/uploads/2012/07/euler-angles.pdf
  *
  * \sa class AngleAxis
  */
 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;
 }

 /** \geometry_module \ingroup Geometry_Module
  *
  *
  * \returns the Euler-angles of the rotation matrix \c *this using the convention defined by the triplet (\a a0,\a a1,\a
  * a2)
  *
  * NB: The returned angles are in non-canonical ranges [0:pi]x[-pi:pi]x[-pi:pi]. For canonical Tait-Bryan/proper Euler
  * ranges, use canonicalEulerAngles.
  *
  * \sa MatrixBase::canonicalEulerAngles
  * \sa class AngleAxis
  */
 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
	// 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

	// IWYU pragma: private
	#include "./InternalHeaderCheck.h"

	namespace Eigen {

	/** \geometry_module \ingroup Geometry_Module
	*
	*
	* \returns the canonical Euler-angles of the rotation matrix \c *this using the convention defined by the triplet (\a
	* a0,\a a1,\a a2)
	*
	* Each of the three parameters \a a0,\a a1,\a a2 represents the respective rotation axis as an integer in {0,1,2}.
	* For instance, in:
	* \code Vector3f ea = mat.eulerAngles(2, 0, 2); \endcode
	* "2" represents the z axis and "0" the x axis, etc. The returned angles are such that
	* we have the following equality:
	* \code
	* mat == AngleAxisf(ea[0], Vector3f::UnitZ())
	* * AngleAxisf(ea[1], Vector3f::UnitX())
	* * AngleAxisf(ea[2], Vector3f::UnitZ()); \endcode
	* This corresponds to the right-multiply conventions (with right hand side frames).
	*
	* For Tait-Bryan angle configurations (a0 != a2), the returned angles are in the ranges [-pi:pi]x[-pi/2:pi/2]x[-pi:pi].
	* For proper Euler angle configurations (a0 == a2), the returned angles are in the ranges [-pi:pi]x[0:pi]x[-pi:pi].
	*
	* The approach used is also described here:
	* https://d3cw3dd2w32x2b.cloudfront.net/wp-content/uploads/2012/07/euler-angles.pdf
	*
	* \sa class AngleAxis
	*/
	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;
	}

	/** \geometry_module \ingroup Geometry_Module
	*
	*
	* \returns the Euler-angles of the rotation matrix \c *this using the convention defined by the triplet (\a a0,\a a1,\a
	* a2)
	*
	* NB: The returned angles are in non-canonical ranges [0:pi]x[-pi:pi]x[-pi:pi]. For canonical Tait-Bryan/proper Euler
	* ranges, use canonicalEulerAngles.
	*
	* \sa MatrixBase::canonicalEulerAngles
	* \sa class AngleAxis
	*/
	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