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

#include <Quaternion.h>

Inheritance diagram for Quaternion< _Scalar, _Options >:

Public Types
typedef Base::AngleAxisType	AngleAxisType
typedef Matrix< Scalar, 3, 3 >	Matrix3
typedef Matrix< Scalar, Dim, Dim >	RotationMatrixType
typedef _Scalar	Scalar
typedef Matrix< Scalar, 3, 1 >	Vector3
Public Member Functions
Vector3	_transformVector (Vector3 v) const
internal::cast_return_type < Quaternion< _Scalar, _Options >, Quaternion < NewScalarType > >::type	cast () const
Coefficients &	coeffs ()
const Coefficients &	coeffs () const
Quaternion< Scalar >	conjugate () const
Scalar	dot (const QuaternionBase< OtherDerived > &other) const
Quaternion< Scalar >	inverse () const
bool	isApprox (const QuaternionBase< OtherDerived > &other, RealScalar prec=NumTraits< Scalar >::dummy_precision()) const
RotationMatrixType	matrix () const
Scalar	norm () const
void	normalize ()
Quaternion< Scalar >	normalized () const
Transform< Scalar, Dim, Isometry >	operator* (const Translation< Scalar, Dim > &t) const
RotationMatrixType	operator* (const UniformScaling< Scalar > &s) const
internal::rotation_base_generic_product_selector < Quaternion< _Scalar, _Options >, OtherDerived, OtherDerived::IsVectorAtCompileTime > ::ReturnType	operator* (const EigenBase< OtherDerived > &e) const
Transform< Scalar, Dim, Mode >	operator* (const Transform< Scalar, Dim, Mode, Options > &t) const
	Quaternion ()
	Quaternion (Scalar w, Scalar x, Scalar y, Scalar z)
	Quaternion (const Scalar *data)
template<class Derived >
	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)
Quaternion< _Scalar, _Options > &	setFromTwoVectors (const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)
QuaternionBase &	setIdentity ()
Quaternion< Scalar >	slerp (Scalar t, const QuaternionBase< OtherDerived > &other) const
Scalar	squaredNorm () const
Matrix3	toRotationMatrix () const
const VectorBlock< const Coefficients, 3 >	vec () const
VectorBlock< Coefficients, 3 >	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< Scalar >	Identity ()
Friends
RotationMatrixType	operator* (const EigenBase< OtherDerived > &l, const Quaternion< _Scalar, _Options > &r)
Transform< Scalar, Dim, Affine >	operator* (const DiagonalMatrix< Scalar, Dim > &l, const Quaternion< _Scalar, _Options > &r)

Detailed Description

template<typename _Scalar, int _Options>
class Eigen::Quaternion< _Scalar, _Options >

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

This is defined in the Geometry module.

 #include <Eigen/Geometry>

Template Parameters:

_Scalar	the scalar type, i.e., the type of the coefficients
_Options	controls the memory alignement of the coeffecients. 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, 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

Member Typedef Documentation

typedef Base::AngleAxisType AngleAxisType

the equivalent angle-axis type

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

typedef Matrix<Scalar,3,3> Matrix3 [inherited]

the equivalent rotation matrix type

typedef Matrix<Scalar,Dim,Dim> RotationMatrixType [inherited]

corresponding linear transformation matrix type

typedef _Scalar Scalar

the scalar type of the coefficients

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

typedef Matrix<Scalar,3,1> Vector3 [inherited]

the type of a 3D vector

Constructor & Destructor Documentation

Quaternion ( ) [inline]

Default constructor leaving the quaternion uninitialized.

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]

Quaternion ( const Scalar * data ) [inline]

Constructs and initialize a quaternion from the array data

Quaternion ( const QuaternionBase< Derived > & other ) [inline]

Copy constructor

Quaternion ( const AngleAxisType & aa ) [inline, explicit]

Constructs and initializes a quaternion from the angle-axis aa

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.

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

Explicit copy constructor with scalar conversion

References Quaternion< _Scalar, _Options >::coeffs().

Member Function Documentation

Vector3 _transformVector ( Vector3 v ) const [inline, inherited]

return the result vector of v through the rotation

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:

Quaternion2: 30n
Via a Matrix3: 24 + 15n

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

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.

Coefficients& coeffs ( ) [inline]

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

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

Referenced by Quaternion< _Scalar, _Options >::Quaternion().

const Coefficients& coeffs ( ) const [inline]

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

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

Quaternion<Scalar> conjugate ( ) const [inherited]

Returns:: the conjugated quaternion; 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:: Quaternion2::inverse()

Scalar dot ( const QuaternionBase< OtherDerived > & other ) const [inline, inherited]

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()

References QuaternionBase< Derived >::coeffs().

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.

References QuaternionBase< Derived >::setFromTwoVectors().

static Quaternion<Scalar> Identity ( ) [inline, static, inherited]

Returns:: a quaternion representing an identity rotation

See also:: MatrixBase::Identity()

Quaternion<Scalar> inverse ( ) const [inherited]

Returns:: the quaternion describing the inverse rotation; 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:: QuaternionBase::conjugate()

Reimplemented from RotationBase< Quaternion< _Scalar, _Options >, 3 >.

bool isApprox	(	const QuaternionBase< OtherDerived > &	other,
		RealScalar	prec = `NumTraits<Scalar>::dummy_precision()`
	)		const `[inline, inherited]`

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

See also:: MatrixBase::isApprox()

References QuaternionBase< Derived >::coeffs().

RotationMatrixType matrix ( ) const [inline, inherited]

Returns:: an equivalent rotation matrix This function is added to be conform with the Transform class' naming scheme.

Scalar norm ( ) const [inline, inherited]

Returns:: the norm of the quaternion's coefficients

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

References QuaternionBase< Derived >::coeffs().

void normalize ( void ) [inline, inherited]

Normalizes the quaternion *this

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

References QuaternionBase< Derived >::coeffs().

Quaternion<Scalar> normalized ( ) const [inline, inherited]

Returns:: a normalized copy of *this

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

References QuaternionBase< Derived >::coeffs().

Transform<Scalar,Dim,Isometry> operator* ( const Translation< Scalar, Dim > & t ) const [inline, inherited]

Returns:: the concatenation of the rotation *this with a translation t

RotationMatrixType operator* ( const UniformScaling< Scalar > & s ) const [inline, inherited]

Returns:: the concatenation of the rotation *this with a uniform scaling s

internal::rotation_base_generic_product_selector<Quaternion< _Scalar, _Options > ,OtherDerived,OtherDerived::IsVectorAtCompileTime>::ReturnType operator* ( const EigenBase< OtherDerived > & e ) const [inline, inherited]

Returns:

the concatenation of the rotation *this with a generic expression e e can be:

a DimxDim linear transformation matrix
a DimxDim diagonal matrix (axis aligned scaling)
a vector of size Dim

Transform<Scalar,Dim,Mode> operator* ( const Transform< Scalar, Dim, Mode, Options > & t ) const [inline, inherited]

Returns:: the concatenation of the rotation *this with a transformation t

Quaternion< _Scalar, _Options > & setFromTwoVectors	(	const MatrixBase< Derived1 > &	a,
		const MatrixBase< Derived2 > &	b
	)		`[inherited]`

Returns:: the quaternion which transform a into b through a rotation

Sets *this to be 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:: a reference to *this.

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

QuaternionBase& setIdentity ( ) [inline, inherited]

See also:: QuaternionBase::Identity(), MatrixBase::setIdentity()

References QuaternionBase< Derived >::coeffs().

Quaternion<Scalar> slerp	(	Scalar	t,
		const QuaternionBase< OtherDerived > &	other
	)		const `[inherited]`

Returns:: an interpolation for a constant motion between other and *this t in [0;1] see http://en.wikipedia.org/wiki/Slerp; the spherical linear interpolation between the two quaternions *this and other at the parameter t

Scalar squaredNorm ( ) const [inline, inherited]

Returns:: the squared norm of the quaternion's coefficients

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

References QuaternionBase< Derived >::coeffs().

Matrix3 toRotationMatrix ( void ) const [inherited]

Returns:: an equivalent 3x3 rotation matrix

Convert the quaternion to a 3x3 rotation matrix. The quaternion is required to be normalized, otherwise the result is undefined.

Reimplemented from RotationBase< Quaternion< _Scalar, _Options >, 3 >.

const VectorBlock<const Coefficients,3> vec ( ) const [inline, inherited]

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

References QuaternionBase< Derived >::coeffs().

VectorBlock<Coefficients,3> vec ( ) [inline, inherited]

Returns:: a vector expression of the imaginary part (x,y,z)

References QuaternionBase< Derived >::coeffs().

Scalar w ( ) const [inline, inherited]

Returns:: the w coefficient

Scalar& w ( ) [inline, inherited]

Returns:: a reference to the w coefficient

Scalar x ( ) const [inline, inherited]

Returns:: the x coefficient

Scalar& x ( ) [inline, inherited]

Returns:: a reference to the x coefficient

Scalar y ( ) const [inline, inherited]

Returns:: the y coefficient

Scalar& y ( ) [inline, inherited]

Returns:: a reference to the y coefficient

Scalar z ( ) const [inline, inherited]

Returns:: the z coefficient

Scalar& z ( ) [inline, inherited]

Returns:: a reference to the z coefficient

Friends And Related Function Documentation

RotationMatrixType operator*	(	const EigenBase< OtherDerived > &	l,
		const Quaternion< _Scalar, _Options > &	r
	)		`[friend, inherited]`

Returns:: the concatenation of a linear transformation l with the rotation r

Transform<Scalar,Dim,Affine> operator*	(	const DiagonalMatrix< Scalar, Dim > &	l,
		const Quaternion< _Scalar, _Options > &	r
	)		`[friend, inherited]`

Returns:: the concatenation of a scaling l with the rotation r

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

Quaternion.h

Public Types

Public Member Functions

Static Public Member Functions

Friends

Detailed Description

template<typename _Scalar, int _Options> class Eigen::Quaternion< _Scalar, _Options >

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

template<typename _Scalar, int _Options>
class Eigen::Quaternion< _Scalar, _Options >