Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis > Class Template Reference

Represents a fixed Euler rotation system. More...

#include <EulerSystem.h>

Public Types
enum	{   AlphaAxisAbs = internal::Abs<AlphaAxis>::value , BetaAxisAbs = internal::Abs<BetaAxis>::value , GammaAxisAbs = internal::Abs<GammaAxis>::value , IsAlphaOpposite = (AlphaAxis < 0) ? 1 : 0 ,   IsBetaOpposite = (BetaAxis < 0) ? 1 : 0 , IsGammaOpposite = (GammaAxis < 0) ? 1 : 0 , IsOdd = ((AlphaAxisAbs) % 3 == (BetaAxisAbs - 1) % 3) ? 0 : 1 , IsEven = IsOdd ? 0 : 1 ,   IsTaitBryan }

Static Public Attributes
static constexpr int	AlphaAxis = _AlphaAxis
static constexpr int	BetaAxis = _BetaAxis
static constexpr int	GammaAxis = _GammaAxis

Private Member Functions
	EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT (internal::IsValidAxis< AlphaAxis >::value, ALPHA_AXIS_IS_INVALID)
	EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT (internal::IsValidAxis< BetaAxis >::value, BETA_AXIS_IS_INVALID)
	EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT (internal::IsValidAxis< GammaAxis >::value, GAMMA_AXIS_IS_INVALID)
	EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT ((unsigned) AlphaAxisAbs !=(unsigned) BetaAxisAbs, ALPHA_AXIS_CANT_BE_EQUAL_TO_BETA_AXIS)
	EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT ((unsigned) BetaAxisAbs !=(unsigned) GammaAxisAbs, BETA_AXIS_CANT_BE_EQUAL_TO_GAMMA_AXIS)

Static Private Member Functions
template<typename Derived >
static void	CalcEulerAngles_imp (Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > &res, const MatrixBase< Derived > &mat, internal::true_type)
template<typename Derived >
static void	CalcEulerAngles_imp (Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > &res, const MatrixBase< Derived > &mat, internal::false_type)
template<typename Scalar >
static void	CalcEulerAngles (EulerAngles< Scalar, EulerSystem > &res, const typename EulerAngles< Scalar, EulerSystem >::Matrix3 &mat)

Static Private Attributes
static const int	I_ = AlphaAxisAbs - 1
static const int	J_ = (AlphaAxisAbs - 1 + 1 + IsOdd) % 3
static const int	K_ = (AlphaAxisAbs - 1 + 2 - IsOdd) % 3

Friends
template<typename Scalar_ , class _System >
class	Eigen::EulerAngles
template<typename System , typename Other , int OtherRows, int OtherCols>
struct	internal::eulerangles_assign_impl

Detailed Description

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>
class Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >

Represents a fixed Euler rotation system.

This meta-class goal is to represent the Euler system in compilation time, for EulerAngles.

You can use this class to get two things:

• Build an Euler system, and then pass it as a template parameter to EulerAngles.
• Query some compile time data about an Euler system. (e.g. Whether it's Tait-Bryan)

Euler rotation is a set of three rotation on fixed axes. (see EulerAngles) This meta-class store constantly those signed axes. (see EulerAxis)

Types of Euler systems

All and only valid 3 dimension Euler rotation over standard signed axes{+X,+Y,+Z,-X,-Y,-Z} are supported:

• all axes X, Y, Z in each valid order (see below what order is valid)
• rotation over the axis is supported both over the positive and negative directions.
• both Tait-Bryan and proper/classic Euler angles (i.e. the opposite).

Since EulerSystem support both positive and negative directions, you may call this rotation distinction in other names:

• right handed or left handed
• counterclockwise or clockwise

Notice all axed combination are valid, and would trigger a static assertion. Same unsigned axes can't be neighbors, e.g. {X,X,Y} is invalid. This yield two and only two classes:

• Tait-Bryan - all unsigned axes are distinct, e.g. {X,Y,Z}
• proper/classic Euler angles - The first and the third unsigned axes is equal, and the second is different, e.g. {X,Y,X}

Intrinsic vs extrinsic Euler systems

Only intrinsic Euler systems are supported for simplicity. If you want to use extrinsic Euler systems, just use the equal intrinsic opposite order for axes and angles. I.e axes (A,B,C) becomes (C,B,A), and angles (a,b,c) becomes (c,b,a).

Convenient user typedefs

Convenient typedefs for EulerSystem exist (only for positive axes Euler systems), in a form of EulerSystem{A}{B}{C}, e.g. EulerSystemXYZ.

Additional reading

More information about Euler angles: https://en.wikipedia.org/wiki/Euler_angles

Template Parameters

_AlphaAxis	the first fixed EulerAxis
_BetaAxis	the second fixed EulerAxis
_GammaAxis	the third fixed EulerAxis

Member Enumeration Documentation

anonymous enum

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

anonymous enum

Enumerator
AlphaAxisAbs	the first rotation axis unsigned
BetaAxisAbs	the second rotation axis unsigned
GammaAxisAbs	the third rotation axis unsigned
IsAlphaOpposite	whether alpha axis is negative
IsBetaOpposite	whether beta axis is negative
IsGammaOpposite	whether gamma axis is negative
IsOdd	whether the Euler system is odd
IsEven	whether the Euler system is even
IsTaitBryan	whether the Euler system is Tait-Bryan

       {
    AlphaAxisAbs = internal::Abs<AlphaAxis>::value, 
    BetaAxisAbs = internal::Abs<BetaAxis>::value,   
    GammaAxisAbs = internal::Abs<GammaAxis>::value, 
    IsAlphaOpposite = (AlphaAxis < 0) ? 1 : 0, 
    IsBetaOpposite = (BetaAxis < 0) ? 1 : 0,   
    IsGammaOpposite = (GammaAxis < 0) ? 1 : 0, 
    // Parity is even if alpha axis X is followed by beta axis Y, or Y is followed
    // by Z, or Z is followed by X; otherwise it is odd.
    IsOdd = ((AlphaAxisAbs) % 3 == (BetaAxisAbs - 1) % 3) ? 0 : 1, 
    IsEven = IsOdd ? 0 : 1,                                        
    IsTaitBryan =
        ((unsigned)AlphaAxisAbs != (unsigned)GammaAxisAbs) ? 1 : 0 
  };

Member Function Documentation

CalcEulerAngles()

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

template<typename Scalar >

static void Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::CalcEulerAngles ( EulerAngles< Scalar, EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis > > &  res, const typename EulerAngles< Scalar, EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis > >::Matrix3 &  mat  )

inlinestaticprivate

                                                                                           {
    CalcEulerAngles_imp(res.angles(), mat,
                        std::conditional_t<IsTaitBryan, internal::true_type, internal::false_type>());
 
    if (IsAlphaOpposite) res.alpha() = -res.alpha();
 
    if (IsBetaOpposite) res.beta() = -res.beta();
 
    if (IsGammaOpposite) res.gamma() = -res.gamma();
  }

References Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::CalcEulerAngles_imp(), Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::IsAlphaOpposite, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::IsBetaOpposite, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::IsGammaOpposite, and res.

CalcEulerAngles_imp() [1/2]

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

template<typename Derived >

static void Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::CalcEulerAngles_imp ( Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > &  res, const MatrixBase< Derived > &  mat, internal::false_type    )

inlinestaticprivate

                                                                                        {
    using std::atan2;
    using std::sqrt;
 
    typedef typename Derived::Scalar Scalar;
 
    const Scalar plusMinus = IsEven ? 1 : -1;
    const Scalar minusPlus = IsOdd ? 1 : -1;
 
    const Scalar Rsum = sqrt((mat(I_, J_) * mat(I_, J_) + mat(I_, K_) * mat(I_, K_) + mat(J_, I_) * mat(J_, I_) +
                              mat(K_, I_) * mat(K_, I_)) /
                             2);
 
    res[1] = atan2(Rsum, mat(I_, I_));
 
    // There is a singularity when sin(beta) == 0
    if (Rsum > 4 * NumTraits<Scalar>::epsilon()) {  // sin(beta) != 0
      res[0] = atan2(mat(J_, I_), minusPlus * mat(K_, I_));
      res[2] = atan2(mat(I_, J_), plusMinus * mat(I_, K_));
    } else if (mat(I_, I_) > 0) {                                       // sin(beta) == 0 and cos(beta) == 1
      Scalar spos = plusMinus * mat(K_, J_) + minusPlus * mat(J_, K_);  // 2*sin(alpha + gamma)
      Scalar cpos = mat(J_, J_) + mat(K_, K_);                          // 2*cos(alpha + gamma)
      res[0] = atan2(spos, cpos);
      res[2] = 0;
    } else {                                                            // sin(beta) == 0 and cos(beta) == -1
      Scalar sneg = plusMinus * mat(K_, J_) + plusMinus * mat(J_, K_);  // 2*sin(alpha - gamma)
      Scalar cneg = mat(J_, J_) - mat(K_, K_);                          // 2*cos(alpha - gamma)
      res[0] = atan2(sneg, cneg);
      res[2] = 0;
    }
  }

References atan2(), Eigen::atan2(), Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::I_, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::IsEven, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::IsOdd, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::J_, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::K_, anonymous_namespace{skew_symmetric_matrix3.cpp}::plusMinus(), res, and sqrt().

CalcEulerAngles_imp() [2/2]

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

template<typename Derived >

static void Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::CalcEulerAngles_imp ( Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > &  res, const MatrixBase< Derived > &  mat, internal::true_type    )

inlinestaticprivate

                                                                                       {
    using std::atan2;
    using std::sqrt;
 
    typedef typename Derived::Scalar Scalar;
 
    const Scalar plusMinus = IsEven ? 1 : -1;
    const Scalar minusPlus = IsOdd ? 1 : -1;
 
    const Scalar Rsum = sqrt((mat(I_, I_) * mat(I_, I_) + mat(I_, J_) * mat(I_, J_) + mat(J_, K_) * mat(J_, K_) +
                              mat(K_, K_) * mat(K_, K_)) /
                             2);
    res[1] = atan2(plusMinus * mat(I_, K_), Rsum);
 
    // There is a singularity when cos(beta) == 0
    if (Rsum > 4 * NumTraits<Scalar>::epsilon()) {  // cos(beta) != 0
      res[0] = atan2(minusPlus * mat(J_, K_), mat(K_, K_));
      res[2] = atan2(minusPlus * mat(I_, J_), mat(I_, I_));
    } else if (plusMinus * mat(I_, K_) > 0) {               // cos(beta) == 0 and sin(beta) == 1
      Scalar spos = mat(J_, I_) + plusMinus * mat(K_, J_);  // 2*sin(alpha + plusMinus * gamma
      Scalar cpos = mat(J_, J_) + minusPlus * mat(K_, I_);  // 2*cos(alpha + plusMinus * gamma)
      Scalar alphaPlusMinusGamma = atan2(spos, cpos);
      res[0] = alphaPlusMinusGamma;
      res[2] = 0;
    } else {                                                              // cos(beta) == 0 and sin(beta) == -1
      Scalar sneg = plusMinus * (mat(K_, J_) + minusPlus * mat(J_, I_));  // 2*sin(alpha + minusPlus*gamma)
      Scalar cneg = mat(J_, J_) + plusMinus * mat(K_, I_);                // 2*cos(alpha + minusPlus*gamma)
      Scalar alphaMinusPlusBeta = atan2(sneg, cneg);
      res[0] = alphaMinusPlusBeta;
      res[2] = 0;
    }
  }

References atan2(), Eigen::atan2(), Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::I_, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::IsEven, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::IsOdd, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::J_, Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::K_, anonymous_namespace{skew_symmetric_matrix3.cpp}::plusMinus(), res, and sqrt().

Referenced by Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::CalcEulerAngles().

EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [1/5]

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT ( (unsigned) AlphaAxisAbs !  = (unsigned) BetaAxisAbs, ALPHA_AXIS_CANT_BE_EQUAL_TO_BETA_AXIS    )

private

EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [2/5]

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT ( (unsigned) BetaAxisAbs !  = (unsigned) GammaAxisAbs, BETA_AXIS_CANT_BE_EQUAL_TO_GAMMA_AXIS    )

private

EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [3/5]

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT ( internal::IsValidAxis< AlphaAxis >::value  , ALPHA_AXIS_IS_INVALID    )

private

EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [4/5]

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT ( internal::IsValidAxis< BetaAxis >::value  , BETA_AXIS_IS_INVALID    )

private

EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [5/5]

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT ( internal::IsValidAxis< GammaAxis >::value  , GAMMA_AXIS_IS_INVALID    )

private

Friends And Related Function Documentation

Eigen::EulerAngles

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

template<typename Scalar_ , class _System >

friend class Eigen::EulerAngles

friend

internal::eulerangles_assign_impl

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

template<typename System , typename Other , int OtherRows, int OtherCols>

friend struct internal::eulerangles_assign_impl

friend

Member Data Documentation

AlphaAxis

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

constexpr int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::AlphaAxis = _AlphaAxis

staticconstexpr

The first rotation axis

BetaAxis

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

constexpr int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::BetaAxis = _BetaAxis

staticconstexpr

The second rotation axis

GammaAxis

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

constexpr int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::GammaAxis = _GammaAxis

staticconstexpr

The third rotation axis

I_

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

const int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::I_ = AlphaAxisAbs - 1

staticprivate

Referenced by Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::CalcEulerAngles_imp().

J_

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

const int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::J_ = (AlphaAxisAbs - 1 + 1 + IsOdd) % 3

staticprivate

Referenced by Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::CalcEulerAngles_imp().

K_

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

const int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::K_ = (AlphaAxisAbs - 1 + 2 - IsOdd) % 3

staticprivate

Referenced by Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::CalcEulerAngles_imp().

---

The documentation for this class was generated from the following file:

• EulerSystem.h