Eigen  3.3.1
 All Classes Namespaces Functions Variables Typedefs Enumerations Enumerator Friends Groups Pages
Geometry module

Detailed Description

This module provides support for:

* #include <Eigen/Geometry>
*

Modules

 Global aligned box typedefs
 

Classes

class  Eigen::AlignedBox< Scalar, Dim >
 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::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::ParametrizedLine< _Scalar, _AmbientDim, Options >
 A parametrized line. More...
 
class  Eigen::Quaternion< Scalar, Options >
 The quaternion class used to represent 3D orientations and rotations. More...
 
class  Eigen::QuaternionBase< Derived >
 Base class for quaternion expressions. More...
 
class  Eigen::Rotation2D< Scalar >
 Represents a rotation/orientation in a 2 dimensional space. More...
 
class  Scaling
 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 Transform< double,
2, Affine > 
Eigen::Affine2d
 
typedef Transform< float,
2, Affine > 
Eigen::Affine2f
 
typedef Transform< double,
3, Affine > 
Eigen::Affine3d
 
typedef Transform< float,
3, Affine > 
Eigen::Affine3f
 
typedef Transform< double,
2, AffineCompact > 
Eigen::AffineCompact2d
 
typedef Transform< float,
2, AffineCompact > 
Eigen::AffineCompact2f
 
typedef Transform< double,
3, AffineCompact > 
Eigen::AffineCompact3d
 
typedef Transform< float,
3, AffineCompact > 
Eigen::AffineCompact3f
 
typedef DiagonalMatrix< double, 2 > Eigen::AlignedScaling2d
 
typedef DiagonalMatrix< float, 2 > Eigen::AlignedScaling2f
 
typedef DiagonalMatrix< double, 3 > Eigen::AlignedScaling3d
 
typedef DiagonalMatrix< float, 3 > Eigen::AlignedScaling3f
 
typedef AngleAxis< double > Eigen::AngleAxisd
 
typedef AngleAxis< float > Eigen::AngleAxisf
 
typedef Transform< double,
2, Isometry > 
Eigen::Isometry2d
 
typedef Transform< float,
2, Isometry > 
Eigen::Isometry2f
 
typedef Transform< double,
3, Isometry > 
Eigen::Isometry3d
 
typedef Transform< float,
3, Isometry > 
Eigen::Isometry3f
 
typedef Transform< double,
2, Projective > 
Eigen::Projective2d
 
typedef Transform< float,
2, Projective > 
Eigen::Projective2f
 
typedef Transform< double,
3, Projective > 
Eigen::Projective3d
 
typedef Transform< float,
3, Projective > 
Eigen::Projective3f
 
typedef Quaternion< double > Eigen::Quaterniond
 
typedef Quaternion< float > Eigen::Quaternionf
 
typedef Map< Quaternion
< double >, Aligned > 
Eigen::QuaternionMapAlignedd
 
typedef Map< Quaternion< float >
, Aligned > 
Eigen::QuaternionMapAlignedf
 
typedef Map< Quaternion
< double >, 0 > 
Eigen::QuaternionMapd
 
typedef Map< Quaternion< float >, 0 > Eigen::QuaternionMapf
 
typedef Rotation2D< double > Eigen::Rotation2Dd
 
typedef Rotation2D< float > Eigen::Rotation2Df
 

Functions

template<typename OtherDerived >
MatrixBase< Derived >::PlainObject Eigen::MatrixBase< Derived >::cross (const MatrixBase< OtherDerived > &other) const
 
template<typename OtherDerived >
const VectorwiseOp
< ExpressionType, Direction >
::CrossReturnType 
Eigen::VectorwiseOp< ExpressionType, Direction >::cross (const MatrixBase< OtherDerived > &other) const
 
template<typename OtherDerived >
MatrixBase< Derived >::PlainObject Eigen::MatrixBase< Derived >::cross3 (const MatrixBase< OtherDerived > &other) const
 
Matrix< Scalar, 3, 1 > Eigen::MatrixBase< Derived >::eulerAngles (Index a0, Index a1, Index a2) const
 
const HNormalizedReturnType Eigen::MatrixBase< Derived >::hnormalized () const
 homogeneous normalization More...
 
const HNormalizedReturnType Eigen::VectorwiseOp< ExpressionType, Direction >::hnormalized () const
 column or row-wise homogeneous normalization More...
 
HomogeneousReturnType Eigen::MatrixBase< Derived >::homogeneous () const
 
HomogeneousReturnType Eigen::VectorwiseOp< ExpressionType, Direction >::homogeneous () const
 
template<typename Derived , typename Scalar >
 operator* (const MatrixBase< Derived > &matrix, const UniformScaling< Scalar > &s)
 
UniformScaling< float > Eigen::Scaling (float s)
 
UniformScaling< double > Eigen::Scaling (double s)
 
template<typename RealScalar >
UniformScaling< std::complex
< RealScalar > > 
Eigen::Scaling (const std::complex< RealScalar > &s)
 
template<typename Scalar >
DiagonalMatrix< Scalar, 2 > Eigen::Scaling (const Scalar &sx, const Scalar &sy)
 
template<typename Scalar >
DiagonalMatrix< Scalar, 3 > Eigen::Scaling (const Scalar &sx, const Scalar &sy, const Scalar &sz)
 
template<typename Derived >
const DiagonalWrapper< const
Derived > 
Eigen::Scaling (const MatrixBase< Derived > &coeffs)
 
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...
 
PlainObject Eigen::MatrixBase< Derived >::unitOrthogonal (void) const
 

Typedef Documentation

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

double precision angle-axis type

typedef AngleAxis<float> Eigen::AngleAxisf

single precision angle-axis type

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

double precision quaternion type

typedef Quaternion<float> Eigen::Quaternionf

single precision quaternion type

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

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

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

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

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

Map an unaligned array of double precision scalars as a quaternion

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

Map an unaligned array of single precision scalars as a quaternion

typedef Rotation2D<double> Eigen::Rotation2Dd

double precision 2D rotation type

typedef Rotation2D<float> Eigen::Rotation2Df

single precision 2D rotation type

Function Documentation

template<typename Derived>
template<typename OtherDerived >
MatrixBase<Derived>::PlainObject Eigen::MatrixBase< Derived >::cross ( const MatrixBase< OtherDerived > &  other) const
inline

This is defined in the Geometry module.

#include <Eigen/Geometry>
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()
template<typename ExpressionType, int Direction>
template<typename OtherDerived >
const VectorwiseOp<ExpressionType,Direction>::CrossReturnType Eigen::VectorwiseOp< ExpressionType, Direction >::cross ( const MatrixBase< OtherDerived > &  other) const

This is defined in the Geometry module.

#include <Eigen/Geometry>
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()
template<typename Derived>
template<typename OtherDerived >
MatrixBase<Derived>::PlainObject Eigen::MatrixBase< Derived >::cross3 ( const MatrixBase< OtherDerived > &  other) const
inline

This is defined in the Geometry module.

#include <Eigen/Geometry>
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()
template<typename Derived >
Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > Eigen::MatrixBase< Derived >::eulerAngles ( Index  a0,
Index  a1,
Index  a2 
) const
inline

This is defined in the Geometry module.

#include <Eigen/Geometry>
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[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
template<typename Derived >
const MatrixBase< Derived >::HNormalizedReturnType Eigen::MatrixBase< Derived >::hnormalized ( ) const
inline

homogeneous normalization

This is defined in the Geometry module.

#include <Eigen/Geometry>
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:

cout << "v = " << v.transpose() << "]^T" << endl;
cout << "v.hnormalized() = " << v.hnormalized().transpose() << "]^T" << endl;
cout << "P*v = " << (P*v).transpose() << "]^T" << endl;
cout << "(P*v).hnormalized() = " << (P*v).hnormalized().transpose() << "]^T" << endl;

Output:

v                   =   0.68 -0.211  0.566  0.597]^T
v.hnormalized()     =   1.14 -0.354  0.949]^T
P*v                 = 0.663 -0.16 -0.13  0.91]^T
(P*v).hnormalized() =  0.729 -0.176 -0.143]^T
See Also
VectorwiseOp::hnormalized()
template<typename ExpressionType , int Direction>
const VectorwiseOp< ExpressionType, Direction >::HNormalizedReturnType Eigen::VectorwiseOp< ExpressionType, Direction >::hnormalized ( ) const
inline

column or row-wise homogeneous normalization

This is defined in the Geometry module.

#include <Eigen/Geometry>
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:

typedef Matrix<double,4,Dynamic> Matrix4Xd;
Matrix4Xd M = Matrix4Xd::Random(4,5);
cout << "The matrix M is:" << endl << M << endl << endl;
cout << "M.colwise().hnormalized():" << endl << M.colwise().hnormalized() << endl << endl;
cout << "P*M:" << endl << P*M << endl << endl;
cout << "(P*M).colwise().hnormalized():" << endl << (P*M).colwise().hnormalized() << endl << endl;

Output:

The matrix M is:
   0.68   0.823  -0.444   -0.27   0.271
 -0.211  -0.605   0.108  0.0268   0.435
  0.566   -0.33 -0.0452   0.904  -0.717
  0.597   0.536   0.258   0.832   0.214

M.colwise().hnormalized():
  1.14   1.53  -1.72 -0.325   1.27
-0.354  -1.13  0.419 0.0322   2.03
 0.949 -0.614 -0.175   1.09  -3.35

P*M:
  0.186  -0.589   0.369    1.33   -1.23
 -0.871  -0.337   0.127  -0.715   0.091
 -0.158 -0.0104   0.312   0.429  -0.478
  0.992   0.777  -0.373   0.468  -0.651

(P*M).colwise().hnormalized():
  0.188  -0.759  -0.989    2.85    1.89
 -0.877  -0.433  -0.342   -1.53   -0.14
  -0.16 -0.0134  -0.837   0.915   0.735

See Also
MatrixBase::hnormalized()
template<typename Derived >
MatrixBase< Derived >::HomogeneousReturnType Eigen::MatrixBase< Derived >::homogeneous ( ) const
inline

This is defined in the Geometry module.

#include <Eigen/Geometry>
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.

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

Example:

cout << "v = [" << v.transpose() << "]^T" << endl;
cout << "h.homogeneous() = [" << v.homogeneous().transpose() << "]^T" << endl;
cout << "(P * v.homogeneous()) = [" << (P * v.homogeneous()).transpose() << "]^T" << endl;
cout << "(P * v.homogeneous()).hnormalized() = [" << (P * v.homogeneous()).eval().hnormalized().transpose() << "]^T" << endl;

Output:

v                                   = [  0.68 -0.211  0.566]^T
h.homogeneous()                     = [  0.68 -0.211  0.566      1]^T
(P * v.homogeneous())               = [  1.27  0.772 0.0154 -0.419]^T
(P * v.homogeneous()).hnormalized() = [  -3.03   -1.84 -0.0367]^T
See Also
VectorwiseOp::homogeneous(), class Homogeneous
template<typename ExpressionType , int Direction>
Homogeneous< ExpressionType, Direction > Eigen::VectorwiseOp< ExpressionType, Direction >::homogeneous ( ) const
inline

This is defined in the Geometry module.

#include <Eigen/Geometry>
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:

typedef Matrix<double,3,Dynamic> Matrix3Xd;
Matrix3Xd M = Matrix3Xd::Random(3,5);
cout << "The matrix M is:" << endl << M << endl << endl;
cout << "M.colwise().homogeneous():" << endl << M.colwise().homogeneous() << endl << endl;
cout << "P * M.colwise().homogeneous():" << endl << P * M.colwise().homogeneous() << endl << endl;
cout << "P * M.colwise().homogeneous().hnormalized(): " << endl << (P * M.colwise().homogeneous()).colwise().hnormalized() << endl << endl;

Output:

The matrix M is:
   0.68   0.597   -0.33   0.108   -0.27
 -0.211   0.823   0.536 -0.0452  0.0268
  0.566  -0.605  -0.444   0.258   0.904

M.colwise().homogeneous():
   0.68   0.597   -0.33   0.108   -0.27
 -0.211   0.823   0.536 -0.0452  0.0268
  0.566  -0.605  -0.444   0.258   0.904
      1       1       1       1       1

P * M.colwise().homogeneous():
0.0832 -0.477  -1.21 -0.545 -0.452
 0.998  0.779  0.695  0.894  0.277
-0.271 -0.608 -0.895 -0.544 -0.874
-0.728 -0.551  0.202  -0.21 -0.469

P * M.colwise().homogeneous().hnormalized(): 
-0.114  0.866     -6    2.6  0.962
 -1.37  -1.41   3.44  -4.27 -0.591
 0.373    1.1  -4.43    2.6   1.86

See Also
MatrixBase::homogeneous(), class Homogeneous
template<typename Derived , typename Scalar >
operator* ( const MatrixBase< Derived > &  matrix,
const UniformScaling< Scalar > &  s 
)
related

Concatenates a linear transformation matrix and a uniform scaling

UniformScaling<float> Eigen::Scaling ( float  s)
inline

Constructs a uniform scaling from scale factor s

UniformScaling<double> Eigen::Scaling ( double  s)
inline

Constructs a uniform scaling from scale factor s

template<typename RealScalar >
UniformScaling<std::complex<RealScalar> > Eigen::Scaling ( const std::complex< RealScalar > &  s)
inline

Constructs a uniform scaling from scale factor s

template<typename Scalar >
DiagonalMatrix<Scalar,2> Eigen::Scaling ( const Scalar &  sx,
const Scalar &  sy 
)
inline

Constructs a 2D axis aligned scaling

template<typename Scalar >
DiagonalMatrix<Scalar,3> Eigen::Scaling ( const Scalar &  sx,
const Scalar &  sy,
const Scalar &  sz 
)
inline

Constructs a 3D axis aligned scaling

template<typename Derived >
const DiagonalWrapper<const Derived> Eigen::Scaling ( const MatrixBase< Derived > &  coeffs)
inline

Constructs an axis aligned scaling expression from vector expression coeffs This is an alias for coeffs.asDiagonal()

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.

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.

Parameters
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.
Returns
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.
template<typename Derived >
MatrixBase< Derived >::PlainObject Eigen::MatrixBase< Derived >::unitOrthogonal ( void  ) const
inline

This is defined in the Geometry module.

#include <Eigen/Geometry>
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()