Eigen-unsupported  3.4.90 (git rev 67eeba6e720c5745abc77ae6c92ce0a44aa7b7ae)
Eigen::EulerAngles< Scalar_, _System > Class Template Reference

## Detailed Description

### template<typename Scalar_, class _System> class Eigen::EulerAngles< Scalar_, _System >

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.
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].
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]
EulerAngles(const MatrixBase<Derived>&)
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. EulerAnglesXYZd, 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 <unsupported/Eigen/EulerAngles>
#include <iostream>
using namespace Eigen;
int main()
{
// A common Euler system by many armies around the world,
// where the first one is the azimuth(the angle from the north -
// the same angle that is show in compass)
// and the second one is elevation(the angle from the horizon)
// and the third one is roll(the angle between the horizontal body
// direction and the plane ground surface)
// Keep remembering we're using radian angles here!
typedef EulerSystem<-EULER_Z, EULER_Y, EULER_X> MyArmySystem;
typedef EulerAngles<double, MyArmySystem> MyArmyAngles;
MyArmyAngles vehicleAngles(
3.14/*PI*/ / 2, /* heading to east, notice that this angle is counter-clockwise */
-0.3, /* going down from a mountain */
0.1); /* slightly rolled to the right */
// Some Euler angles representation that our plane use.
EulerAnglesZYZd planeAngles(0.78474, 0.5271, -0.513794);
MyArmyAngles planeAnglesInMyArmyAngles(planeAngles);
std::cout << "vehicle angles(MyArmy): " << vehicleAngles << std::endl;
std::cout << "plane angles(ZYZ): " << planeAngles << std::endl;
std::cout << "plane angles(MyArmy): " << planeAnglesInMyArmyAngles << std::endl;
// Now lets rotate the plane a little bit
std::cout << "==========================================================\n";
std::cout << "rotating plane now!\n";
std::cout << "==========================================================\n";
Quaterniond planeRotated = AngleAxisd(-0.342, Vector3d::UnitY()) * planeAngles;
planeAngles = planeRotated;
planeAnglesInMyArmyAngles = planeRotated;
std::cout << "new plane angles(ZYZ): " << planeAngles << std::endl;
std::cout << "new plane angles(MyArmy): " << planeAnglesInMyArmyAngles << std::endl;
return 0;
}
static const BasisReturnType UnitY()
@ EULER_X
Definition: EulerSystem.h:65
@ EULER_Z
Definition: EulerSystem.h:67
@ EULER_Y
Definition: EulerSystem.h:66
Quaternion< double > Quaterniond
AngleAxis< double > AngleAxisd
Namespace containing all symbols from the Eigen library.

Output:

vehicle angles(MyArmy):     1.57 -0.3  0.1
plane angles(ZYZ):          0.78474    0.5271 -0.513794
plane angles(MyArmy):     -0.206273  0.453463 -0.278617
==========================================================
rotating plane now!
==========================================================
new plane angles(ZYZ):      1.44358 0.366507 -1.23637
new plane angles(MyArmy):  -0.18648  0.117896 -0.347841


If you're want to get more idea about how Euler system work in Eigen see EulerSystem.

Template Parameters
 Scalar_ the scalar type, i.e. the type of the angles. _System the EulerSystem to use, which represents the axes of rotation.
Inheritance diagram for Eigen::EulerAngles< Scalar_, _System >:

## Public Types

typedef AngleAxis< ScalarAngleAxisType

typedef Matrix< Scalar, 3, 3 > Matrix3

typedef Quaternion< ScalarQuaternionType

typedef Scalar_ Scalar

typedef _System System

typedef Matrix< Scalar, 3, 1 > Vector3

## Public Member Functions

Scalaralpha ()

Scalar alpha () const

Vector3angles ()

const Vector3angles () const

Scalarbeta ()

Scalar beta () const

template<typename NewScalarType >
EulerAngles< NewScalarType, Systemcast () const

EulerAngles ()

template<typename Derived >
EulerAngles (const MatrixBase< Derived > &other)

template<typename Derived >
EulerAngles (const RotationBase< Derived, 3 > &rot)

EulerAngles (const Scalar &alpha, const Scalar &beta, const Scalar &gamma)

EulerAngles (const Scalar *data)

Scalargamma ()

Scalar gamma () const

EulerAngles inverse () const

bool isApprox (const EulerAngles &other, const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const

operator QuaternionType () const

EulerAngles operator- () const

template<class Derived >
EulerAnglesoperator= (const MatrixBase< Derived > &other)

template<typename Derived >
EulerAnglesoperator= (const RotationBase< Derived, 3 > &rot)

Matrix3 toRotationMatrix () const

## Static Public Member Functions

static Vector3 AlphaAxisVector ()

static Vector3 BetaAxisVector ()

static Vector3 GammaAxisVector ()

## ◆ AngleAxisType

template<typename Scalar_ , class _System >
 typedef AngleAxis Eigen::EulerAngles< Scalar_, _System >::AngleAxisType

the equivalent angle-axis type

## ◆ Matrix3

template<typename Scalar_ , class _System >
 typedef Matrix Eigen::EulerAngles< Scalar_, _System >::Matrix3

the equivalent rotation matrix type

## ◆ QuaternionType

template<typename Scalar_ , class _System >
 typedef Quaternion Eigen::EulerAngles< Scalar_, _System >::QuaternionType

the equivalent quaternion type

## ◆ Scalar

template<typename Scalar_ , class _System >
 typedef Scalar_ Eigen::EulerAngles< Scalar_, _System >::Scalar

the scalar type of the angles

## ◆ System

template<typename Scalar_ , class _System >
 typedef _System Eigen::EulerAngles< Scalar_, _System >::System

the EulerSystem to use, which represents the axes of rotation.

## ◆ Vector3

template<typename Scalar_ , class _System >
 typedef Matrix Eigen::EulerAngles< Scalar_, _System >::Vector3

the equivalent 3 dimension vector type

## ◆ EulerAngles() [1/5]

template<typename Scalar_ , class _System >
 Eigen::EulerAngles< Scalar_, _System >::EulerAngles ( )
inline

Default constructor without initialization.

## ◆ EulerAngles() [2/5]

template<typename Scalar_ , class _System >
 Eigen::EulerAngles< Scalar_, _System >::EulerAngles ( const Scalar & alpha, const Scalar & beta, const Scalar & gamma )
inline

Constructs and initialize an EulerAngles (alpha, beta, gamma).

## ◆ EulerAngles() [3/5]

template<typename Scalar_ , class _System >
 Eigen::EulerAngles< Scalar_, _System >::EulerAngles ( const Scalar * data )
inlineexplicit

Constructs and initialize an EulerAngles from the array data {alpha, beta, gamma}

## ◆ EulerAngles() [4/5]

template<typename Scalar_ , class _System >
template<typename Derived >
 Eigen::EulerAngles< Scalar_, _System >::EulerAngles ( const MatrixBase< Derived > & other )
inlineexplicit

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 other is a 3x3 rotation matrix, the angles range rules will be as follow:
Alpha and gamma angles will be in the range [-PI, PI].
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]

## ◆ EulerAngles() [5/5]

template<typename Scalar_ , class _System >
template<typename Derived >
 Eigen::EulerAngles< Scalar_, _System >::EulerAngles ( const RotationBase< Derived, 3 > & rot )
inline

Constructs and initialize Euler angles from a rotation rot.

Note
If 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].
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]

## ◆ alpha() [1/2]

template<typename Scalar_ , class _System >
 Scalar& Eigen::EulerAngles< Scalar_, _System >::alpha ( )
inline
Returns
A read-write reference to the angle of the first angle.

## ◆ alpha() [2/2]

template<typename Scalar_ , class _System >
 Scalar Eigen::EulerAngles< Scalar_, _System >::alpha ( ) const
inline
Returns
The value of the first angle.

## ◆ AlphaAxisVector()

template<typename Scalar_ , class _System >
 static Vector3 Eigen::EulerAngles< Scalar_, _System >::AlphaAxisVector ( )
inlinestatic
Returns
the axis vector of the first (alpha) rotation

## ◆ angles() [1/2]

template<typename Scalar_ , class _System >
 Vector3& Eigen::EulerAngles< Scalar_, _System >::angles ( )
inline
Returns
A read-write reference to the angle values stored in a vector (alpha, beta, gamma).

## ◆ angles() [2/2]

template<typename Scalar_ , class _System >
 const Vector3& Eigen::EulerAngles< Scalar_, _System >::angles ( ) const
inline
Returns
The angle values stored in a vector (alpha, beta, gamma).

## ◆ beta() [1/2]

template<typename Scalar_ , class _System >
 Scalar& Eigen::EulerAngles< Scalar_, _System >::beta ( )
inline
Returns
A read-write reference to the angle of the second angle.

## ◆ beta() [2/2]

template<typename Scalar_ , class _System >
 Scalar Eigen::EulerAngles< Scalar_, _System >::beta ( ) const
inline
Returns
The value of the second angle.

## ◆ BetaAxisVector()

template<typename Scalar_ , class _System >
 static Vector3 Eigen::EulerAngles< Scalar_, _System >::BetaAxisVector ( )
inlinestatic
Returns
the axis vector of the second (beta) rotation

## ◆ cast()

template<typename Scalar_ , class _System >
template<typename NewScalarType >
 EulerAngles Eigen::EulerAngles< Scalar_, _System >::cast ( ) const
inline
Returns
*this with scalar type casted to NewScalarType

## ◆ gamma() [1/2]

template<typename Scalar_ , class _System >
 Scalar& Eigen::EulerAngles< Scalar_, _System >::gamma ( )
inline
Returns
A read-write reference to the angle of the third angle.

## ◆ gamma() [2/2]

template<typename Scalar_ , class _System >
 Scalar Eigen::EulerAngles< Scalar_, _System >::gamma ( ) const
inline
Returns
The value of the third angle.

## ◆ GammaAxisVector()

template<typename Scalar_ , class _System >
 static Vector3 Eigen::EulerAngles< Scalar_, _System >::GammaAxisVector ( )
inlinestatic
Returns
the axis vector of the third (gamma) rotation

## ◆ inverse()

template<typename Scalar_ , class _System >
 EulerAngles Eigen::EulerAngles< Scalar_, _System >::inverse ( ) const
inline
Returns
The Euler angles rotation inverse (which is as same as the negative), (-alpha, -beta, -gamma).

## ◆ isApprox()

template<typename Scalar_ , class _System >
 bool Eigen::EulerAngles< Scalar_, _System >::isApprox ( const EulerAngles< Scalar_, _System > & other, const RealScalar & prec = NumTraits::dummy_precision() ) const
inline
Returns
true if *this is approximately equal to other, within the precision determined by prec.
MatrixBase::isApprox()

## ◆ operator QuaternionType()

template<typename Scalar_ , class _System >
 Eigen::EulerAngles< Scalar_, _System >::operator QuaternionType ( ) const
inline

Convert the Euler angles to quaternion.

## ◆ operator-()

template<typename Scalar_ , class _System >
 EulerAngles Eigen::EulerAngles< Scalar_, _System >::operator- ( ) const
inline
Returns
The Euler angles rotation negative (which is as same as the inverse), (-alpha, -beta, -gamma).

## ◆ operator=() [1/2]

template<typename Scalar_ , class _System >
template<class Derived >
 EulerAngles& Eigen::EulerAngles< Scalar_, _System >::operator= ( const MatrixBase< Derived > & other )
inline

Set *this from either:

• a 3x3 rotation matrix expression(i.e. pure orthogonal matrix with determinant of +1),
• a 3D vector expression representing Euler angles.

## ◆ operator=() [2/2]

template<typename Scalar_ , class _System >
template<typename Derived >
 EulerAngles& Eigen::EulerAngles< Scalar_, _System >::operator= ( const RotationBase< Derived, 3 > & rot )
inline

Set *this from a rotation.