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

#include <Quaternion.h>

Inheritance diagram for Eigen::Quaternion< _Scalar >:

Collaboration diagram for Eigen::Quaternion< _Scalar >:

Public Types
typedef _Scalar	Scalar

typedef Matrix< Scalar, 4, 1 >	Coefficients

typedef Matrix< Scalar, 3, 1 >	Vector3

typedef Matrix< Scalar, 3, 3 >	Matrix3

typedef AngleAxis< Scalar >	AngleAxisType

typedef _Scalar	Scalar

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

typedef Base::AngleAxisType	AngleAxisType

Public Types inherited from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >
enum

typedef internal::traits< Quaternion< _Scalar, _Options > >::Scalar	Scalar

typedef NumTraits< Scalar >::Real	RealScalar

typedef internal::traits< Quaternion< _Scalar, _Options > >::Coefficients	Coefficients

typedef Matrix< Scalar, 3, 1 >	Vector3

typedef Matrix< Scalar, 3, 3 >	Matrix3

typedef AngleAxis< Scalar >	AngleAxisType

Public Types inherited from Eigen::RotationBase< Quaternion< _Scalar, _Options >, 3 >
enum

enum

typedef ei_traits< Quaternion< _Scalar, _Options > >::Scalar	Scalar

typedef internal::traits< Quaternion< _Scalar, _Options > >::Scalar	Scalar

typedef Matrix< Scalar, Dim, Dim >	RotationMatrixType

typedef Matrix< Scalar, Dim, Dim >	RotationMatrixType

typedef Matrix< Scalar, Dim, 1 >	VectorType

Public Types inherited from Eigen::RotationBase< Quaternion< _Scalar >, 3 >
enum

enum

typedef ei_traits< Quaternion< _Scalar > >::Scalar	Scalar

typedef internal::traits< Quaternion< _Scalar > >::Scalar	Scalar

typedef Matrix< Scalar, Dim, Dim >	RotationMatrixType

typedef Matrix< Scalar, Dim, Dim >	RotationMatrixType

typedef Matrix< Scalar, Dim, 1 >	VectorType

Public Member Functions
Scalar	x () const

Scalar	y () const

Scalar	z () const

Scalar	w () const

Scalar &	x ()

Scalar &	y ()

Scalar &	z ()

Scalar &	w ()

const Block< const Coefficients, 3, 1 >	vec () const

Block< Coefficients, 3, 1 >	vec ()

const Coefficients &	coeffs () const

Coefficients &	coeffs ()

	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)

Quaternion &	operator= (const Quaternion &other)

Quaternion &	operator= (const AngleAxisType &aa)

template<typename Derived >
Quaternion &	operator= (const MatrixBase< Derived > &m)

Quaternion &	setIdentity ()

Scalar	squaredNorm () const

Scalar	norm () const

void	normalize ()

Quaternion	normalized () const

Scalar	eigen2_dot (const Quaternion &other) const

Scalar	angularDistance (const Quaternion &other) const

Matrix3	toRotationMatrix (void) const

template<typename Derived1 , typename Derived2 >
Quaternion &	setFromTwoVectors (const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)

Quaternion	operator* (const Quaternion &q) const

Quaternion &	operator*= (const Quaternion &q)

Quaternion	inverse (void) const

Quaternion	conjugate (void) const

Quaternion	slerp (Scalar t, const Quaternion &other) const

template<typename Derived >
Vector3	operator* (const MatrixBase< Derived > &vec) const

template<typename NewScalarType >
internal::cast_return_type< Quaternion, Quaternion< NewScalarType > >::type	cast () const

template<typename OtherScalarType >
	Quaternion (const Quaternion< OtherScalarType > &other)

bool	isApprox (const Quaternion &other, typename NumTraits< Scalar >::Real prec=precision< Scalar >()) const

	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)

Coefficients &	coeffs ()

const Coefficients &	coeffs () const

template<typename Derived >
Quaternion< Scalar > &	operator= (const MatrixBase< Derived > &xpr)

template<typename Derived1 , typename Derived2 >
Quaternion< Scalar > &	setFromTwoVectors (const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)

template<typename Derived1 , typename Derived2 >
Quaternion< Scalar, Options >	FromTwoVectors (const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)

Public Member Functions inherited from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >
Scalar	x () const

Scalar &	x ()

Scalar	y () const

Scalar &	y ()

Scalar	z () const

Scalar &	z ()

Scalar	w () const

Scalar &	w ()

const VectorBlock< const Coefficients, 3 >	vec () const

VectorBlock< Coefficients, 3 >	vec ()

const internal::traits< Quaternion< _Scalar, _Options > >::Coefficients &	coeffs () const

internal::traits< Quaternion< _Scalar, _Options > >::Coefficients &	coeffs ()

EIGEN_STRONG_INLINE QuaternionBase< Quaternion< _Scalar, _Options > > &	operator= (const QuaternionBase< Quaternion< _Scalar, _Options > > &other)

EIGEN_STRONG_INLINE Quaternion< _Scalar, _Options > &	operator= (const QuaternionBase< OtherDerived > &other)

Quaternion< _Scalar, _Options > &	operator= (const AngleAxisType &aa)

Quaternion< _Scalar, _Options > &	operator= (const MatrixBase< OtherDerived > &m)

QuaternionBase &	setIdentity ()

Scalar	squaredNorm () const

Scalar	norm () const

void	normalize ()

Quaternion< Scalar >	normalized () const

Scalar	dot (const QuaternionBase< OtherDerived > &other) const

Scalar	angularDistance (const QuaternionBase< OtherDerived > &other) const

Matrix3	toRotationMatrix () const

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

EIGEN_STRONG_INLINE Quaternion< Scalar >	operator* (const QuaternionBase< OtherDerived > &q) const

EIGEN_STRONG_INLINE Quaternion< _Scalar, _Options > &	*operator=** (const QuaternionBase< OtherDerived > &q)

Quaternion< Scalar >	inverse () const

Quaternion< Scalar >	conjugate () const

Quaternion< Scalar >	slerp (const Scalar &t, const QuaternionBase< OtherDerived > &other) const

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

EIGEN_STRONG_INLINE Vector3	_transformVector (Vector3 v) const

internal::cast_return_type< Quaternion< _Scalar, _Options >, Quaternion< NewScalarType > >::type	cast () const

Public Member Functions inherited from Eigen::RotationBase< Quaternion< _Scalar, _Options >, 3 >
const Quaternion< _Scalar, _Options > &	derived () const

Quaternion< _Scalar, _Options > &	derived ()

const Quaternion< _Scalar, _Options > &	derived () const

Quaternion< _Scalar, _Options > &	derived ()

RotationMatrixType	toRotationMatrix () const

RotationMatrixType	toRotationMatrix () const

Quaternion< _Scalar, _Options >	inverse () const

Quaternion< _Scalar, _Options >	inverse () const

Transform< Scalar, Dim >	operator* (const Translation< Scalar, Dim > &t) const

RotationMatrixType	operator* (const Scaling< Scalar, Dim > &s) const

Transform< Scalar, Dim >	operator* (const Transform< Scalar, Dim > &t) const

Transform< Scalar, Dim, Isometry >	operator* (const Translation< Scalar, Dim > &t) const

RotationMatrixType	operator* (const UniformScaling< Scalar > &s) const

EIGEN_STRONG_INLINE 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

RotationMatrixType	matrix () const

VectorType	_transformVector (const OtherVectorType &v) const

Public Member Functions inherited from Eigen::RotationBase< Quaternion< _Scalar >, 3 >
const Quaternion< _Scalar > &	derived () const

Quaternion< _Scalar > &	derived ()

const Quaternion< _Scalar > &	derived () const

Quaternion< _Scalar > &	derived ()

RotationMatrixType	toRotationMatrix () const

RotationMatrixType	toRotationMatrix () const

Quaternion< _Scalar >	inverse () const

Quaternion< _Scalar >	inverse () const

Transform< Scalar, Dim >	operator* (const Translation< Scalar, Dim > &t) const

RotationMatrixType	operator* (const Scaling< Scalar, Dim > &s) const

Transform< Scalar, Dim >	operator* (const Transform< Scalar, Dim > &t) const

Transform< Scalar, Dim, Isometry >	operator* (const Translation< Scalar, Dim > &t) const

RotationMatrixType	operator* (const UniformScaling< Scalar > &s) const

EIGEN_STRONG_INLINE internal::rotation_base_generic_product_selector< Quaternion< _Scalar >, OtherDerived, OtherDerived::IsVectorAtCompileTime >::ReturnType	operator* (const EigenBase< OtherDerived > &e) const

Transform< Scalar, Dim, Mode >	operator* (const Transform< Scalar, Dim, Mode, Options > &t) const

RotationMatrixType	matrix () const

VectorType	_transformVector (const OtherVectorType &v) const

Static Public Member Functions
static Quaternion	Identity ()

template<typename Derived1 , typename Derived2 >
static Quaternion	FromTwoVectors (const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)

Static Public Member Functions inherited from Eigen::QuaternionBase< Quaternion< _Scalar, _Options > >
static Quaternion< Scalar >	Identity ()

Static Protected Member Functions
static EIGEN_STRONG_INLINE void	_check_template_params ()

Protected Attributes
Coefficients	m_coeffs

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 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

Definition at line 47 of file Quaternion.h.

Member Typedef Documentation

template<typename _Scalar>

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

the equivalent angle-axis type

Definition at line 66 of file Quaternion.h.

template<typename _Scalar>

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

the type of the Coefficients 4-vector

Definition at line 60 of file Quaternion.h.

template<typename _Scalar>

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

the equivalent rotation matrix type

Definition at line 64 of file Quaternion.h.

template<typename _Scalar>

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

the scalar type of the coefficients

Definition at line 57 of file Quaternion.h.

template<typename _Scalar>

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

the type of a 3D vector

Definition at line 62 of file 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 Quaternion.h.

99 {}

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 Quaternion.h.

109 { m_coeffs << x, y, z, w; }

Eigen::Quaternion::y

Scalar y() const

Definition: Quaternion.h:71

Eigen::Quaternion::z

Scalar z() const

Definition: Quaternion.h:73

Eigen::Quaternion::x

Scalar x() const

Definition: Quaternion.h:69

Eigen::Quaternion::w

Scalar w() const

Definition: Quaternion.h:75

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( const Quaternion< _Scalar > & other )

inline

Copy constructor

Definition at line 112 of file Quaternion.h.

112 { m_coeffs = other.m_coeffs; }

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( const AngleAxisType & aa )

inlineexplicit

Constructs and initializes a quaternion from the angle-axis aa

Definition at line 115 of file Quaternion.h.

115 { *this = aa; }

template<typename _Scalar>

template<typename Derived >

Eigen::Quaternion< _Scalar >::Quaternion ( const MatrixBase< Derived > & other )

inlineexplicit

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 Quaternion.h.

123 { *this = other; }

template<typename _Scalar>

template<typename OtherScalarType >

Eigen::Quaternion< _Scalar >::Quaternion ( const Quaternion< OtherScalarType > & other )

inlineexplicit

Copy constructor with scalar type conversion

Definition at line 192 of file Quaternion.h.

193 { m_coeffs = other.coeffs().template cast<Scalar>(); }

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( )

inline

Default constructor leaving the quaternion uninitialized.

Definition at line 243 of file Quaternion.h.

243 {}

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 252 of file Quaternion.h.

252 : m_coeffs(x, y, z, w){}

Eigen::Quaternion::y

Scalar y() const

Definition: Quaternion.h:71

Eigen::Quaternion::z

Scalar z() const

Definition: Quaternion.h:73

Eigen::Quaternion::x

Scalar x() const

Definition: Quaternion.h:69

Eigen::Quaternion::w

Scalar w() const

Definition: Quaternion.h:75

template<typename _Scalar>

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

inline

Constructs and initialize a quaternion from the array data

Definition at line 255 of file Quaternion.h.

255 : m_coeffs(data) {}

template<typename _Scalar>

template<class Derived >

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

inline

Copy constructor

Definition at line 258 of file Quaternion.h.

258 { this->Base::operator=(other); }

template<typename _Scalar>

Eigen::Quaternion< _Scalar >::Quaternion ( const AngleAxisType & aa )

inlineexplicit

Constructs and initializes a quaternion from the angle-axis aa

Definition at line 261 of file Quaternion.h.

261 { *this = aa; }

template<typename _Scalar>

template<typename Derived >

Eigen::Quaternion< _Scalar >::Quaternion ( const MatrixBase< Derived > & other )

inlineexplicit

Constructs and initializes a quaternion from either:

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

Definition at line 268 of file Quaternion.h.

268 { *this = other; }

template<typename _Scalar>

template<typename OtherScalar , int OtherOptions>

Eigen::Quaternion< _Scalar >::Quaternion ( const Quaternion< OtherScalar, OtherOptions > & other )

inlineexplicit

Explicit copy constructor with scalar conversion

Definition at line 272 of file Quaternion.h.

273 { m_coeffs = other.coeffs().template cast<Scalar>(); }

Member Function Documentation

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 Quaternion.h.

 {
   double d = ei_abs(this->eigen2_dot(other));
   if (d>=1.0)
     return 0;
   return Scalar(2) * std::acos(d);
 }

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.

Definition at line 187 of file Quaternion.h.

188 { return typename internal::cast_return_type<Quaternion,Quaternion<NewScalarType> >::type(*this); }

template<typename _Scalar>

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

inline

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

Definition at line 93 of file Quaternion.h.

93 { return m_coeffs; }

template<typename _Scalar>

Coefficients& Eigen::Quaternion< _Scalar >::coeffs ( )

inline

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

Definition at line 96 of file Quaternion.h.

96 { return m_coeffs; }

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()

Definition at line 395 of file Quaternion.h.

 {
   return Quaternion(this->w(),-this->x(),-this->y(),-this->z());
 }

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 Quaternion.h.

161 { return m_coeffs.eigen2_dot(other.m_coeffs); }

template<typename _Scalar>

template<typename Derived1 , typename Derived2 >

Quaternion<Scalar,Options> Eigen::Quaternion< _Scalar >::FromTwoVectors	(	const MatrixBase< Derived1 > &	a,
		const MatrixBase< Derived2 > &	b
	)

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 623 of file Quaternion.h.

 {
     Quaternion quat;
     quat.setFromTwoVectors(a, b);
     return quat;
 }

template<typename _Scalar>

static Quaternion Eigen::Quaternion< _Scalar >::Identity ( )

inlinestatic

Returns: a quaternion representing an identity rotation

See also: MatrixBase::Identity()

Definition at line 133 of file Quaternion.h.

133 { return Quaternion(1, 0, 0, 0); }

Eigen::Quaternion::Quaternion

Quaternion()

Definition: Quaternion.h:99

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()

Definition at line 375 of file Quaternion.h.

 {
   // FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite()  ??
   Scalar n2 = this->squaredNorm();
   if (n2 > 0)
     return Quaternion(conjugate().coeffs() / n2);
   else
   {
     // return an invalid result to flag the error
     return Quaternion(Coefficients::Zero());
   }
 }

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 Quaternion.h.

200 { return m_coeffs.isApprox(other.m_coeffs, prec); }

template<typename _Scalar>

Scalar Eigen::Quaternion< _Scalar >::norm ( ) const

inline

Returns: the norm of the quaternion's coefficients

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

Definition at line 147 of file Quaternion.h.

147 { return m_coeffs.norm(); }

template<typename _Scalar>

void Eigen::Quaternion< _Scalar >::normalize ( void )

inline

Normalizes the quaternion *this

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

Definition at line 151 of file Quaternion.h.

151 { m_coeffs.normalize(); }

template<typename _Scalar>

Quaternion Eigen::Quaternion< _Scalar >::normalized ( ) const

inline

Returns: a normalized version of *this

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

Definition at line 154 of file Quaternion.h.

154 { return Quaternion(m_coeffs.normalized()); }

Eigen::Quaternion::Quaternion

Quaternion()

Definition: Quaternion.h:99

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 Quaternion.h.

 {
   return ei_quaternion_product(*this,other);
 }

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 Quaternion.h.

 {
     // Note that this algorithm comes from the optimization by hand
     // of the conversion to a Matrix followed by a Matrix/Vector product.
     // It appears to be much faster than the common algorithm found
     // in the litterature (30 versus 39 flops). It also requires two
     // Vector3 as temporaries.
     Vector3 uv;
     uv = 2 * this->vec().cross(v);
     return v + this->w() * uv + this->vec().cross(uv);
 }

template<typename Scalar >

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

inline

See also: operator*(Quaternion)

Definition at line 235 of file Quaternion.h.

 {
   return (*this = *this * other);
 }

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

Definition at line 272 of file Quaternion.h.

 {
   Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
   this->w() = ei_cos(ha);
   this->vec() = ei_sin(ha) * aa.axis();
   return *this;
 }

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 Quaternion.h.

 {
   ei_quaternion_assign_impl<Derived>::run(*this, xpr.derived());
   return *this;
 }

template<typename _Scalar>

template<typename Derived1 , typename Derived2 >

Quaternion<Scalar>& Eigen::Quaternion< _Scalar >::setFromTwoVectors	(	const MatrixBase< Derived1 > &	a,
		const MatrixBase< Derived2 > &	b
	)

inline

Sets *this to be a quaternion representing a rotation sending the vector a to the vector b.

Returns: a reference to *this.

Note that the two input vectors do not have to be normalized.

Definition at line 338 of file Quaternion.h.

 {
   Vector3 v0 = a.normalized();
   Vector3 v1 = b.normalized();
   Scalar c = v0.eigen2_dot(v1);
 
   // if dot == 1, vectors are the same
   if (ei_isApprox(c,Scalar(1)))
   {
     // set to identity
     this->w() = 1; this->vec().setZero();
     return *this;
   }
   // if dot == -1, vectors are opposites
   if (ei_isApprox(c,Scalar(-1)))
   {
     this->vec() = v0.unitOrthogonal();
     this->w() = 0;
     return *this;
   }
 
   Vector3 axis = v0.cross(v1);
   Scalar s = ei_sqrt((Scalar(1)+c)*Scalar(2));
   Scalar invs = Scalar(1)/s;
   this->vec() = axis * invs;
   this->w() = s * Scalar(0.5);
 
   return *this;
 }

template<typename _Scalar>

Quaternion& Eigen::Quaternion< _Scalar >::setIdentity ( )

inline

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

Definition at line 137 of file Quaternion.h.

137 { m_coeffs << 0, 0, 0, 1; return *this; }

template<typename Scalar >

Quaternion< Scalar > Eigen::Quaternion< Scalar >::slerp	(	Scalar	t,
		const Quaternion< _Scalar > &	other
	)		const

Returns: the spherical linear interpolation between the two quaternions *this and other at the parameter t

Definition at line 416 of file Quaternion.h.

 {
   static const Scalar one = Scalar(1) - machine_epsilon<Scalar>();
   Scalar d = this->eigen2_dot(other);
   Scalar absD = ei_abs(d);
 
   Scalar scale0;
   Scalar scale1;
 
   if (absD>=one)
   {
     scale0 = Scalar(1) - t;
     scale1 = t;
   }
   else
   {
     // theta is the angle between the 2 quaternions
     Scalar theta = std::acos(absD);
     Scalar sinTheta = ei_sin(theta);
 
     scale0 = ei_sin( ( Scalar(1) - t ) * theta) / sinTheta;
     scale1 = ei_sin( ( t * theta) ) / sinTheta;
     if (d<0)
       scale1 = -scale1;
   }
 
   return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
 }

template<typename _Scalar>

Scalar Eigen::Quaternion< _Scalar >::squaredNorm ( ) const

inline

Returns: the squared norm of the quaternion's coefficients

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

Definition at line 142 of file Quaternion.h.

142 { return m_coeffs.squaredNorm(); }

template<typename Scalar >

Quaternion< Scalar >::Matrix3 Eigen::Quaternion< Scalar >::toRotationMatrix ( void ) const

inline

Convert the quaternion to a 3x3 rotation matrix

Definition at line 296 of file Quaternion.h.

 {
   // NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!)
   // if not inlined then the cost of the return by value is huge ~ +35%,
   // however, not inlining this function is an order of magnitude slower, so
   // it has to be inlined, and so the return by value is not an issue
   Matrix3 res;
 
   const Scalar tx  = Scalar(2)*this->x();
   const Scalar ty  = Scalar(2)*this->y();
   const Scalar tz  = Scalar(2)*this->z();
   const Scalar twx = tx*this->w();
   const Scalar twy = ty*this->w();
   const Scalar twz = tz*this->w();
   const Scalar txx = tx*this->x();
   const Scalar txy = ty*this->x();
   const Scalar txz = tz*this->x();
   const Scalar tyy = ty*this->y();
   const Scalar tyz = tz*this->y();
   const Scalar tzz = tz*this->z();
 
   res.coeffRef(0,0) = Scalar(1)-(tyy+tzz);
   res.coeffRef(0,1) = txy-twz;
   res.coeffRef(0,2) = txz+twy;
   res.coeffRef(1,0) = txy+twz;
   res.coeffRef(1,1) = Scalar(1)-(txx+tzz);
   res.coeffRef(1,2) = tyz-twx;
   res.coeffRef(2,0) = txz-twy;
   res.coeffRef(2,1) = tyz+twx;
   res.coeffRef(2,2) = Scalar(1)-(txx+tyy);
 
   return res;
 }

template<typename _Scalar>

const Block<const Coefficients,3,1> Eigen::Quaternion< _Scalar >::vec ( ) const

inline

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

Definition at line 87 of file Quaternion.h.

87 { return m_coeffs.template start<3>(); }

template<typename _Scalar>

Block<Coefficients,3,1> Eigen::Quaternion< _Scalar >::vec ( )

inline

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

Definition at line 90 of file Quaternion.h.

90 { return m_coeffs.template start<3>(); }

template<typename _Scalar>

Scalar Eigen::Quaternion< _Scalar >::w ( ) const

inline

Returns: the w coefficient

Definition at line 75 of file Quaternion.h.

75 { return m_coeffs.coeff(3); }

template<typename _Scalar>

Scalar& Eigen::Quaternion< _Scalar >::w ( )

inline

Returns: a reference to the w coefficient

Definition at line 84 of file Quaternion.h.

84 { return m_coeffs.coeffRef(3); }

template<typename _Scalar>

Scalar Eigen::Quaternion< _Scalar >::x ( ) const

inline

Returns: the x coefficient

Definition at line 69 of file Quaternion.h.

69 { return m_coeffs.coeff(0); }

template<typename _Scalar>

Scalar& Eigen::Quaternion< _Scalar >::x ( )

inline

Returns: a reference to the x coefficient

Definition at line 78 of file Quaternion.h.

78 { return m_coeffs.coeffRef(0); }

template<typename _Scalar>

Scalar Eigen::Quaternion< _Scalar >::y ( ) const

inline

Returns: the y coefficient

Definition at line 71 of file Quaternion.h.

71 { return m_coeffs.coeff(1); }

template<typename _Scalar>

Scalar& Eigen::Quaternion< _Scalar >::y ( )

inline

Returns: a reference to the y coefficient

Definition at line 80 of file Quaternion.h.

80 { return m_coeffs.coeffRef(1); }

template<typename _Scalar>

Scalar Eigen::Quaternion< _Scalar >::z ( ) const

inline

Returns: the z coefficient

Definition at line 73 of file Quaternion.h.

73 { return m_coeffs.coeff(2); }

template<typename _Scalar>

Scalar& Eigen::Quaternion< _Scalar >::z ( )

inline

Returns: a reference to the z coefficient

Definition at line 82 of file Quaternion.h.

82 { return m_coeffs.coeffRef(2); }

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

C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Eigen2Support/Geometry/Quaternion.h

Public Types

Public Member Functions

Static Public Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

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

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

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