Eigen  3.2.1
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Geometry module

Detailed Description

This module provides support for:

* #include <Eigen/Geometry>


 Global aligned box typedefs


class  AlignedBox< _Scalar, _AmbientDim >
 An axis aligned box. More...
class  AngleAxis< Scalar >
 Represents a 3D rotation as a rotation angle around an arbitrary 3D axis. More...
class  Homogeneous< MatrixType, Direction >
 Expression of one (or a set of) homogeneous vector(s) More...
class  Hyperplane< _Scalar, _AmbientDim, Options >
 A hyperplane. More...
class  Map< const Quaternion< _Scalar >, _Options >
 Quaternion expression mapping a constant memory buffer. More...
class  Map< Quaternion< _Scalar >, _Options >
 Expression of a quaternion from a memory buffer. More...
class  ParametrizedLine< _Scalar, _AmbientDim, Options >
 A parametrized line. More...
class  Quaternion< Scalar, Options >
 The quaternion class used to represent 3D orientations and rotations. More...
class  QuaternionBase< Derived >
 Base class for quaternion expressions. More...
class  Rotation2D< Scalar >
 Represents a rotation/orientation in a 2 dimensional space. More...
class  Scaling
 Represents a generic uniform scaling transformation. More...
class  Transform< Scalar, Dim, Mode, _Options >
 Represents an homogeneous transformation in a N dimensional space. More...
class  Translation< Scalar, Dim >
 Represents a translation transformation. More...


typedef Transform< double,
2, Affine > 
typedef Transform< float,
2, Affine > 
typedef Transform< double,
3, Affine > 
typedef Transform< float,
3, Affine > 
typedef Transform< double,
2, AffineCompact > 
typedef Transform< float,
2, AffineCompact > 
typedef Transform< double,
3, AffineCompact > 
typedef Transform< float,
3, AffineCompact > 
typedef DiagonalMatrix< double, 2 > AlignedScaling2d
typedef DiagonalMatrix< float, 2 > AlignedScaling2f
typedef DiagonalMatrix< double, 3 > AlignedScaling3d
typedef DiagonalMatrix< float, 3 > AlignedScaling3f
typedef AngleAxis< double > AngleAxisd
typedef AngleAxis< float > AngleAxisf
typedef Transform< double,
2, Isometry > 
typedef Transform< float,
2, Isometry > 
typedef Transform< double,
3, Isometry > 
typedef Transform< float,
3, Isometry > 
typedef Transform< double,
2, Projective > 
typedef Transform< float,
2, Projective > 
typedef Transform< double,
3, Projective > 
typedef Transform< float,
3, Projective > 
typedef Quaternion< double > Quaterniond
typedef Quaternion< float > Quaternionf
typedef Map< Quaternion
< double >, Aligned > 
typedef Map< Quaternion< float >
, Aligned > 
typedef Map< Quaternion
< double >, 0 > 
typedef Map< Quaternion< float >, 0 > QuaternionMapf
typedef Rotation2D< double > Rotation2Dd
typedef Rotation2D< float > Rotation2Df


Matrix< Scalar, 3, 1 > eulerAngles (Index a0, Index a1, Index a2) const
template<typename Derived , typename OtherDerived >
< Derived, OtherDerived >
umeyama (const MatrixBase< Derived > &src, const MatrixBase< OtherDerived > &dst, bool with_scaling=true)
 Returns the transformation between two point sets. More...

Typedef Documentation

typedef Transform<double,2,Affine> Affine2d
typedef Transform<float,2,Affine> Affine2f
typedef Transform<double,3,Affine> Affine3d
typedef Transform<float,3,Affine> Affine3f
typedef Transform<double,2,AffineCompact> AffineCompact2d
typedef Transform<float,2,AffineCompact> AffineCompact2f
typedef Transform<double,3,AffineCompact> AffineCompact3d
typedef Transform<float,3,AffineCompact> AffineCompact3f
typedef DiagonalMatrix<double,2> AlignedScaling2d
typedef DiagonalMatrix<float, 2> AlignedScaling2f
typedef DiagonalMatrix<double,3> AlignedScaling3d
typedef DiagonalMatrix<float, 3> AlignedScaling3f
typedef AngleAxis<double> AngleAxisd

double precision angle-axis type

typedef AngleAxis<float> AngleAxisf

single precision angle-axis type

typedef Transform<double,2,Isometry> Isometry2d
typedef Transform<float,2,Isometry> Isometry2f
typedef Transform<double,3,Isometry> Isometry3d
typedef Transform<float,3,Isometry> Isometry3f
typedef Transform<double,2,Projective> Projective2d
typedef Transform<float,2,Projective> Projective2f
typedef Transform<double,3,Projective> Projective3d
typedef Transform<float,3,Projective> Projective3f
typedef Quaternion<double> Quaterniond

double precision quaternion type

typedef Quaternion<float> Quaternionf

single precision quaternion type

typedef Map<Quaternion<double>, Aligned> QuaternionMapAlignedd

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

typedef Map<Quaternion<float>, Aligned> QuaternionMapAlignedf

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

typedef Map<Quaternion<double>, 0> QuaternionMapd

Map an unaligned array of double precision scalars as a quaternion

typedef Map<Quaternion<float>, 0> QuaternionMapf

Map an unaligned array of single precision scalars as a quaternion

typedef Rotation2D<double> Rotation2Dd

double precision 2D rotation type

typedef Rotation2D<float> Rotation2Df

single precision 2D rotation type

Function Documentation

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

This is defined in the Geometry module.

#include <Eigen/Geometry>
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[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
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.

This is defined in the Geometry module.

#include <Eigen/Geometry>

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.

srcSource points $ \mathbf{x} = \left( x_1, \hdots, x_n \right) $.
dstDestination points $ \mathbf{y} = \left( y_1, \hdots, y_n \right) $.
with_scalingSets $ c=1 $ when false is passed.
The homogeneous transformation

\begin{align*} T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} \end{align*}

minimizing the resudiual above. This transformation is always returned as an Eigen::Matrix.

References DenseBase< Derived >::colwise(), Eigen::ComputeFullU, Eigen::ComputeFullV, JacobiSVD< MatrixType, QRPreconditioner >::matrixU(), JacobiSVD< MatrixType, QRPreconditioner >::matrixV(), DenseBase< Derived >::rowwise(), and JacobiSVD< MatrixType, QRPreconditioner >::singularValues().