Base class for quaternion expressions. More...

#include <Quaternion.h>

Inheritance diagram for Eigen::QuaternionBase< Derived >:

Collaboration diagram for Eigen::QuaternionBase< Derived >:

Public Types
enum	{ Flags = Eigen::internal::traits<Derived>::Flags }

typedef internal::traits< Derived >::Scalar	Scalar

typedef NumTraits< Scalar >::Real	RealScalar

typedef internal::traits< Derived >::Coefficients	Coefficients

typedef Matrix< Scalar, 3, 1 >	Vector3

typedef Matrix< Scalar, 3, 3 >	Matrix3

typedef AngleAxis< Scalar >	AngleAxisType

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

enum

typedef ei_traits< Derived >::Scalar	Scalar

typedef internal::traits< Derived >::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 VectorBlock< const Coefficients, 3 >	vec () const

VectorBlock< Coefficients, 3 >	vec ()

const internal::traits< Derived >::Coefficients &	coeffs () const

internal::traits< Derived >::Coefficients &	coeffs ()

EIGEN_STRONG_INLINE QuaternionBase< Derived > &	operator= (const QuaternionBase< Derived > &other)

template<class OtherDerived >
EIGEN_STRONG_INLINE Derived &	operator= (const QuaternionBase< OtherDerived > &other)

Derived &	operator= (const AngleAxisType &aa)

template<class OtherDerived >
Derived &	operator= (const MatrixBase< OtherDerived > &m)

QuaternionBase &	setIdentity ()

Scalar	squaredNorm () const

Scalar	norm () const

void	normalize ()

Quaternion< Scalar >	normalized () const

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

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

Matrix3	toRotationMatrix () const

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

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

template<class OtherDerived >
EIGEN_STRONG_INLINE Derived &	operator*= (const QuaternionBase< OtherDerived > &q)

Quaternion< Scalar >	inverse () const

Quaternion< Scalar >	conjugate () const

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

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

EIGEN_STRONG_INLINE Vector3	_transformVector (Vector3 v) const

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

template<class OtherDerived >
EIGEN_STRONG_INLINE Quaternion< typename internal::traits< Derived >::Scalar >	operator* (const QuaternionBase< OtherDerived > &other) const

template<class MatrixDerived >
Derived &	operator= (const MatrixBase< MatrixDerived > &xpr)

template<class OtherDerived >
internal::traits< Derived >::Scalar	angularDistance (const QuaternionBase< OtherDerived > &other) const

template<class OtherDerived >
Quaternion< typename internal::traits< Derived >::Scalar >	slerp (const Scalar &t, const QuaternionBase< OtherDerived > &other) const

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

Derived &	derived ()

const Derived &	derived () const

Derived &	derived ()

RotationMatrixType	toRotationMatrix () const

RotationMatrixType	toRotationMatrix () const

Derived	inverse () const

Derived	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< Derived, 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< Scalar >	Identity ()

Detailed Description

template<class Derived>
class Eigen::QuaternionBase< Derived >

Base class for quaternion expressions.

Template Parameters

Derived derived type (CRTP)

See also: class Quaternion

Definition at line 233 of file ForwardDeclarations.h.

Member Typedef Documentation

template<class Derived>

typedef AngleAxis<Scalar> Eigen::QuaternionBase< Derived >::AngleAxisType

the equivalent angle-axis type

Definition at line 55 of file Quaternion.h.

template<class Derived>

typedef Matrix<Scalar,3,3> Eigen::QuaternionBase< Derived >::Matrix3

the equivalent rotation matrix type

Definition at line 53 of file Quaternion.h.

template<class Derived>

typedef Matrix<Scalar,3,1> Eigen::QuaternionBase< Derived >::Vector3

the type of a 3D vector

Definition at line 51 of file Quaternion.h.

Member Function Documentation

template<class Derived >

EIGEN_STRONG_INLINE QuaternionBase< Derived >::Vector3 Eigen::QuaternionBase< Derived >::_transformVector ( Vector3 v ) const

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

Definition at line 466 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 = this->vec().cross(v);
     uv += uv;
     return v + this->w() * uv + this->vec().cross(uv);
 }

template<class Derived>

template<class OtherDerived >

internal::traits<Derived>::Scalar Eigen::QuaternionBase< Derived >::angularDistance ( const QuaternionBase< OtherDerived > & other ) const

inline

Returns: the angle (in radian) between two rotations

See also: dot()

Definition at line 670 of file Quaternion.h.

 {
   using std::acos;
   using std::abs;
   double d = abs(this->dot(other));
   if (d>=1.0)
     return Scalar(0);
   return static_cast<Scalar>(2 * acos(d));
 }

template<class Derived>

template<typename NewScalarType >

internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type Eigen::QuaternionBase< Derived >::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 176 of file Quaternion.h.

   {
     return typename internal::cast_return_type<Derived,Quaternion<NewScalarType> >::type(derived());
   }

template<class Derived>

const internal::traits<Derived>::Coefficients& Eigen::QuaternionBase< Derived >::coeffs ( ) const

inline

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

Definition at line 84 of file Quaternion.h.

84 { return derived().coeffs(); }

template<class Derived>

internal::traits<Derived>::Coefficients& Eigen::QuaternionBase< Derived >::coeffs ( )

inline

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

Definition at line 87 of file Quaternion.h.

87 { return derived().coeffs(); }

template<class Derived >

Quaternion< typename internal::traits< Derived >::Scalar > Eigen::QuaternionBase< Derived >::conjugate ( void ) const

inline

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()

Definition at line 659 of file Quaternion.h.

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

template<class Derived>

template<class OtherDerived >

Scalar Eigen::QuaternionBase< Derived >::dot ( const QuaternionBase< OtherDerived > & 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 133 of file Quaternion.h.

133 { return coeffs().dot(other.coeffs()); }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived>

static Quaternion<Scalar> Eigen::QuaternionBase< Derived >::Identity ( )

inlinestatic

Returns: a quaternion representing an identity rotation

See also: MatrixBase::Identity()

Definition at line 105 of file Quaternion.h.

105 { return Quaternion<Scalar>(1, 0, 0, 0); }

template<class Derived >

Quaternion< typename internal::traits< Derived >::Scalar > Eigen::QuaternionBase< Derived >::inverse ( void ) const

inline

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()

Definition at line 638 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<Scalar>(conjugate().coeffs() / n2);
   else
   {
     // return an invalid result to flag the error
     return Quaternion<Scalar>(Coefficients::Zero());
   }
 }

template<class Derived>

template<class OtherDerived >

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

inline

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

See also: MatrixBase::isApprox()

Definition at line 164 of file Quaternion.h.

165 { return coeffs().isApprox(other.coeffs(), prec); }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived>

Scalar Eigen::QuaternionBase< Derived >::norm ( ) const

inline

Returns: the norm of the quaternion's coefficients

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

Definition at line 119 of file Quaternion.h.

119 { return coeffs().norm(); }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived>

void Eigen::QuaternionBase< Derived >::normalize ( void )

inline

Normalizes the quaternion *this

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

Definition at line 123 of file Quaternion.h.

123 { coeffs().normalize(); }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived>

Quaternion<Scalar> Eigen::QuaternionBase< Derived >::normalized ( ) const

inline

Returns: a normalized copy of *this

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

Definition at line 126 of file Quaternion.h.

126 { return Quaternion<Scalar>(coeffs().normalized()); }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived>

template<class OtherDerived >

EIGEN_STRONG_INLINE Quaternion<typename internal::traits<Derived>::Scalar> Eigen::QuaternionBase< Derived >::operator* ( const QuaternionBase< OtherDerived > & other ) const

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

Definition at line 439 of file Quaternion.h.

 {
   EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename OtherDerived::Scalar>::value),
    YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
   return internal::quat_product<Architecture::Target, Derived, OtherDerived,
                          typename internal::traits<Derived>::Scalar,
                          internal::traits<Derived>::IsAligned && internal::traits<OtherDerived>::IsAligned>::run(*this, other);
 }

template<class Derived >

template<class OtherDerived >

EIGEN_STRONG_INLINE Derived & Eigen::QuaternionBase< Derived >::operator*= ( const QuaternionBase< OtherDerived > & other )

See also: operator*(Quaternion)

Definition at line 451 of file Quaternion.h.

 {
   derived() = derived() * other.derived();
   return derived();
 }

template<class Derived>

EIGEN_STRONG_INLINE Derived & Eigen::QuaternionBase< Derived >::operator= ( const AngleAxisType & aa )

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

Definition at line 496 of file Quaternion.h.

 {
   using std::cos;
   using std::sin;
   Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
   this->w() = cos(ha);
   this->vec() = sin(ha) * aa.axis();
   return derived();
 }

template<class Derived>

template<class MatrixDerived >

Derived& Eigen::QuaternionBase< Derived >::operator= ( const MatrixBase< MatrixDerived > & 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 514 of file Quaternion.h.

 {
   EIGEN_STATIC_ASSERT((internal::is_same<typename Derived::Scalar, typename MatrixDerived::Scalar>::value),
    YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
   internal::quaternionbase_assign_impl<MatrixDerived>::run(*this, xpr.derived());
   return derived();
 }

template<class Derived >

template<typename Derived1 , typename Derived2 >

Derived & Eigen::QuaternionBase< Derived >::setFromTwoVectors	(	const MatrixBase< Derived1 > &	a,
		const MatrixBase< Derived2 > &	b
	)

inline

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.

Definition at line 573 of file Quaternion.h.

 {
   using std::max;
   using std::sqrt;
   Vector3 v0 = a.normalized();
   Vector3 v1 = b.normalized();
   Scalar c = v1.dot(v0);
 
   // if dot == -1, vectors are nearly opposites
   // => accuraletly compute the rotation axis by computing the
   //    intersection of the two planes. This is done by solving:
   //       x^T v0 = 0
   //       x^T v1 = 0
   //    under the constraint:
   //       ||x|| = 1
   //    which yields a singular value problem
   if (c < Scalar(-1)+NumTraits<Scalar>::dummy_precision())
   {
     c = max<Scalar>(c,-1);
     Matrix<Scalar,2,3> m; m << v0.transpose(), v1.transpose();
     JacobiSVD<Matrix<Scalar,2,3> > svd(m, ComputeFullV);
     Vector3 axis = svd.matrixV().col(2);
 
     Scalar w2 = (Scalar(1)+c)*Scalar(0.5);
     this->w() = sqrt(w2);
     this->vec() = axis * sqrt(Scalar(1) - w2);
     return derived();
   }
   Vector3 axis = v0.cross(v1);
   Scalar s = sqrt((Scalar(1)+c)*Scalar(2));
   Scalar invs = Scalar(1)/s;
   this->vec() = axis * invs;
   this->w() = s * Scalar(0.5);
 
   return derived();
 }

template<class Derived>

QuaternionBase& Eigen::QuaternionBase< Derived >::setIdentity ( )

inline

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

Definition at line 109 of file Quaternion.h.

109 { coeffs() << 0, 0, 0, 1; return *this; }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived>

template<class OtherDerived >

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

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

template<class Derived>

template<class OtherDerived >

Quaternion<typename internal::traits<Derived>::Scalar> Eigen::QuaternionBase< Derived >::slerp	(	const Scalar &	t,
		const QuaternionBase< OtherDerived > &	other
	)		const

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

Definition at line 686 of file Quaternion.h.

 {
   using std::acos;
   using std::sin;
   using std::abs;
   static const Scalar one = Scalar(1) - NumTraits<Scalar>::epsilon();
   Scalar d = this->dot(other);
   Scalar absD = 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 = acos(absD);
     Scalar sinTheta = sin(theta);
 
     scale0 = sin( ( Scalar(1) - t ) * theta) / sinTheta;
     scale1 = sin( ( t * theta) ) / sinTheta;
   }
   if(d<0) scale1 = -scale1;
 
   return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
 }

template<class Derived>

Scalar Eigen::QuaternionBase< Derived >::squaredNorm ( ) const

inline

Returns: the squared norm of the quaternion's coefficients

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

Definition at line 114 of file Quaternion.h.

114 { return coeffs().squaredNorm(); }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived >

QuaternionBase< Derived >::Matrix3 Eigen::QuaternionBase< Derived >::toRotationMatrix ( void ) const

inline

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.

Definition at line 527 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<class Derived>

const VectorBlock<const Coefficients,3> Eigen::QuaternionBase< Derived >::vec ( ) const

inline

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

Definition at line 78 of file Quaternion.h.

78 { return coeffs().template head<3>(); }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived>

VectorBlock<Coefficients,3> Eigen::QuaternionBase< Derived >::vec ( )

inline

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

Definition at line 81 of file Quaternion.h.

81 { return coeffs().template head<3>(); }

Eigen::QuaternionBase::coeffs

const internal::traits< Derived >::Coefficients & coeffs() const

Definition: Quaternion.h:84

template<class Derived>

Scalar Eigen::QuaternionBase< Derived >::w ( ) const

inline

Returns: the w coefficient

Definition at line 66 of file Quaternion.h.

66 { return this->derived().coeffs().coeff(3); }

template<class Derived>

Scalar& Eigen::QuaternionBase< Derived >::w ( )

inline

Returns: a reference to the w coefficient

Definition at line 75 of file Quaternion.h.

75 { return this->derived().coeffs().coeffRef(3); }

template<class Derived>

Scalar Eigen::QuaternionBase< Derived >::x ( ) const

inline

Returns: the x coefficient

Definition at line 60 of file Quaternion.h.

60 { return this->derived().coeffs().coeff(0); }

template<class Derived>

Scalar& Eigen::QuaternionBase< Derived >::x ( )

inline

Returns: a reference to the x coefficient

Definition at line 69 of file Quaternion.h.

69 { return this->derived().coeffs().coeffRef(0); }

template<class Derived>

Scalar Eigen::QuaternionBase< Derived >::y ( ) const

inline

Returns: the y coefficient

Definition at line 62 of file Quaternion.h.

62 { return this->derived().coeffs().coeff(1); }

template<class Derived>

Scalar& Eigen::QuaternionBase< Derived >::y ( )

inline

Returns: a reference to the y coefficient

Definition at line 71 of file Quaternion.h.

71 { return this->derived().coeffs().coeffRef(1); }

template<class Derived>

Scalar Eigen::QuaternionBase< Derived >::z ( ) const

inline

Returns: the z coefficient

Definition at line 64 of file Quaternion.h.

64 { return this->derived().coeffs().coeff(2); }

template<class Derived>

Scalar& Eigen::QuaternionBase< Derived >::z ( )

inline

Returns: a reference to the z coefficient

Definition at line 73 of file Quaternion.h.

73 { return this->derived().coeffs().coeffRef(2); }

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

C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Core/util/ForwardDeclarations.h
C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Geometry/Quaternion.h

Public Types

Public Member Functions

Static Public Member Functions

Detailed Description

template<class Derived> class Eigen::QuaternionBase< Derived >

Member Typedef Documentation

Member Function Documentation

template<class Derived>
class Eigen::QuaternionBase< Derived >