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Quaternion< Scalar, Options > Class Template Reference

Detailed Description

template<typename Scalar, int Options = AutoAlign>
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
_Scalarthe scalar type, i.e., the type of the coefficients
_Optionscontrols 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
+ Inheritance diagram for Quaternion< Scalar, Options >:

Public Types

typedef Matrix< Scalar, 3, 3 > Matrix3
 
typedef Matrix< Scalar, Dim, Dim > RotationMatrixType
 
typedef Matrix< Scalar, 3, 1 > Vector3
 

Public Member Functions

Vector3 _transformVector (const Vector3 &v) const
 
internal::cast_return_type
< Quaternion< _Scalar,
_Options >, Quaternion
< NewScalarType > >::type 
cast () const
 
Quaternion< Scalar > conjugate () const
 
Scalar dot (const QuaternionBase< OtherDerived > &other) const
 
template<typename Derived1 , typename Derived2 >
Quaternion< Scalar, Options > FromTwoVectors (const MatrixBase< Derived1 > &a, const MatrixBase< Derived2 > &b)
 
Quaternion< Scalar > inverse () const
 
bool isApprox (const QuaternionBase< OtherDerived > &other, const RealScalar &prec=NumTraits< Scalar >::dummy_precision()) const
 
RotationMatrixType matrix () const
 
Scalar norm () const
 
void normalize ()
 
Quaternion< Scalar > normalized () const
 
Transform< Scalar, Dim, Isometryoperator* (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 (const Scalar &w, const Scalar &x, const Scalar &y, const 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)
 
QuaternionBasesetIdentity ()
 
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

static Quaternion< Scalar > Identity ()
 

Member Typedef Documentation

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 Matrix<Scalar,3,1> Vector3
inherited

the type of a 3D vector

Constructor & Destructor Documentation

Quaternion ( )
inline

Default constructor leaving the quaternion uninitialized.

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]
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)
inlineexplicit

Constructs and initializes a quaternion from the angle-axis aa

Quaternion ( const MatrixBase< Derived > &  other)
inlineexplicit

Constructs and initializes a quaternion from either:

  • a rotation matrix expression,
  • a 4D vector expression representing quaternion coefficients.
Quaternion ( const Quaternion< OtherScalar, OtherOptions > &  other)
inlineexplicit

Explicit copy constructor with scalar conversion

Member Function Documentation

Vector3 _transformVector ( const Vector3 v) const
inlineinherited

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
inlineinherited
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.

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
inlineinherited
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 
)

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.

static Quaternion<Scalar> Identity ( )
inlinestaticinherited
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()
bool isApprox ( const QuaternionBase< OtherDerived > &  other,
const RealScalar &  prec = NumTraits<Scalar>::dummy_precision() 
) const
inlineinherited
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
inlineinherited
Returns
an equivalent rotation matrix This function is added to be conform with the Transform class' naming scheme.
Scalar norm ( ) const
inlineinherited
Returns
the norm of the quaternion's coefficients
See Also
QuaternionBase::squaredNorm(), MatrixBase::norm()

References QuaternionBase< Derived >::coeffs().

void normalize ( void  )
inlineinherited

Normalizes the quaternion *this

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

References QuaternionBase< Derived >::coeffs().

Quaternion<Scalar> normalized ( ) const
inlineinherited
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
inlineinherited
Returns
the concatenation of the rotation *this with a translation t
RotationMatrixType operator* ( const UniformScaling< Scalar > &  s) const
inlineinherited
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
inlineinherited
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
inlineinherited
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 ( )
inlineinherited
See Also
QuaternionBase::Identity(), MatrixBase::setIdentity()

References QuaternionBase< Derived >::coeffs().

Scalar squaredNorm ( ) const
inlineinherited
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.

const VectorBlock<const Coefficients,3> vec ( ) const
inlineinherited
Returns
a read-only vector expression of the imaginary part (x,y,z)

References QuaternionBase< Derived >::coeffs().

VectorBlock<Coefficients,3> vec ( )
inlineinherited
Returns
a vector expression of the imaginary part (x,y,z)

References QuaternionBase< Derived >::coeffs().

Scalar w ( ) const
inlineinherited
Returns
the w coefficient
Scalar& w ( )
inlineinherited
Returns
a reference to the w coefficient
Scalar x ( ) const
inlineinherited
Returns
the x coefficient
Scalar& x ( )
inlineinherited
Returns
a reference to the x coefficient
Scalar y ( ) const
inlineinherited
Returns
the y coefficient
Scalar& y ( )
inlineinherited
Returns
a reference to the y coefficient
Scalar z ( ) const
inlineinherited
Returns
the z coefficient
Scalar& z ( )
inlineinherited
Returns
a reference to the z coefficient

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