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// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2008 Gael Guennebaud // Copyright (C) 2023 Juraj Oršulić, University of Zagreb // // 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 EIGEN_DEVICE_FUNC inline Matrix::Scalar, 3, 1> MatrixBase::canonicalEulerAngles( Index a0, Index a1, Index a2) const { /* Implemented from Graphics Gems IV */ EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Derived, 3, 3) Matrix 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 EIGEN_DEVICE_FUNC inline Matrix::Scalar, 3, 1> MatrixBase::eulerAngles( Index a0, Index a1, Index a2) const { /* Implemented from Graphics Gems IV */ EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Derived, 3, 3) Matrix 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