Represents a fixed Euler rotation system. More...

Public Types
enum	{ AlphaAxisAbs , BetaAxisAbs , GammaAxisAbs , IsAlphaOpposite , IsBetaOpposite , IsGammaOpposite , IsOdd , IsEven , IsTaitBryan }

Static Public Attributes
static constexpr int	AlphaAxis

static constexpr int	BetaAxis

static constexpr int	GammaAxis

Private Member Functions
	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)

	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)

Static Private Member Functions
template<typename Scalar >
static void	CalcEulerAngles (EulerAngles< Scalar, EulerSystem > &res, const typename EulerAngles< Scalar, EulerSystem >::Matrix3 &mat)

template<typename Derived >
static void	CalcEulerAngles_imp (Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > &res, const MatrixBase< Derived > &mat, internal::false_type)

template<typename Derived >
static void	CalcEulerAngles_imp (Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > &res, const MatrixBase< Derived > &mat, internal::true_type)

Static Private Attributes
static const int	I_

static const int	J_

static const int	K_

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

Definition at line 128 of file EulerSystem.h.

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

Definition at line 143 of file EulerSystem.h.

     {
       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

Definition at line 259 of file EulerSystem.h.

     {
       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();
     }

◆ 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

Definition at line 224 of file EulerSystem.h.

     {
       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;
       }
     }

◆ 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

Definition at line 189 of file EulerSystem.h.

     {
       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;
       }
     }

◆ 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

Member Data Documentation

◆ AlphaAxis

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

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

staticconstexpr

The first rotation axis

Definition at line 135 of file EulerSystem.h.

◆ BetaAxis

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

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

staticconstexpr

The second rotation axis

Definition at line 138 of file EulerSystem.h.

◆ GammaAxis

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

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

staticconstexpr

The third rotation axis

Definition at line 141 of file EulerSystem.h.

◆ I_

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

const int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::I_

staticprivate

Definition at line 182 of file EulerSystem.h.

◆ J_

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

const int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::J_

staticprivate

Definition at line 183 of file EulerSystem.h.

◆ K_

template<int _AlphaAxis, int _BetaAxis, int _GammaAxis>

const int Eigen::EulerSystem< _AlphaAxis, _BetaAxis, _GammaAxis >::K_

staticprivate

Definition at line 184 of file EulerSystem.h.

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

EulerSystem.h

Public Types

Static Public Attributes

Private Member Functions

Static Private Member Functions

Static Private Attributes

Detailed Description

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

Types of Euler systems

Intrinsic vs extrinsic Euler systems

Convenient user typedefs

Additional reading

Member Enumeration Documentation

◆ anonymous enum

Member Function Documentation

◆ CalcEulerAngles()

◆ CalcEulerAngles_imp() [1/2]

◆ CalcEulerAngles_imp() [2/2]

◆ EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [1/5]

◆ EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [2/5]

◆ EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [3/5]

◆ EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [4/5]

◆ EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT() [5/5]

Member Data Documentation

◆ AlphaAxis

◆ BetaAxis

◆ GammaAxis

◆ I_

◆ J_

◆ K_

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