unsupported/Eigen/src/EulerAngles/EulerAngles.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2015 Tal Hadad <tal_hd@hotmail.com>
//
// 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_EULERANGLESCLASS_H  // TODO: Fix previous "EIGEN_EULERANGLES_H" definition?
#define EIGEN_EULERANGLESCLASS_H

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

namespace Eigen {
/** \class EulerAngles
 *
 * \ingroup EulerAngles_Module
 *
 * \brief Represents a rotation in a 3 dimensional space as three Euler angles.
 *
 * Euler rotation is a set of three rotation of three angles over three fixed axes, defined by the EulerSystem given as
 * a template parameter.
 *
 * Here is how intrinsic Euler angles works:
 *  - first, rotate the axes system over the alpha axis in angle alpha
 *  - then, rotate the axes system over the beta axis(which was rotated in the first stage) in angle beta
 *  - then, rotate the axes system over the gamma axis(which was rotated in the two stages above) in angle gamma
 *
 * \note This class support only intrinsic Euler angles for simplicity,
 *  see EulerSystem how to easily overcome this for extrinsic systems.
 *
 * ### Rotation representation and conversions ###
 *
 * It has been proved(see Wikipedia link below) that every rotation can be represented
 *  by Euler angles, but there is no single representation (e.g. unlike rotation matrices).
 * Therefore, you can convert from Eigen rotation and to them
 *  (including rotation matrices, which is not called "rotations" by Eigen design).
 *
 * Euler angles usually used for:
 *  - convenient human representation of rotation, especially in interactive GUI.
 *  - gimbal systems and robotics
 *  - efficient encoding(i.e. 3 floats only) of rotation for network protocols.
 *
 * However, Euler angles are slow comparing to quaternion or matrices,
 *  because their unnatural math definition, although it's simple for human.
 * To overcome this, this class provide easy movement from the math friendly representation
 *  to the human friendly representation, and vise-versa.
 *
 * All the user need to do is a safe simple C++ type conversion,
 *  and this class take care for the math.
 * Additionally, some axes related computation is done in compile time.
 *
 * #### Euler angles ranges in conversions ####
 * Rotations representation as EulerAngles are not single (unlike matrices),
 *  and even have infinite EulerAngles representations.<BR>
 * For example, add or subtract 2*PI from either angle of EulerAngles
 *  and you'll get the same rotation.
 * This is the general reason for infinite representation,
 *  but it's not the only general reason for not having a single representation.
 *
 * When converting rotation to EulerAngles, this class convert it to specific ranges
 * When converting some rotation to EulerAngles, the rules for ranges are as follow:
 * - If the rotation we converting from is an EulerAngles
 *  (even when it represented as RotationBase explicitly), angles ranges are __undefined__.
 * - otherwise, alpha and gamma angles will be in the range [-PI, PI].<BR>
 *   As for Beta angle:
 *    - If the system is Tait-Bryan, the beta angle will be in the range [-PI/2, PI/2].
 *    - otherwise:
 *      - If the beta axis is positive, the beta angle will be in the range [0, PI]
 *      - If the beta axis is negative, the beta angle will be in the range [-PI, 0]
 *
 * \sa EulerAngles(const MatrixBase<Derived>&)
 * \sa EulerAngles(const RotationBase<Derived, 3>&)
 *
 * ### Convenient user typedefs ###
 *
 * Convenient typedefs for EulerAngles exist for float and double scalar,
 *  in a form of EulerAngles{A}{B}{C}{scalar},
 *  e.g. \ref EulerAnglesXYZd, \ref EulerAnglesZYZf.
 *
 * Only for positive axes{+x,+y,+z} Euler systems are have convenient typedef.
 * If you need negative axes{-x,-y,-z}, it is recommended to create you own typedef with
 *  a word that represent what you need.
 *
 * ### Example ###
 *
 * \include EulerAngles.cpp
 * Output: \verbinclude EulerAngles.out
 *
 * ### Additional reading ###
 *
 * If you're want to get more idea about how Euler system work in Eigen see EulerSystem.
 *
 * More information about Euler angles: https://en.wikipedia.org/wiki/Euler_angles
 *
 * \tparam Scalar_ the scalar type, i.e. the type of the angles.
 *
 * \tparam _System the EulerSystem to use, which represents the axes of rotation.
 */
template <typename Scalar_, class _System>
class EulerAngles : public RotationBase<EulerAngles<Scalar_, _System>, 3> {
 public:
  typedef RotationBase<EulerAngles<Scalar_, _System>, 3> Base;

  /** the scalar type of the angles */
  typedef Scalar_ Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;

  /** the EulerSystem to use, which represents the axes of rotation. */
  typedef _System System;

  typedef Matrix<Scalar, 3, 3> Matrix3;      /*!< the equivalent rotation matrix type */
  typedef Matrix<Scalar, 3, 1> Vector3;      /*!< the equivalent 3 dimension vector type */
  typedef Quaternion<Scalar> QuaternionType; /*!< the equivalent quaternion type */
  typedef AngleAxis<Scalar> AngleAxisType;   /*!< the equivalent angle-axis type */

  /** \returns the axis vector of the first (alpha) rotation */
  static Vector3 AlphaAxisVector() {
    const Vector3& u = Vector3::Unit(System::AlphaAxisAbs - 1);
    return System::IsAlphaOpposite ? -u : u;
  }

  /** \returns the axis vector of the second (beta) rotation */
  static Vector3 BetaAxisVector() {
    const Vector3& u = Vector3::Unit(System::BetaAxisAbs - 1);
    return System::IsBetaOpposite ? -u : u;
  }

  /** \returns the axis vector of the third (gamma) rotation */
  static Vector3 GammaAxisVector() {
    const Vector3& u = Vector3::Unit(System::GammaAxisAbs - 1);
    return System::IsGammaOpposite ? -u : u;
  }

 private:
  Vector3 m_angles;

 public:
  /** Default constructor without initialization. */
  EulerAngles() {}
  /** Constructs and initialize an EulerAngles (\p alpha, \p beta, \p gamma). */
  EulerAngles(const Scalar& alpha, const Scalar& beta, const Scalar& gamma) : m_angles(alpha, beta, gamma) {}

  // TODO: Test this constructor
  /** Constructs and initialize an EulerAngles from the array data {alpha, beta, gamma} */
  explicit EulerAngles(const Scalar* data) : m_angles(data) {}

  /** Constructs and initializes an EulerAngles from either:
   *  - a 3x3 rotation matrix expression(i.e. pure orthogonal matrix with determinant of +1),
   *  - a 3D vector expression representing Euler angles.
   *
   * \note If \p other is a 3x3 rotation matrix, the angles range rules will be as follow:<BR>
   *  Alpha and gamma angles will be in the range [-PI, PI].<BR>
   *  As for Beta angle:
   *   - If the system is Tait-Bryan, the beta angle will be in the range [-PI/2, PI/2].
   *   - otherwise:
   *     - If the beta axis is positive, the beta angle will be in the range [0, PI]
   *     - If the beta axis is negative, the beta angle will be in the range [-PI, 0]
   */
  template <typename Derived>
  explicit EulerAngles(const MatrixBase<Derived>& other) {
    *this = other;
  }

  /** Constructs and initialize Euler angles from a rotation \p rot.
   *
   * \note If \p rot is an EulerAngles (even when it represented as RotationBase explicitly),
   *  angles ranges are __undefined__.
   *  Otherwise, alpha and gamma angles will be in the range [-PI, PI].<BR>
   *  As for Beta angle:
   *   - If the system is Tait-Bryan, the beta angle will be in the range [-PI/2, PI/2].
   *   - otherwise:
   *     - If the beta axis is positive, the beta angle will be in the range [0, PI]
   *     - If the beta axis is negative, the beta angle will be in the range [-PI, 0]
   */
  template <typename Derived>
  EulerAngles(const RotationBase<Derived, 3>& rot) {
    System::CalcEulerAngles(*this, rot.toRotationMatrix());
  }

  /*EulerAngles(const QuaternionType& q)
  {
    // TODO: Implement it in a faster way for quaternions
    // According to http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToEuler/
    //  we can compute only the needed matrix cells and then convert to euler angles. (see ZYX example below)
    // Currently we compute all matrix cells from quaternion.

    // Special case only for ZYX
    //Scalar y2 = q.y() * q.y();
    //m_angles[0] = std::atan2(2*(q.w()*q.z() + q.x()*q.y()), (1 - 2*(y2 + q.z()*q.z())));
    //m_angles[1] = std::asin( 2*(q.w()*q.y() - q.z()*q.x()));
    //m_angles[2] = std::atan2(2*(q.w()*q.x() + q.y()*q.z()), (1 - 2*(q.x()*q.x() + y2)));
  }*/

  /** \returns The angle values stored in a vector (alpha, beta, gamma). */
  const Vector3& angles() const { return m_angles; }
  /** \returns A read-write reference to the angle values stored in a vector (alpha, beta, gamma). */
  Vector3& angles() { return m_angles; }

  /** \returns The value of the first angle. */
  Scalar alpha() const { return m_angles[0]; }
  /** \returns A read-write reference to the angle of the first angle. */
  Scalar& alpha() { return m_angles[0]; }

  /** \returns The value of the second angle. */
  Scalar beta() const { return m_angles[1]; }
  /** \returns A read-write reference to the angle of the second angle. */
  Scalar& beta() { return m_angles[1]; }

  /** \returns The value of the third angle. */
  Scalar gamma() const { return m_angles[2]; }
  /** \returns A read-write reference to the angle of the third angle. */
  Scalar& gamma() { return m_angles[2]; }

  /** \returns The Euler angles rotation inverse (which is as same as the negative),
   *  (-alpha, -beta, -gamma).
   */
  EulerAngles inverse() const {
    EulerAngles res;
    res.m_angles = -m_angles;
    return res;
  }

  /** \returns The Euler angles rotation negative (which is as same as the inverse),
   *  (-alpha, -beta, -gamma).
   */
  EulerAngles operator-() const { return inverse(); }

  /** Set \c *this from either:
   *  - a 3x3 rotation matrix expression(i.e. pure orthogonal matrix with determinant of +1),
   *  - a 3D vector expression representing Euler angles.
   *
   * See EulerAngles(const MatrixBase<Derived, 3>&) for more information about
   *  angles ranges output.
   */
  template <class Derived>
  EulerAngles& operator=(const MatrixBase<Derived>& other) {
    EIGEN_STATIC_ASSERT(
        (internal::is_same<Scalar, typename Derived::Scalar>::value),
        YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)

    internal::eulerangles_assign_impl<System, Derived>::run(*this, other.derived());
    return *this;
  }

  // TODO: Assign and construct from another EulerAngles (with different system)

  /** Set \c *this from a rotation.
   *
   * See EulerAngles(const RotationBase<Derived, 3>&) for more information about
   *  angles ranges output.
   */
  template <typename Derived>
  EulerAngles& operator=(const RotationBase<Derived, 3>& rot) {
    System::CalcEulerAngles(*this, rot.toRotationMatrix());
    return *this;
  }

  /** \returns \c true if \c *this is approximately equal to \a other, within the precision
   * determined by \a prec.
   *
   * \sa MatrixBase::isApprox() */
  bool isApprox(const EulerAngles& other, const RealScalar& prec = NumTraits<Scalar>::dummy_precision()) const {
    return angles().isApprox(other.angles(), prec);
  }

  /** \returns an equivalent 3x3 rotation matrix. */
  Matrix3 toRotationMatrix() const {
    // TODO: Calc it faster
    return static_cast<QuaternionType>(*this).toRotationMatrix();
  }

  /** Convert the Euler angles to quaternion. */
  operator QuaternionType() const {
    return AngleAxisType(alpha(), AlphaAxisVector()) * AngleAxisType(beta(), BetaAxisVector()) *
           AngleAxisType(gamma(), GammaAxisVector());
  }

  friend std::ostream& operator<<(std::ostream& s, const EulerAngles<Scalar, System>& eulerAngles) {
    s << eulerAngles.angles().transpose();
    return s;
  }

  /** \returns \c *this with scalar type casted to \a NewScalarType */
  template <typename NewScalarType>
  EulerAngles<NewScalarType, System> cast() const {
    EulerAngles<NewScalarType, System> e;
    e.angles() = angles().template cast<NewScalarType>();
    return e;
  }
};

#define EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(AXES, SCALAR_TYPE, SCALAR_POSTFIX) \
  /** \ingroup EulerAngles_Module */                                         \
  typedef EulerAngles<SCALAR_TYPE, EulerSystem##AXES> EulerAngles##AXES##SCALAR_POSTFIX;

#define EIGEN_EULER_ANGLES_TYPEDEFS(SCALAR_TYPE, SCALAR_POSTFIX)      \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(XYZ, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(XYX, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(XZY, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(XZX, SCALAR_TYPE, SCALAR_POSTFIX) \
                                                                      \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(YZX, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(YZY, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(YXZ, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(YXY, SCALAR_TYPE, SCALAR_POSTFIX) \
                                                                      \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(ZXY, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(ZXZ, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(ZYX, SCALAR_TYPE, SCALAR_POSTFIX) \
  EIGEN_EULER_ANGLES_SINGLE_TYPEDEF(ZYZ, SCALAR_TYPE, SCALAR_POSTFIX)

EIGEN_EULER_ANGLES_TYPEDEFS(float, f)
EIGEN_EULER_ANGLES_TYPEDEFS(double, d)

namespace internal {
template <typename Scalar_, class _System>
struct traits<EulerAngles<Scalar_, _System> > {
  typedef Scalar_ Scalar;
};

// set from a rotation matrix
template <class System, class Other>
struct eulerangles_assign_impl<System, Other, 3, 3> {
  typedef typename Other::Scalar Scalar;
  static void run(EulerAngles<Scalar, System>& e, const Other& m) { System::CalcEulerAngles(e, m); }
};

// set from a vector of Euler angles
template <class System, class Other>
struct eulerangles_assign_impl<System, Other, 3, 1> {
  typedef typename Other::Scalar Scalar;
  static void run(EulerAngles<Scalar, System>& e, const Other& vec) { e.angles() = vec; }
};
}  // namespace internal
}  // namespace Eigen

#endif  // EIGEN_EULERANGLESCLASS_H