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Geometry_Module
Geometry » Reference

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

◆ Affine2d

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

◆ Affine2f

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

◆ Affine3d

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

◆ Affine3f

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

◆ AffineCompact2d

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

◆ AffineCompact2f

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

◆ AffineCompact3d

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

◆ AffineCompact3f

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

◆ AngleAxisd

typedef AngleAxis<double> Eigen::AngleAxisd

double precision angle-axis type

◆ AngleAxisf

typedef AngleAxis<float> Eigen::AngleAxisf

single precision angle-axis type

◆ Isometry2d

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

◆ Isometry2f

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

◆ Isometry3d

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

◆ Isometry3f

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

◆ Projective2d

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

◆ Projective2f

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

◆ Projective3d

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

◆ Projective3f

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

◆ 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);
mat
MatrixXf mat
Definition Tutorial_AdvancedInitialization_CommaTemporary.cpp:1

"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());
Eigen::AngleAxisf
AngleAxis< float > AngleAxisf
Definition AngleAxis.h:157

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 ( ) const
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);
head
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE FixedSegmentReturnType< internal::get_fixed_value< NType >::value >::Type head(NType n)
Definition BlockMethods.h:1208
size
Scalar Scalar int size
Definition benchVecAdd.cpp:17

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 
          (
          )
           const
        

      

  		
  
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 
          (
          )
           const
        

      

  		
  
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 
          (
          )
           const
        

      

  		
  
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

  

    
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 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() 











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