The quaternion class used to represent 3D orientations and rotations. More...

#include <Quaternion.h>

Inheritance diagram for Eigen::Quaternion< _Scalar >:

Public Types
typedef AngleAxis< Scalar >	AngleAxisType
typedef Base::AngleAxisType	AngleAxisType
typedef Matrix< Scalar, 4, 1 >	Coefficients
typedef internal::traits < Quaternion >::Coefficients	Coefficients
typedef Matrix< Scalar, 3, 3 >	Matrix3
typedef _Scalar	Scalar
typedef _Scalar	Scalar
typedef Matrix< Scalar, 3, 1 >	Vector3
Public Member Functions
Scalar	angularDistance (const Quaternion &other) const
template<typename NewScalarType >
internal::cast_return_type < Quaternion, Quaternion < NewScalarType > >::type	cast () const
const Coefficients &	coeffs () const
Coefficients &	coeffs ()
Coefficients &	coeffs ()
const Coefficients &	coeffs () const
Quaternion	conjugate (void) const
Scalar	eigen2_dot (const Quaternion &other) const
Quaternion	inverse (void) const
bool	isApprox (const Quaternion &other, typename NumTraits< Scalar >::Real prec=precision< Scalar >()) const
Scalar	norm () const
void	normalize ()
Quaternion	normalized () const
Quaternion	operator* (const Quaternion &q) const
template<typename Derived >
Vector3	operator* (const MatrixBase< Derived > &vec) const
Quaternion &	operator*= (const Quaternion &q)
Quaternion &	operator= (const Quaternion &other)
Quaternion &	operator= (const AngleAxisType &aa)
template<typename Derived >
Quaternion &	operator= (const MatrixBase< Derived > &m)
	Quaternion ()
	Quaternion (Scalar w, Scalar x, Scalar y, Scalar z)
	Quaternion (const Quaternion &other)
	Quaternion (const AngleAxisType &aa)
template<typename Derived >
	Quaternion (const MatrixBase< Derived > &other)
template<typename OtherScalarType >
	Quaternion (const Quaternion< OtherScalarType > &other)
	Quaternion ()
	Quaternion (const Scalar &w, const Scalar &x, const Scalar &y, const Scalar &z)
	Quaternion (const Scalar *data)
template<class Derived >
EIGEN_STRONG_INLINE	Quaternion (const QuaternionBase< Derived > &other)
	Quaternion (const AngleAxisType &aa)
template<typename Derived >
	Quaternion (const MatrixBase< Derived > &other)
template<typename OtherScalar , int OtherOptions>
	Quaternion (const Quaternion< OtherScalar, OtherOptions > &other)
template<typename Derived1 , typename Derived2 >
Quaternion &	setFromTwoVectors (const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)
Quaternion &	setIdentity ()
Quaternion	slerp (Scalar t, const Quaternion &other) const
Scalar	squaredNorm () const
Matrix3	toRotationMatrix (void) const
const Block< const Coefficients, 3, 1 >	vec () const
Block< Coefficients, 3, 1 >	vec ()
Scalar	w () const
Scalar &	w ()
Scalar	x () const
Scalar &	x ()
Scalar	y () const
Scalar &	y ()
Scalar	z () const
Scalar &	z ()
Static Public Member Functions
template<typename Derived1 , typename Derived2 >
static Quaternion	FromTwoVectors (const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)
static Quaternion	Identity ()
Static Protected Member Functions
static EIGEN_STRONG_INLINE void	_check_template_params ()
Protected Attributes
Coefficients	m_coeffs
Private Types
enum	{ IsAligned = internal::traits<Quaternion>::IsAligned }
typedef RotationBase < Quaternion< _Scalar >, 3 >	Base
typedef QuaternionBase < Quaternion< _Scalar, _Options > >	Base

Detailed Description

template<typename _Scalar>
class Eigen::Quaternion< _Scalar >

The quaternion class used to represent 3D orientations and rotations.

Parameters:

_Scalar the scalar type, i.e., the type of the coefficients

This class represents a quaternion $ w+xi+yj+zk $ that is a convenient representation of orientations and rotations of objects in three dimensions. Compared to other representations like Euler angles or 3x3 matrices, quatertions offer the following advantages:

compact storage (4 scalars)
efficient to compose (28 flops),
stable spherical interpolation

The following two typedefs are provided for convenience:

Quaternionf for float
Quaterniond for double

See also:: class AngleAxis, class Transform

Template Parameters:

_Scalar	the scalar type, i.e., the type of the coefficients
_Options	controls the memory alignment of the coefficients. Can be # AutoAlign or # DontAlign. Default is AutoAlign.

This class represents a quaternion $ w+xi+yj+zk $ that is a convenient representation of orientations and rotations of objects in three dimensions. Compared to other representations like Euler angles or 3x3 matrices, quaternions offer the following advantages:

compact storage (4 scalars)
efficient to compose (28 flops),
stable spherical interpolation

The following two typedefs are provided for convenience:

Quaternionf for float
Quaterniond for double

Warning:: Operations interpreting the quaternion as rotation have undefined behavior if the quaternion is not normalized.

See also:: class AngleAxis, class Transform

Definition at line 47 of file Eigen2Support/Geometry/Quaternion.h.

Member Typedef Documentation

template<typename _Scalar>

typedef AngleAxis<Scalar> Eigen::Quaternion< _Scalar >::AngleAxisType

the equivalent angle-axis type

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 66 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

typedef Base::AngleAxisType Eigen::Quaternion< _Scalar >::AngleAxisType

the equivalent angle-axis type

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 238 of file Geometry/Quaternion.h.

template<typename _Scalar>

typedef RotationBase<Quaternion<_Scalar>,3> Eigen::Quaternion< _Scalar >::Base [private]

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 49 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

typedef QuaternionBase<Quaternion<_Scalar,_Options> > Eigen::Quaternion< _Scalar >::Base [private]

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 228 of file Geometry/Quaternion.h.

template<typename _Scalar>

typedef Matrix<Scalar, 4, 1> Eigen::Quaternion< _Scalar >::Coefficients

the type of the Coefficients 4-vector

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 60 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

typedef internal::traits<Quaternion>::Coefficients Eigen::Quaternion< _Scalar >::Coefficients

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 237 of file Geometry/Quaternion.h.

template<typename _Scalar>

typedef Matrix<Scalar,3,3> Eigen::Quaternion< _Scalar >::Matrix3

the equivalent rotation matrix type

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 64 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

typedef _Scalar Eigen::Quaternion< _Scalar >::Scalar

the scalar type of the coefficients

Reimplemented from Eigen::RotationBase< Quaternion< _Scalar >, 3 >.

Definition at line 57 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

typedef _Scalar Eigen::Quaternion< _Scalar >::Scalar

the scalar type of the coefficients

Reimplemented from Eigen::RotationBase< Quaternion< _Scalar >, 3 >.

Definition at line 232 of file Geometry/Quaternion.h.

template<typename _Scalar>

typedef Matrix<Scalar,3,1> Eigen::Quaternion< _Scalar >::Vector3

the type of a 3D vector

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 62 of file Eigen2Support/Geometry/Quaternion.h.

Member Enumeration Documentation

template<typename _Scalar>

anonymous enum [private]

Enumerator:

IsAligned

Definition at line 229 of file Geometry/Quaternion.h.

Constructor & Destructor Documentation

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( ) [inline]

Default constructor leaving the quaternion uninitialized.

Definition at line 99 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion	(	Scalar	w,
		Scalar	x,
		Scalar	y,
		Scalar	z
	)		`[inline]`

Constructs and initializes the quaternion $ w+xi+yj+zk $ from its four coefficients w, x, y and z.

Warning:: Note the order of the arguments: the real w coefficient first, while internally the coefficients are stored in the following order: [x, y, z, w]

Definition at line 108 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( const Quaternion< _Scalar > & other ) [inline]

Copy constructor

Definition at line 112 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( const AngleAxisType & aa ) [inline, explicit]

Constructs and initializes a quaternion from the angle-axis aa

Definition at line 115 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

template<typename Derived >

Eigen::Quaternion< _Scalar >::Quaternion ( const MatrixBase< Derived > & other ) [inline, explicit]

Constructs and initializes a quaternion from either:

a rotation matrix expression,
a 4D vector expression representing quaternion coefficients.
See also:
operator=(MatrixBase<Derived>)

Definition at line 123 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

template<typename OtherScalarType >

Eigen::Quaternion< _Scalar >::Quaternion ( const Quaternion< OtherScalarType > & other ) [inline, explicit]

Copy constructor with scalar type conversion

Definition at line 192 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( ) [inline]

Default constructor leaving the quaternion uninitialized.

Definition at line 241 of file Geometry/Quaternion.h.

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion	(	const Scalar &	w,
		const Scalar &	x,
		const Scalar &	y,
		const Scalar &	z
	)		`[inline]`

Constructs and initializes the quaternion $ w+xi+yj+zk $ from its four coefficients w, x, y and z.

Warning:: Note the order of the arguments: the real w coefficient first, while internally the coefficients are stored in the following order: [x, y, z, w]

Definition at line 250 of file Geometry/Quaternion.h.

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( const Scalar * data ) [inline]

Constructs and initialize a quaternion from the array data

Definition at line 253 of file Geometry/Quaternion.h.

template<typename _Scalar>

template<class Derived >

EIGEN_STRONG_INLINE Eigen::Quaternion< _Scalar >::Quaternion ( const QuaternionBase< Derived > & other ) [inline]

Copy constructor

Definition at line 256 of file Geometry/Quaternion.h.

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( const AngleAxisType & aa ) [inline, explicit]

Constructs and initializes a quaternion from the angle-axis aa

Definition at line 259 of file Geometry/Quaternion.h.

template<typename _Scalar>

template<typename Derived >

Eigen::Quaternion< _Scalar >::Quaternion ( const MatrixBase< Derived > & other ) [inline, explicit]

Constructs and initializes a quaternion from either:

a rotation matrix expression,
a 4D vector expression representing quaternion coefficients.

Definition at line 266 of file Geometry/Quaternion.h.

template<typename _Scalar>

template<typename OtherScalar , int OtherOptions>

Eigen::Quaternion< _Scalar >::Quaternion ( const Quaternion< OtherScalar, OtherOptions > & other ) [inline, explicit]

Explicit copy constructor with scalar conversion

Definition at line 270 of file Geometry/Quaternion.h.

Member Function Documentation

template<typename _Scalar>

static EIGEN_STRONG_INLINE void Eigen::Quaternion< _Scalar >::_check_template_params ( ) [inline, static, protected]

Definition at line 285 of file Geometry/Quaternion.h.

template<typename Scalar >

Scalar Eigen::Quaternion< Scalar >::angularDistance ( const Quaternion< _Scalar > & other ) const [inline]

Returns:: the angle (in radian) between two rotations

See also:: eigen2_dot()

Definition at line 404 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

template<typename NewScalarType >

internal::cast_return_type<Quaternion,Quaternion<NewScalarType> >::type Eigen::Quaternion< _Scalar >::cast ( ) const [inline]

Returns:: *this with scalar type casted to NewScalarType

Note that if NewScalarType is equal to the current scalar type of *this then this function smartly returns a const reference to *this.

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 187 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

const Coefficients& Eigen::Quaternion< _Scalar >::coeffs ( ) const [inline]

Returns:: a read-only vector expression of the coefficients (x,y,z,w)

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 93 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Coefficients& Eigen::Quaternion< _Scalar >::coeffs ( ) [inline]

Returns:: a vector expression of the coefficients (x,y,z,w)

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 96 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Coefficients& Eigen::Quaternion< _Scalar >::coeffs ( ) [inline]

Returns:: a vector expression of the coefficients (x,y,z,w)

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 276 of file Geometry/Quaternion.h.

template<typename _Scalar>

const Coefficients& Eigen::Quaternion< _Scalar >::coeffs ( ) const [inline]

Returns:: a read-only vector expression of the coefficients (x,y,z,w)

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 277 of file Geometry/Quaternion.h.

template<typename Scalar >

Quaternion< Scalar > Eigen::Quaternion< Scalar >::conjugate ( void ) const [inline]

Returns:: the conjugate of the *this which is equal to the multiplicative inverse if the quaternion is normalized. The conjugate of a quaternion represents the opposite rotation.

See also:: Quaternion::inverse()

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 395 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Scalar Eigen::Quaternion< _Scalar >::eigen2_dot ( const Quaternion< _Scalar > & other ) const [inline]

Returns:: the dot product of *this and other Geometrically speaking, the dot product of two unit quaternions corresponds to the cosine of half the angle between the two rotations.

See also:: angularDistance()

Definition at line 161 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar , int Options>

template<typename Derived1 , typename Derived2 >

Quaternion< Scalar, Options > Eigen::Quaternion< Scalar, Options >::FromTwoVectors	(	const MatrixBase< Derived1 > &	a,
		const MatrixBase< Derived2 > &	b
	)		`[static]`

Returns a quaternion representing a rotation between the two arbitrary vectors a and b. In other words, the built rotation represent a rotation sending the line of direction a to the line of direction b, both lines passing through the origin.

Returns:: resulting quaternion

Note that the two input vectors do not have to be normalized, and do not need to have the same norm.

Definition at line 621 of file Geometry/Quaternion.h.

template<typename _Scalar>

static Quaternion Eigen::Quaternion< _Scalar >::Identity ( ) [inline, static]

Returns:: a quaternion representing an identity rotation

See also:: MatrixBase::Identity()

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 133 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar >

Quaternion< Scalar > Eigen::Quaternion< Scalar >::inverse ( void ) const [inline]

Returns:: the multiplicative inverse of *this Note that in most cases, i.e., if you simply want the opposite rotation, and/or the quaternion is normalized, then it is enough to use the conjugate.

See also:: Quaternion::conjugate()

Reimplemented from Eigen::RotationBase< Quaternion< _Scalar >, 3 >.

Definition at line 375 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

bool Eigen::Quaternion< _Scalar >::isApprox	(	const Quaternion< _Scalar > &	other,
		typename NumTraits< Scalar >::Real	prec = `precision<Scalar>()`
	)		const `[inline]`

Returns:: true if *this is approximately equal to other, within the precision determined by prec.

See also:: MatrixBase::isApprox()

Definition at line 199 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Scalar Eigen::Quaternion< _Scalar >::norm ( ) const [inline]

Returns:: the norm of the quaternion's coefficients

See also:: Quaternion::squaredNorm(), MatrixBase::norm()

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 147 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

void Eigen::Quaternion< _Scalar >::normalize ( void ) [inline]

Normalizes the quaternion *this

See also:: normalized(), MatrixBase::normalize()

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 151 of file Eigen2Support/Geometry/Quaternion.h.

template<typename _Scalar>

Quaternion Eigen::Quaternion< _Scalar >::normalized ( ) const [inline]

Returns:: a normalized version of *this

See also:: normalize(), MatrixBase::normalized()

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 154 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar >

Quaternion< Scalar > Eigen::Quaternion< Scalar >::operator* ( const Quaternion< _Scalar > & other ) const [inline]

Returns:: the concatenation of two rotations as a quaternion-quaternion product

Definition at line 228 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar >

template<typename Derived >

Quaternion< Scalar >::Vector3 Eigen::Quaternion< Scalar >::operator* ( const MatrixBase< Derived > & v ) const [inline]

Rotation of a vector by a quaternion.

Remarks:

If the quaternion is used to rotate several points (>1) then it is much more efficient to first convert it to a 3x3 Matrix. Comparison of the operation cost for n transformations:

Quaternion: 30n
Via a Matrix3: 24 + 15n

Definition at line 250 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar >

Quaternion< Scalar > & Eigen::Quaternion< Scalar >::operator*= ( const Quaternion< _Scalar > & other ) [inline]

See also:: operator*(Quaternion)

Definition at line 235 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar >

Quaternion< Scalar > & Eigen::Quaternion< Scalar >::operator= ( const Quaternion< _Scalar > & other ) [inline]

Definition at line 263 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar >

Quaternion< Scalar > & Eigen::Quaternion< Scalar >::operator= ( const AngleAxisType & aa ) [inline]

Set *this from an angle-axis aa and returns a reference to *this

Reimplemented from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >.

Definition at line 272 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar >

template<typename Derived >

Quaternion< Scalar > & Eigen::Quaternion< Scalar >::operator= ( const MatrixBase< Derived > & xpr ) [inline]

Set *this from the expression xpr:

if xpr is a 4x1 vector, then xpr is assumed to be a quaternion
if xpr is a 3x3 matrix, then xpr is assumed to be rotation matrix and xpr is converted to a quaternion

Definition at line 287 of file Eigen2Support/Geometry/Quaternion.h.

template<typename Scalar >

template<typename Derived1 , typename Derived2 >