Visual Servoing Platform  version 3.2.0 under development (2018-12-17)
vpRzyxVector Class Reference

#include <visp3/core/vpRzyxVector.h>

+ Inheritance diagram for vpRzyxVector:

Public Member Functions

 vpRzyxVector ()
 
 vpRzyxVector (const vpRzyxVector &rzyx)
 
 vpRzyxVector (const double phi, const double theta, const double psi)
 
 vpRzyxVector (const vpRotationMatrix &R)
 
 vpRzyxVector (const vpThetaUVector &tu)
 
 vpRzyxVector (const vpColVector &rzyx)
 
 vpRzyxVector (const std::vector< double > &rzyx)
 
virtual ~vpRzyxVector ()
 
vpRzyxVector buildFrom (const vpRotationMatrix &R)
 
vpRzyxVector buildFrom (const vpThetaUVector &R)
 
vpRzyxVector buildFrom (const vpColVector &rxyz)
 
vpRzyxVector buildFrom (const std::vector< double > &rxyz)
 
void buildFrom (const double phi, const double theta, const double psi)
 
vpRzyxVectoroperator= (const vpColVector &rzyx)
 
vpRzyxVectoroperator= (double x)
 
Inherited functionalities from vpRotationVector
double & operator[] (unsigned int i)
 
const double & operator[] (unsigned int i) const
 
vpColVector operator* (double x) const
 
double sumSquare () const
 
vpRowVector t () const
 
std::vector< double > toStdVector ()
 
Inherited functionalities from vpArray2D
unsigned int getCols () const
 
double getMaxValue () const
 
double getMinValue () const
 
unsigned int getRows () const
 
unsigned int size () const
 
void resize (const unsigned int nrows, const unsigned int ncols, const bool flagNullify=true, const bool recopy_=true)
 
vpArray2D< double > hadamard (const vpArray2D< double > &m) const
 

Static Public Member Functions

Inherited I/O from vpArray2D with Static Public Member Functions
static bool load (const std::string &filename, vpArray2D< double > &A, const bool binary=false, char *header=NULL)
 
static bool loadYAML (const std::string &filename, vpArray2D< double > &A, char *header=NULL)
 
static bool save (const std::string &filename, const vpArray2D< double > &A, const bool binary=false, const char *header="")
 
static bool saveYAML (const std::string &filename, const vpArray2D< double > &A, const char *header="")
 

Public Attributes

double * data
 

Protected Attributes

unsigned int rowNum
 
unsigned int colNum
 
double ** rowPtrs
 
unsigned int dsize
 

Related Functions

(Note that these are not member functions.)

vpColVector operator* (const double &x, const vpRotationVector &v)
 
enum  vpGEMMmethod
 

Detailed Description

Implementation of a rotation vector as $R(z,y,x)$ Euler angle minimal representation.

Class that consider the case of the Euler $(\varphi,\theta,\psi)$ angle using the z-y-x convention, where $(\varphi,\theta,\psi)$ are respectively the rotation angles around the $z$, $y$ and $x$ axis.

\[R_{zyx}(\varphi,\theta,\psi) = R_z(\varphi) \; R_y(\theta) \; R_x(\psi)\]

with

\[ R_{z}(\varphi) = \left( \begin{array}{ccc} \cos \varphi & -\sin\varphi & 0\\ \sin\varphi &\cos \varphi& 0 \\ 0 & 0 & 1 \end{array} \right) \; R_{y}(\theta) = \left( \begin{array}{ccc} \cos \theta & 0 & \sin\theta\\ 0 & 1 & 0 \\ -\sin\theta & 0 &\cos \theta \end{array} \right) \; R_{x}(\psi) = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 &\cos \psi & -\sin\psi \\ 0 &\sin \psi & \cos\psi \\ \end{array} \right) \]

The rotation matrix corresponding to the z-y-x convention is given by:

\[ R_{zyx}(\varphi,\theta,\psi) = \left( \begin{array}{ccc} \cos\varphi \cos\theta & -\sin\varphi \cos\psi + \cos\varphi\sin\theta\sin\psi & \sin\varphi \sin\psi +\cos\varphi\sin\theta\cos\psi \\ \sin\varphi \cos\theta & \cos\varphi\cos\psi + \sin\varphi\sin\theta \sin\psi & -\cos\varphi \sin\psi +\sin\varphi\sin\theta\cos\psi \\ -\sin\theta & \cos\theta \sin\psi & \cos\theta \cos\psi \end{array} \right) \]

The vpRzyxVector class is derived from vpRotationVector.

The code below shows first how to initialize this representation of Euler angles, than how to contruct a rotation matrix from a vpRzyxVector and finaly how to extract the vpRzyxVector Euler angles from the build rotation matrix.

#include <visp3/core/vpMath.h>
#include <visp3/core/vpRotationMatrix.h>
#include <visp3/core/vpRzyxVector.h>
int main()
{
// Initialise the Euler angles
rzyx[0] = vpMath::rad( 45.f); // phi angle in rad/s around z axis
rzyx[1] = vpMath::rad(-30.f); // theta angle in rad/s around y axis
rzyx[2] = vpMath::rad( 90.f); // psi angle in rad/s around x axis
// Construct a rotation matrix from the Euler angles
// Extract the Euler angles around z,y,x axis from a rotation matrix
rzyx.buildFrom(R);
// Print the extracted Euler angles. Values are the same than the
// one used for initialization
std::cout << rzyx;
// Since the rotation vector is 3 values column vector, the
// transpose operation produce a row vector.
vpRowVector rzyx_t = rzyx.t();
// Print the transpose row vector
std::cout << rzyx_t << std::endl;
}
Examples:
testPoseRansac2.cpp.

Definition at line 158 of file vpRzyxVector.h.

Constructor & Destructor Documentation

vpRzyxVector::vpRzyxVector ( )

Default constructor that initialize all the 3 angles to zero.

Definition at line 51 of file vpRzyxVector.cpp.

vpRzyxVector::vpRzyxVector ( const vpRzyxVector rzyx)

Copy constructor.

Definition at line 54 of file vpRzyxVector.cpp.

vpRzyxVector::vpRzyxVector ( const double  phi,
const double  theta,
const double  psi 
)

Constructor from 3 angles (in radian).

Parameters
phi: $\varphi$ angle around the $z$ axis.
theta: $\theta$ angle around the $y$ axis.
psi: $\psi$ angle around the $x$ axis.

Definition at line 62 of file vpRzyxVector.cpp.

References buildFrom().

vpRzyxVector::vpRzyxVector ( const vpRotationMatrix R)
explicit

Constructor that initialize $R_{zyx}=(\varphi,\theta,\psi)$ Euler angles from a rotation matrix.

Parameters
R: Rotation matrix used to initialize the Euler angles.

Definition at line 72 of file vpRzyxVector.cpp.

References buildFrom().

vpRzyxVector::vpRzyxVector ( const vpThetaUVector tu)
explicit

Constructor that initialize $R_{zyx}=(\varphi,\theta,\psi)$ Euler angles vector from a $\theta {\bf u}$ vector.

Parameters
tu: $\theta {\bf u}$ representation of a rotation used here as input to initialize the Euler angles.

Definition at line 80 of file vpRzyxVector.cpp.

References buildFrom().

vpRzyxVector::vpRzyxVector ( const vpColVector rzyx)
explicit

Copy constructor from a 3-dimension vector.

Definition at line 83 of file vpRzyxVector.cpp.

References buildFrom().

vpRzyxVector::vpRzyxVector ( const std::vector< double > &  rzyx)
explicit

Copy constructor from a 3-dimension vector.

Definition at line 89 of file vpRzyxVector.cpp.

References buildFrom().

virtual vpRzyxVector::~vpRzyxVector ( )
inlinevirtual

Destructor.

Definition at line 174 of file vpRzyxVector.h.

References vpRotationVector::operator=().

Member Function Documentation

vpRzyxVector vpRzyxVector::buildFrom ( const vpRotationMatrix R)

Convert a rotation matrix into a $R_{zyx}=(\varphi,\theta,\psi)$ Euler angles vector.

Source: R. Paul, Robot Manipulators: Mathematics, Programming, and Control. MIT Press, 1981, p. 71

Parameters
R: Rotation matrix used as input.
Returns
$R_{zyx}=(\varphi,\theta,\psi)$ Euler angles vector.

Definition at line 104 of file vpRzyxVector.cpp.

Referenced by buildFrom(), and vpRzyxVector().

vpRzyxVector vpRzyxVector::buildFrom ( const vpThetaUVector tu)

Convert a $\theta {\bf u}$ vector into a $R_{zyx}=(\varphi,\theta,\psi)$ Euler angles vector.

Parameters
tu: $\theta {\bf u}$ representation of a rotation used here as input.
Returns
$R_{zyx}=(\varphi,\theta,\psi)$ Euler angles vector.

Definition at line 134 of file vpRzyxVector.cpp.

References vpRotationMatrix::buildFrom(), and buildFrom().

vpRzyxVector vpRzyxVector::buildFrom ( const vpColVector rzyx)

Construct a $R_{zyx}=(\varphi,\theta,\psi)$ Euler angles vectorfrom a 3-dim vector.

Definition at line 159 of file vpRzyxVector.cpp.

References vpArray2D< double >::data, vpException::dimensionError, and vpArray2D< Type >::size().

vpRzyxVector vpRzyxVector::buildFrom ( const std::vector< double > &  rzyx)

Construct a $R_{zyx}=(\varphi,\theta,\psi)$ Euler angles vector from a 3-dim vector.

Definition at line 175 of file vpRzyxVector.cpp.

References vpArray2D< double >::data, and vpException::dimensionError.

void vpRzyxVector::buildFrom ( const double  phi,
const double  theta,
const double  psi 
)

Construction from 3 angles (in radian).

Parameters
phi: $\varphi$ angle around the $z$ axis.
theta: $\theta$ angle around the $y$ axis.
psi: $\psi$ angle around the $x$ axis.

Definition at line 149 of file vpRzyxVector.cpp.

References vpArray2D< double >::data.

double vpArray2D< double >::getMaxValue ( ) const
inherited

Return the array max value.

Examples:
servoMomentImage.cpp, testArray2D.cpp, and testMatrix.cpp.
double vpArray2D< double >::getMinValue ( ) const
inherited

Return the array min value.

Examples:
servoMomentImage.cpp, testArray2D.cpp, and testMatrix.cpp.
vpArray2D<double > vpArray2D< double >::hadamard ( const vpArray2D< double > &  m) const
inherited

Compute the Hadamard product (element wise matrix multiplication).

Parameters
m: Second matrix;
Returns
m1.hadamard(m2) The Hadamard product : $ m1 \circ m2 = (m1 \circ m2)_{i,j} = (m1)_{i,j} (m2)_{i,j} $
Examples:
testArray2D.cpp.
static bool vpArray2D< double >::load ( const std::string &  filename,
vpArray2D< double > &  A,
const bool  binary = false,
char *  header = NULL 
)
inlinestaticinherited

Load a matrix from a file.

Parameters
filename: Absolute file name.
A: Array to be loaded
binary: If true the matrix is loaded from a binary file, else from a text file.
header: Header of the file is loaded in this parameter.
Returns
Returns true if success.
See also
save()

Definition at line 316 of file vpArray2D.h.

References vpException::badValue, and vpArray2D< Type >::resize().

static bool vpArray2D< double >::loadYAML ( const std::string &  filename,
vpArray2D< double > &  A,
char *  header = NULL 
)
inlinestaticinherited

Load an array from a YAML-formatted file.

Parameters
filename: absolute file name.
A: array to be loaded from the file.
header: header of the file is loaded in this parameter.
Returns
Returns true on success.
See also
saveYAML()
Examples:
servoFrankaPBVS.cpp, and tutorial-hand-eye-calibration.cpp.

Definition at line 424 of file vpArray2D.h.

References vpArray2D< Type >::resize().

vpColVector vpRotationVector::operator* ( double  x) const
inherited

Operator that allows to multiply each element of a rotation vector by a scalar.

Parameters
x: The scalar.
Returns
The rotation vector multiplied by the scalar as a column vector. The current rotation vector (*this) is unchanged.

Definition at line 89 of file vpRotationVector.cpp.

References vpArray2D< double >::dsize.

Referenced by vpQuaternionVector::~vpQuaternionVector().

vpRzyxVector & vpRzyxVector::operator= ( const vpColVector rzyx)

Copy operator that initializes a $R_{zyx}=(\varphi,\theta,\psi)$ Euler angles vector from a 3-dimension column vector.

Parameters
rzyx: 3-dimension vector containing the values of the rotation vector.
#include <visp3/core/vpRzyxVector.h>
int main()
{
v[0] = 0.1;
v[1] = 0.2;
v[2] = 0.3;
rzyx = v;
// rzyx is now equal to v : 0.1, 0.2, 0.3
}

Definition at line 237 of file vpRzyxVector.cpp.

References vpArray2D< double >::data, vpException::dimensionError, and vpArray2D< Type >::size().

vpRzyxVector & vpRzyxVector::operator= ( double  v)

Initialize each element of the vector to the same angle value v.

Parameters
v: Angle value to set for each element of the vector.
#include <visp3/core/vpMath.h>
#include <visp3/core/vpRzyxVector.h>
int main()
{
// Initialise the rotation vector
v = vpMath::rad( 45.f); // All the 3 angles are set to 45 degrees
}

Definition at line 206 of file vpRzyxVector.cpp.

References vpArray2D< double >::data, and vpArray2D< double >::dsize.

double& vpRotationVector::operator[] ( unsigned int  i)
inlineinherited

Operator that allows to set the value of an element of the rotation vector: r[i] = value

Definition at line 127 of file vpRotationVector.h.

const double& vpRotationVector::operator[] ( unsigned int  i) const
inlineinherited

Operator that allows to get the value of an element of the rotation vector: value = r[i]

Definition at line 132 of file vpRotationVector.h.

void vpArray2D< double >::resize ( const unsigned int  nrows,
const unsigned int  ncols,
const bool  flagNullify = true,
const bool  recopy_ = true 
)
inlineinherited

Set the size of the array and initialize all the values to zero.

Parameters
nrows: number of rows.
ncols: number of column.
flagNullify: if true, then the array is re-initialized to 0 after resize. If false, the initial values from the common part of the array (common part between old and new version of the array) are kept. Default value is true.
recopy_: if true, will perform an explicit recopy of the old data if needed and if flagNullify is set to false.
Examples:
testArray2D.cpp, testMatrix.cpp, testMatrixDeterminant.cpp, testMatrixInverse.cpp, testMatrixPseudoInverse.cpp, and testSvd.cpp.

Definition at line 171 of file vpArray2D.h.

References vpArray2D< Type >::colNum, vpArray2D< Type >::dsize, vpException::memoryAllocationError, vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

Referenced by vpMatrix::diag(), vpMatrix::eye(), vpMatrix::init(), vpMatrix::operator=(), and vpMatrix::stack().

static bool vpArray2D< double >::save ( const std::string &  filename,
const vpArray2D< double > &  A,
const bool  binary = false,
const char *  header = "" 
)
inlinestaticinherited

Save a matrix to a file.

Parameters
filename: Absolute file name.
A: Array to be saved.
binary: If true the matrix is saved in a binary file, else a text file.
header: Optional line that will be saved at the beginning of the file.
Returns
Returns true if success.

Warning : If you save the matrix as in a text file the precision is less than if you save it in a binary file.

See also
load()

Definition at line 506 of file vpArray2D.h.

References vpArray2D< Type >::getCols(), and vpArray2D< Type >::getRows().

static bool vpArray2D< double >::saveYAML ( const std::string &  filename,
const vpArray2D< double > &  A,
const char *  header = "" 
)
inlinestaticinherited

Save an array in a YAML-formatted file.

Parameters
filename: absolute file name.
A: array to be saved in the file.
header: optional lines that will be saved at the beginning of the file. Should be YAML-formatted and will adapt to the indentation if any.
Returns
Returns true if success.

Here is an example of outputs.

vpArray2D::saveYAML("matrix.yml", M, "example: a YAML-formatted header");
vpArray2D::saveYAML("matrixIndent.yml", M, "example:\n - a YAML-formatted
header\n - with inner indentation");

Content of matrix.yml:

example: a YAML-formatted header
rows: 3
cols: 4
- [0, 0, 0, 0]
- [0, 0, 0, 0]
- [0, 0, 0, 0]

Content of matrixIndent.yml:

example:
- a YAML-formatted header
- with inner indentation
rows: 3
cols: 4
- [0, 0, 0, 0]
- [0, 0, 0, 0]
- [0, 0, 0, 0]
See also
loadYAML()
Examples:
tutorial-chessboard-pose.cpp, tutorial-franka-acquire-calib-data.cpp, and tutorial-hand-eye-calibration.cpp.

Definition at line 595 of file vpArray2D.h.

References vpArray2D< Type >::getCols(), and vpArray2D< Type >::getRows().

double vpRotationVector::sumSquare ( ) const
inherited

Return the sum square of all the elements $r_{i}$ of the rotation vector r(m).

Returns
The value

\[\sum{i=0}^{m} r_i^{2}\]

.
Examples:
servoFrankaPBVS.cpp.

Definition at line 115 of file vpRotationVector.cpp.

References vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

vpRowVector vpRotationVector::t ( ) const
inherited

Return the transpose of the rotation vector.

Examples:
testViper650.cpp, and testViper850.cpp.

Definition at line 56 of file vpRotationVector.cpp.

References vpArray2D< double >::data, and vpArray2D< double >::dsize.

std::vector< double > vpRotationVector::toStdVector ( )
inherited

Converts the vpRotationVector to a std::vector.

Returns
The corresponding std::vector<double>.

Definition at line 70 of file vpRotationVector.cpp.

References vpArray2D< double >::data, and vpArray2D< double >::size().

Friends And Related Function Documentation

vpColVector operator* ( const double &  x,
const vpRotationVector v 
)
related

Allows to multiply a scalar by rotaion vector.

Definition at line 102 of file vpRotationVector.cpp.

enum vpGEMMmethod
related

Enumeration of the operations applied on matrices in vpGEMM() function.

Operations are :

  • VP_GEMM_A_T to use the transpose matrix of A instead of the matrix A
  • VP_GEMM_B_T to use the transpose matrix of B instead of the matrix B
  • VP_GEMM_C_T to use the transpose matrix of C instead of the matrix C

Definition at line 57 of file vpGEMM.h.

Member Data Documentation

unsigned int vpArray2D< double >::colNum
protectedinherited

Number of columns in the array.

Definition at line 76 of file vpArray2D.h.

Referenced by vpMatrix::AAt(), vpMatrix::AtA(), vpColVector::clear(), vpMatrix::detByLU(), vpMatrix::diag(), vpMatrix::eigenValues(), vpMatrix::expm(), vpMatrix::eye(), vpMatrix::getRow(), vpColVector::hadamard(), vpMatrix::hadamard(), vpMatrix::infinityNorm(), vpSubColVector::init(), vpSubRowVector::init(), vpSubMatrix::init(), vpRowVector::insert(), vpMatrix::insert(), vpMatrix::inverseByCholesky(), vpMatrix::inverseTriangular(), vpRotationMatrix::operator*(), vpRowVector::operator*(), vpMatrix::operator*(), vpRotationMatrix::operator*=(), vpRowVector::operator*=(), vpMatrix::operator*=(), vpRowVector::operator+(), vpRowVector::operator+=(), vpMatrix::operator+=(), vpRowVector::operator-(), vpRowVector::operator-=(), vpMatrix::operator-=(), vpRowVector::operator/(), vpMatrix::operator/(), vpRowVector::operator/=(), vpMatrix::operator/=(), vpMatrix::operator<<(), vpColVector::operator<<(), vpSubRowVector::operator=(), vpSubMatrix::operator=(), vpRowVector::operator=(), vpColVector::operator=(), vpMatrix::operator=(), vpColVector::operator==(), vpMatrix::qr(), vpMatrix::qrPivot(), vpRowVector::reshape(), vpMatrix::setIdentity(), vpMatrix::solveByQR(), vpMatrix::solveBySVD(), vpMatrix::stack(), vpRowVector::stack(), vpMatrix::stackColumns(), vpMatrix::stackRows(), vpRowVector::sum(), vpMatrix::sum(), vpRowVector::sumSquare(), vpMatrix::sumSquare(), vpRowVector::t(), vpMatrix::t(), vpMatrix::transpose(), vpColVector::vpColVector(), and vpMatrix::vpMatrix().

double * vpArray2D< double >::data
inherited

Address of the first element of the data array.

Examples:
testArray2D.cpp, testDisplacement.cpp, testImageFilter.cpp, testMatrix.cpp, testPoseVector.cpp, testTranslationVector.cpp, and tutorial-matlab.cpp.

Definition at line 84 of file vpArray2D.h.

Referenced by vpMatrix::AtA(), vpQuaternionVector::buildFrom(), vpHomogeneousMatrix::buildFrom(), vpRzyzVector::buildFrom(), vpThetaUVector::buildFrom(), vpRxyzVector::buildFrom(), buildFrom(), vpSubColVector::checkParentStatus(), vpSubRowVector::checkParentStatus(), vpSubMatrix::checkParentStatus(), vpColVector::clear(), vpHomogeneousMatrix::convert(), vpMatrix::detByLU(), vpTranslationVector::euclideanNorm(), vpRowVector::euclideanNorm(), vpMatrix::euclideanNorm(), vpMatrix::expm(), vpThetaUVector::extract(), vpMatrix::getRow(), vpThetaUVector::getTheta(), vpThetaUVector::getU(), vpColVector::hadamard(), vpMatrix::hadamard(), vpSubColVector::init(), vpSubRowVector::init(), vpSubMatrix::init(), vpColVector::insert(), vpMatrix::insert(), vpMatrix::inverseByCholesky(), vpTranslationVector::operator*(), vpRowVector::operator*(), vpColVector::operator*(), vpHomography::operator*(), vpMatrix::operator*(), vpTranslationVector::operator-(), vpRowVector::operator-(), vpColVector::operator-(), vpTranslationVector::operator/(), vpRowVector::operator/(), vpColVector::operator/(), vpHomography::operator/(), vpHomography::operator/=(), vpSubColVector::operator=(), vpSubRowVector::operator=(), vpQuaternionVector::operator=(), vpTranslationVector::operator=(), vpRowVector::operator=(), vpRzyzVector::operator=(), vpRxyzVector::operator=(), operator=(), vpColVector::operator=(), vpMatrix::operator=(), vpThetaUVector::operator=(), vpColVector::operator==(), vpColVector::operator[](), vpMatrix::qr(), vpMatrix::qrPivot(), vpRowVector::reshape(), vpColVector::reshape(), vpQuaternionVector::set(), vpMatrix::stack(), vpMatrix::stackRows(), vpColVector::sum(), vpColVector::sumSquare(), vpRotationVector::t(), vpTranslationVector::t(), vpPoseVector::t(), vpRowVector::t(), vpColVector::t(), vpRotationVector::toStdVector(), vpRowVector::toStdVector(), vpColVector::toStdVector(), vpColVector::vpColVector(), vpMatrix::vpMatrix(), vpQuaternionVector::w(), vpQuaternionVector::x(), vpQuaternionVector::y(), vpQuaternionVector::z(), vpSubColVector::~vpSubColVector(), vpSubMatrix::~vpSubMatrix(), and vpSubRowVector::~vpSubRowVector().

unsigned int vpArray2D< double >::rowNum
protectedinherited

Number of rows in the array.

Definition at line 74 of file vpArray2D.h.

Referenced by vpMatrix::AAt(), vpMatrix::AtA(), vpColVector::clear(), vpMatrix::detByLU(), vpMatrix::diag(), vpMatrix::eigenValues(), vpMatrix::expm(), vpColVector::extract(), vpMatrix::eye(), vpColVector::hadamard(), vpMatrix::hadamard(), vpColVector::infinityNorm(), vpMatrix::infinityNorm(), vpSubColVector::init(), vpSubRowVector::init(), vpSubMatrix::init(), vpMatrix::insert(), vpMatrix::inverseByCholesky(), vpMatrix::inverseTriangular(), vpRotationMatrix::operator*(), vpTranslationVector::operator*(), vpHomogeneousMatrix::operator*(), vpColVector::operator*(), vpMatrix::operator*(), vpRotationMatrix::operator*=(), vpTranslationVector::operator*=(), vpColVector::operator*=(), vpMatrix::operator*=(), vpColVector::operator+(), vpColVector::operator+=(), vpMatrix::operator+=(), vpColVector::operator-(), vpColVector::operator-=(), vpMatrix::operator-=(), vpColVector::operator/(), vpMatrix::operator/(), vpTranslationVector::operator/=(), vpColVector::operator/=(), vpMatrix::operator/=(), vpMatrix::operator<<(), vpColVector::operator<<(), vpSubColVector::operator=(), vpSubRowVector::operator=(), vpSubMatrix::operator=(), vpTranslationVector::operator=(), vpRowVector::operator=(), vpColVector::operator=(), vpMatrix::operator=(), vpColVector::operator==(), vpMatrix::qr(), vpMatrix::qrPivot(), vpColVector::reshape(), vpMatrix::setIdentity(), vpMatrix::stack(), vpColVector::stack(), vpMatrix::stackColumns(), vpMatrix::stackRows(), vpColVector::sum(), vpMatrix::sum(), vpRotationVector::sumSquare(), vpTranslationVector::sumSquare(), vpColVector::sumSquare(), vpMatrix::sumSquare(), vpTranslationVector::t(), vpPoseVector::t(), vpColVector::t(), vpMatrix::t(), vpMatrix::transpose(), vpColVector::vpColVector(), and vpMatrix::vpMatrix().