Modules
	Global aligned box typedefs

Classes
class	Eigen::Map< const Quaternion< _Scalar >, _Options >
	Quaternion expression mapping a constant memory buffer. More...

class	Eigen::Map< Quaternion< _Scalar >, _Options >
	Expression of a quaternion from a memory buffer. More...

class	Eigen::AlignedBox< _Scalar, _AmbientDim >
	An axis aligned box. More...

class	Eigen::AngleAxis< _Scalar >
	Represents a 3D rotation as a rotation angle around an arbitrary 3D axis. More...

class	Eigen::Homogeneous< MatrixType, _Direction >
	Expression of one (or a set of) homogeneous vector(s) More...

class	Eigen::Hyperplane< _Scalar, _AmbientDim, _Options >
	A hyperplane. More...

class	Eigen::ParametrizedLine< _Scalar, _AmbientDim, _Options >
	A parametrized line. More...

class	Eigen::QuaternionBase< Derived >
	Base class for quaternion expressions. More...

class	Eigen::Quaternion< _Scalar, _Options >
	The quaternion class used to represent 3D orientations and rotations. More...

class	Eigen::Rotation2D< _Scalar >
	Represents a rotation/orientation in a 2 dimensional space. More...

class	Eigen::UniformScaling< _Scalar >
	Represents a generic uniform scaling transformation. More...

class	Eigen::Transform< _Scalar, _Dim, _Mode, _Options >
	Represents an homogeneous transformation in a N dimensional space. More...

class	Eigen::Translation< _Scalar, _Dim >
	Represents a translation transformation. More...

Typedefs
typedef AngleAxis< float >	Eigen::AngleAxisf

typedef AngleAxis< double >	Eigen::AngleAxisd

typedef Quaternion< float >	Eigen::Quaternionf

typedef Quaternion< double >	Eigen::Quaterniond

typedef Map< Quaternion< float >, 0 >	Eigen::QuaternionMapf

typedef Map< Quaternion< double >, 0 >	Eigen::QuaternionMapd

typedef Map< Quaternion< float >, Aligned >	Eigen::QuaternionMapAlignedf

typedef Map< Quaternion< double >, Aligned >	Eigen::QuaternionMapAlignedd

typedef Rotation2D< float >	Eigen::Rotation2Df

typedef Rotation2D< double >	Eigen::Rotation2Dd

typedef Transform< float, 2, Isometry >	Eigen::Isometry2f

typedef Transform< float, 3, Isometry >	Eigen::Isometry3f

typedef Transform< double, 2, Isometry >	Eigen::Isometry2d

typedef Transform< double, 3, Isometry >	Eigen::Isometry3d

typedef Transform< float, 2, Affine >	Eigen::Affine2f

typedef Transform< float, 3, Affine >	Eigen::Affine3f

typedef Transform< double, 2, Affine >	Eigen::Affine2d

typedef Transform< double, 3, Affine >	Eigen::Affine3d

typedef Transform< float, 2, AffineCompact >	Eigen::AffineCompact2f

typedef Transform< float, 3, AffineCompact >	Eigen::AffineCompact3f

typedef Transform< double, 2, AffineCompact >	Eigen::AffineCompact2d

typedef Transform< double, 3, AffineCompact >	Eigen::AffineCompact3d

typedef Transform< float, 2, Projective >	Eigen::Projective2f

typedef Transform< float, 3, Projective >	Eigen::Projective3f

typedef Transform< double, 2, Projective >	Eigen::Projective2d

typedef Transform< double, 3, Projective >	Eigen::Projective3d

Functions
template<typename Derived , typename OtherDerived >
internal::umeyama_transform_matrix_type< Derived, OtherDerived >::type	Eigen::umeyama (const MatrixBase< Derived > &src, const MatrixBase< OtherDerived > &dst, bool with_scaling=true)
	Returns the transformation between two point sets. More...

EIGEN_DEVICE_FUNC Matrix< Scalar, 3, 1 >	Eigen::MatrixBase< Derived >::eulerAngles (Index a0, Index a1, Index a2) const

EIGEN_DEVICE_FUNC HomogeneousReturnType	Eigen::MatrixBase< Derived >::homogeneous () const

EIGEN_DEVICE_FUNC HomogeneousReturnType	Eigen::VectorwiseOp< ExpressionType, Direction >::homogeneous () const

EIGEN_DEVICE_FUNC const HNormalizedReturnType	Eigen::MatrixBase< Derived >::hnormalized () const
	homogeneous normalization More...

EIGEN_DEVICE_FUNC const HNormalizedReturnType	Eigen::VectorwiseOp< ExpressionType, Direction >::hnormalized () const
	column or row-wise homogeneous normalization More...

template<typename OtherDerived >
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE MatrixBase< Derived >::template cross_product_return_type< OtherDerived >::type	Eigen::MatrixBase< Derived >::cross (const MatrixBase< OtherDerived > &other) const

template<typename OtherDerived >
EIGEN_DEVICE_FUNC PlainObject	Eigen::MatrixBase< Derived >::cross3 (const MatrixBase< OtherDerived > &other) const

template<typename OtherDerived >
EIGEN_DEVICE_FUNC const CrossReturnType	Eigen::VectorwiseOp< ExpressionType, Direction >::cross (const MatrixBase< OtherDerived > &other) const

EIGEN_DEVICE_FUNC PlainObject	Eigen::MatrixBase< Derived >::unitOrthogonal (void) const

Detailed Description

Typedef Documentation

◆ AngleAxisd

typedef AngleAxis<double> Eigen::AngleAxisd

double precision angle-axis type

◆ AngleAxisf

typedef AngleAxis<float> Eigen::AngleAxisf

single precision angle-axis type

◆ Quaterniond

typedef Quaternion<double> Eigen::Quaterniond

double precision quaternion type

◆ Quaternionf

typedef Quaternion<float> Eigen::Quaternionf

single precision quaternion type

◆ QuaternionMapAlignedd

typedef Map<Quaternion<double>, Aligned> Eigen::QuaternionMapAlignedd

Map a 16-byte aligned array of double precision scalars as a quaternion

◆ QuaternionMapAlignedf

typedef Map<Quaternion<float>, Aligned> Eigen::QuaternionMapAlignedf

Map a 16-byte aligned array of single precision scalars as a quaternion

◆ QuaternionMapd

typedef Map<Quaternion<double>, 0> Eigen::QuaternionMapd

Map an unaligned array of double precision scalars as a quaternion

◆ QuaternionMapf

typedef Map<Quaternion<float>, 0> Eigen::QuaternionMapf

Map an unaligned array of single precision scalars as a quaternion

◆ Rotation2Dd

typedef Rotation2D<double> Eigen::Rotation2Dd

double precision 2D rotation type

◆ Rotation2Df

typedef Rotation2D<float> Eigen::Rotation2Df

single precision 2D rotation type

Function Documentation

◆ cross() [1/2]

template<typename ExpressionType , int Direction>

template<typename OtherDerived >

EIGEN_DEVICE_FUNC const VectorwiseOp< ExpressionType, Direction >::CrossReturnType Eigen::VectorwiseOp< ExpressionType, Direction >::cross ( const MatrixBase< OtherDerived > & other ) const

\geometry_module

Returns: a matrix expression of the cross product of each column or row of the referenced expression with the other vector.

The referenced matrix must have one dimension equal to 3. The result matrix has the same dimensions than the referenced one.

See also: MatrixBase::cross()

◆ cross() [2/2]

template<typename Derived >

template<typename OtherDerived >

EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE MatrixBase<Derived>::template cross_product_return_type<OtherDerived>::type Eigen::MatrixBase< Derived >::cross ( const MatrixBase< OtherDerived > & other ) const

\geometry_module

Returns: the cross product of *this and other

Here is a very good explanation of cross-product: http://xkcd.com/199/

With complex numbers, the cross product is implemented as $(\mathbf{a}+i\mathbf{b}) \times (\mathbf{c}+i\mathbf{d}) = (\mathbf{a} \times \mathbf{c} - \mathbf{b} \times \mathbf{d}) - i(\mathbf{a} \times \mathbf{d} - \mathbf{b} \times \mathbf{c})$

See also: MatrixBase::cross3()

◆ cross3()

template<typename Derived >

template<typename OtherDerived >

EIGEN_DEVICE_FUNC MatrixBase< Derived >::PlainObject Eigen::MatrixBase< Derived >::cross3 ( const MatrixBase< OtherDerived > & other ) const

inline

\geometry_module

Returns: the cross product of *this and other using only the x, y, and z coefficients

The size of *this and other must be four. This function is especially useful when using 4D vectors instead of 3D ones to get advantage of SSE/AltiVec vectorization.

See also: MatrixBase::cross()

◆ eulerAngles()

template<typename Derived >

EIGEN_DEVICE_FUNC Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > Eigen::MatrixBase< Derived >::eulerAngles	(	Index	a0,
		Index	a1,
		Index	a2
	)		const

inline

\geometry_module

Returns: the Euler-angles of the rotation matrix *this using the convention defined by the triplet (a0,a1,a2)

Each of the three parameters a0,a1,a2 represents the respective rotation axis as an integer in {0,1,2}. For instance, in:

Vector3f ea = mat.eulerAngles(2, 0, 2);

"2" represents the z axis and "0" the x axis, etc. The returned angles are such that we have the following equality:

mat == AngleAxisf(ea[0], Vector3f::UnitZ())
     * AngleAxisf(ea[1], Vector3f::UnitX())
     * AngleAxisf(ea[2], Vector3f::UnitZ()); 

This corresponds to the right-multiply conventions (with right hand side frames).

The returned angles are in the ranges [0:pi]x[-pi:pi]x[-pi:pi].

See also: class AngleAxis

◆ hnormalized() [1/2]

template<typename Derived >

EIGEN_DEVICE_FUNC const MatrixBase< Derived >::HNormalizedReturnType Eigen::MatrixBase< Derived >::hnormalized

inline

homogeneous normalization

\geometry_module

Returns: a vector expression of the N-1 first coefficients of *this divided by that last coefficient.

This can be used to convert homogeneous coordinates to affine coordinates.

It is essentially a shortcut for:

this->head(this->size()-1)/this->coeff(this->size()-1);

Example:

Output:

See also: VectorwiseOp::hnormalized()

◆ hnormalized() [2/2]

template<typename ExpressionType , int Direction>

EIGEN_DEVICE_FUNC const VectorwiseOp< ExpressionType, Direction >::HNormalizedReturnType Eigen::VectorwiseOp< ExpressionType, Direction >::hnormalized

inline

column or row-wise homogeneous normalization

\geometry_module

Returns: an expression of the first N-1 coefficients of each column (or row) of *this divided by the last coefficient of each column (or row).

This can be used to convert homogeneous coordinates to affine coordinates.

It is conceptually equivalent to calling MatrixBase::hnormalized() to each column (or row) of *this.

Example:

Output:

See also: MatrixBase::hnormalized()

◆ homogeneous() [1/2]

template<typename Derived >

EIGEN_DEVICE_FUNC MatrixBase< Derived >::HomogeneousReturnType Eigen::MatrixBase< Derived >::homogeneous

inline

\geometry_module

Returns: a vector expression that is one longer than the vector argument, with the value 1 symbolically appended as the last coefficient.

This can be used to convert affine coordinates to homogeneous coordinates.

\only_for_vectors

Example:

Output:

See also: VectorwiseOp::homogeneous(), class Homogeneous

◆ homogeneous() [2/2]

template<typename ExpressionType , int Direction>

EIGEN_DEVICE_FUNC Homogeneous< ExpressionType, Direction > Eigen::VectorwiseOp< ExpressionType, Direction >::homogeneous

inline

\geometry_module

Returns: an expression where the value 1 is symbolically appended as the final coefficient to each column (or row) of the matrix.

This can be used to convert affine coordinates to homogeneous coordinates.

Example:

Output:

See also: MatrixBase::homogeneous(), class Homogeneous

◆ umeyama()

template<typename Derived , typename OtherDerived >

internal::umeyama_transform_matrix_type<Derived, OtherDerived>::type Eigen::umeyama	(	const MatrixBase< Derived > &	src,
		const MatrixBase< OtherDerived > &	dst,
		bool	with_scaling = `true`
	)

Returns the transformation between two point sets.

\geometry_module

The algorithm is based on: "Least-squares estimation of transformation parameters between two point patterns", Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573

It estimates parameters $c, \mathbf{R},$ and $\mathbf{t}$ such that

$\begin{align*} \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2 \end{align*}$

is minimized.

The algorithm is based on the analysis of the covariance matrix $\Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d}$ of the input point sets $\mathbf{x}$ and $\mathbf{y}$ where $d$ is corresponding to the dimension (which is typically small). The analysis is involving the SVD having a complexity of $O(d^3)$ though the actual computational effort lies in the covariance matrix computation which has an asymptotic lower bound of $O(dm)$ when the input point sets have dimension $d \times m$ .

Currently the method is working only for floating point matrices.

Todo:: Should the return type of umeyama() become a Transform?

Parameters

src	Source points $\mathbf{x} = \left( x_1, \hdots, x_n \right)$ .
dst	Destination points $\mathbf{y} = \left( y_1, \hdots, y_n \right)$ .
with_scaling	Sets when `false` is passed.

Returns: The homogeneous transformation
$\begin{align*} T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} \end{align*}$
minimizing the residual above. This transformation is always returned as an Eigen::Matrix.

◆ unitOrthogonal()

template<typename Derived >

EIGEN_DEVICE_FUNC MatrixBase< Derived >::PlainObject Eigen::MatrixBase< Derived >::unitOrthogonal ( void ) const

inline

\geometry_module

Returns: a unit vector which is orthogonal to *this

The size of *this must be at least 2. If the size is exactly 2, then the returned vector is a counter clock wise rotation of *this, i.e., (-y,x).normalized().

See also: cross()

Modules

Classes

Typedefs

Functions

Detailed Description

Typedef Documentation

◆ AngleAxisd

◆ AngleAxisf

◆ Quaterniond

◆ Quaternionf

◆ QuaternionMapAlignedd

◆ QuaternionMapAlignedf

◆ QuaternionMapd

◆ QuaternionMapf

◆ Rotation2Dd

◆ Rotation2Df

Function Documentation

◆ cross() [1/2]

◆ cross() [2/2]

◆ cross3()

◆ eulerAngles()

◆ hnormalized() [1/2]

◆ hnormalized() [2/2]

◆ homogeneous() [1/2]

◆ homogeneous() [2/2]

◆ umeyama()

◆ unitOrthogonal()