ThetaUVector¶

class ThetaUVector(*args, **kwargs)¶

Bases: RotationVector

Implementation of a rotation vector as \(\theta {\bf u}\) axis-angle minimal representation.

Class that consider the case of the \(\theta {\bf u}\) parameterization for the rotation.

The vpThetaUVector class is derived from vpRotationVector .

The \(\theta {\bf u}\) representation is one of the minimal representation of a rotation matrix, where \({\bf u} = (u_{x} \; u_{y} \; u_{z})^{\top}\) is a unit vector representing the rotation axis and \(\theta\) is the rotation angle.

From the \(\theta {\bf u}\) representation it is possible to build the rotation matrix \({\bf R}\) using the Rodrigues formula:

\[{\bf R} = {\bf I}_{3} + (1 - \cos{ \theta}) \; {\bf u u}^{\top} + \sin{ \theta} \; [{\bf u}]_{\times} \]

with \({\bf I}_{3}\) the identity matrix of dimension \(3\times3\) and \([{\bf u}]_{\times}\) the skew matrix:

\[\begin{split}[{\bf u}]_{\times} = \left( \begin{array}{ccc} 0 & -u_{z} & u_{y} \\u_{z} & 0 & -u_{x} \\-u_{y} & u_{x} & 0 \end{array} \right) \end{split}\]

From the implementation point of view, it is nothing more than an array of three floats with values in [rad].

You can set values [rad] accessing each element:

vpThetaUVector tu;
tu[0] = M_PI_4;
tu[1] = M_PI_2;
tu[2] = M_PI;

You can also initialize the vector using operator<<(double) :

tu << M_PI_4, M_PI_2, M_PI;

Or you can also initialize the vector from a list of doubles if ViSP is build with c++11 enabled:

tu = {M_PI_4, M_PI_2, M_PI};

To get the values [rad] use:

double tux = tu[0];
double tuy = tu[1];
double tuz = tu[2];

The code below shows first how to initialize a \(\theta {\bf u}\) vector, than how to construct a rotation matrix from a vpThetaUVector and finally how to extract the theta U angles from the build rotation matrix.

#include <iostream>
#include <visp3/core/vpMath.h>
#include <visp3/core/vpRotationMatrix.h>
#include <visp3/core/vpThetaUVector.h>

int main()
{
  vpThetaUVector tu;

  // Initialise the theta U rotation vector
  tu[0] = vpMath::rad( 45.f);
  tu[1] = vpMath::rad(-30.f);
  tu[2] = vpMath::rad( 90.f);

  // Construct a rotation matrix from the theta U angles
  vpRotationMatrix R(tu);

  // Extract the theta U angles from a rotation matrix
  tu.buildFrom(R);

  // Print the extracted theta U angles. Values are the same than the
  // one used for initialization
  std::cout << tu;

  // Since the rotation vector is 3 values column vector, the
  // transpose operation produce a row vector.
  vpRowVector tu_t = tu.t();

  // Print the transpose row vector
  std::cout << tu_t << std::endl;
}

Overloaded function.

__init__(self: visp._visp.core.ThetaUVector) -> None

Default constructor that initialize all the 3 angles to zero.

__init__(self: visp._visp.core.ThetaUVector, tu: visp._visp.core.ThetaUVector) -> None

Copy constructor.

__init__(self: visp._visp.core.ThetaUVector, M: visp._visp.core.HomogeneousMatrix) -> None

Initialize a \(\theta {\bf u}\) vector from an homogeneous matrix.

__init__(self: visp._visp.core.ThetaUVector, p: visp._visp.core.PoseVector) -> None

Initialize a \(\theta {\bf u}\) vector from a pose vector.

__init__(self: visp._visp.core.ThetaUVector, R: visp._visp.core.RotationMatrix) -> None

Initialize a \(\theta {\bf u}\) vector from a rotation matrix.

__init__(self: visp._visp.core.ThetaUVector, rzyx: visp._visp.core.RzyxVector) -> None

Initialize a \(\theta {\bf u}\) vector from an Euler z-y-x representation vector.

__init__(self: visp._visp.core.ThetaUVector, rzyz: visp._visp.core.RzyzVector) -> None

Initialize a \(\theta {\bf u}\) vector from an Euler z-y-z representation vector.

__init__(self: visp._visp.core.ThetaUVector, rxyz: visp._visp.core.RxyzVector) -> None

Initialize a \(\theta {\bf u}\) vector from an Euler x-y-z representation vector.

__init__(self: visp._visp.core.ThetaUVector, q: visp._visp.core.QuaternionVector) -> None

Initialize a \(\theta {\bf u}\) vector from a quaternion representation vector.

__init__(self: visp._visp.core.ThetaUVector, tu: visp._visp.core.ColVector) -> None

Copy constructor from a 3-dimension vector.

__init__(self: visp._visp.core.ThetaUVector, tu: list[float]) -> None

Build a \(\theta {\bf u}\) vector from a vector of 3 angles in radian.

__init__(self: visp._visp.core.ThetaUVector, tux: float, tuy: float, tuz: float) -> None

Build a \(\theta {\bf u}\) vector from 3 angles in radians.

#include <visp3/core/vpThetaUVector.cpp>

int main()
{
  vpThetaUVector tu(0, M_PI_2, M_PI);
  std::cout << "tu: " << tu.t() << std::endl;
}

It produces the following printings:

tu: 0  1.570796327  3.141592654

Methods

`__init__`	Overloaded function.
`buildFrom`	Overloaded function.
`extract`	Extract the rotation angle \(\theta\) and the unit vector \(\bf u\) from the \(\theta {\bf u}\) representation.
`getTheta`	Get the rotation angle \(\theta\) from the \(\theta {\bf u}\) representation.
`getU`	Get the unit vector \(\bf u\) from the \(\theta {\bf u}\) representation.

Inherited Methods

`insert`	Overloaded function.
`getCols`	Return the number of columns of the 2D array.
`insertStatic`	Insert array B in array A at the given position.
`toStdVector`	return: The corresponding std::vector<double>.
`save`	Overloaded function.
`reshape`
`getRows`	Return the number of rows of the 2D array.
`size`	Return the number of elements of the 2D array.
`getMinValue`	Return the array min value.
`sumSquare`	Return the sum square of all the elements \(r_{i}\) of the rotation vector r(m).
`getMaxValue`	Return the array max value.
`hadamard`	param m: Second matrix;
`saveYAML`	Save an array in a YAML-formatted file.
`conv2`	Overloaded function.
`t`	Overloaded function.
`numpy`	Numpy view of the underlying array data.
`resize`	Set the size of the array and initialize all the values to zero.

Operators

`__doc__`
`__init__`	Overloaded function.
`__module__`
`__mul__`	Perform rotation chaining / rotation multiplication using the theta.u rotation representation.

Attributes

`__annotations__`
`__hash__`

__eq__(*args, **kwargs)¶

Overloaded function.

__eq__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Equal to comparison operator of a 2D array.

__eq__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Equal to comparison operator of a 2D array.

__eq__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Equal to comparison operator of a 2D array.

__getitem__(*args, **kwargs)¶

Overloaded function.

__getitem__(self: visp._visp.core.ArrayDouble2D, arg0: tuple[int, int]) -> float
__getitem__(self: visp._visp.core.ArrayDouble2D, arg0: int) -> numpy.ndarray[numpy.float64]
__getitem__(self: visp._visp.core.ArrayDouble2D, arg0: slice) -> numpy.ndarray[numpy.float64]
__getitem__(self: visp._visp.core.ArrayDouble2D, arg0: tuple) -> numpy.ndarray[numpy.float64]

__init__(*args, **kwargs)¶

Overloaded function.

__init__(self: visp._visp.core.ThetaUVector) -> None

Default constructor that initialize all the 3 angles to zero.

__init__(self: visp._visp.core.ThetaUVector, tu: visp._visp.core.ThetaUVector) -> None

Copy constructor.

__init__(self: visp._visp.core.ThetaUVector, M: visp._visp.core.HomogeneousMatrix) -> None

Initialize a \(\theta {\bf u}\) vector from an homogeneous matrix.

__init__(self: visp._visp.core.ThetaUVector, p: visp._visp.core.PoseVector) -> None

Initialize a \(\theta {\bf u}\) vector from a pose vector.

__init__(self: visp._visp.core.ThetaUVector, R: visp._visp.core.RotationMatrix) -> None

Initialize a \(\theta {\bf u}\) vector from a rotation matrix.

__init__(self: visp._visp.core.ThetaUVector, rzyx: visp._visp.core.RzyxVector) -> None

Initialize a \(\theta {\bf u}\) vector from an Euler z-y-x representation vector.

__init__(self: visp._visp.core.ThetaUVector, rzyz: visp._visp.core.RzyzVector) -> None

Initialize a \(\theta {\bf u}\) vector from an Euler z-y-z representation vector.

__init__(self: visp._visp.core.ThetaUVector, rxyz: visp._visp.core.RxyzVector) -> None

Initialize a \(\theta {\bf u}\) vector from an Euler x-y-z representation vector.

__init__(self: visp._visp.core.ThetaUVector, q: visp._visp.core.QuaternionVector) -> None

Initialize a \(\theta {\bf u}\) vector from a quaternion representation vector.

__init__(self: visp._visp.core.ThetaUVector, tu: visp._visp.core.ColVector) -> None

Copy constructor from a 3-dimension vector.

__init__(self: visp._visp.core.ThetaUVector, tu: list[float]) -> None

Build a \(\theta {\bf u}\) vector from a vector of 3 angles in radian.

__init__(self: visp._visp.core.ThetaUVector, tux: float, tuy: float, tuz: float) -> None

Build a \(\theta {\bf u}\) vector from 3 angles in radians.

#include <visp3/core/vpThetaUVector.cpp>

int main()
{
  vpThetaUVector tu(0, M_PI_2, M_PI);
  std::cout << "tu: " << tu.t() << std::endl;
}

It produces the following printings:

tu: 0  1.570796327  3.141592654

__mul__(self, tu_b: visp._visp.core.ThetaUVector) → visp._visp.core.ThetaUVector¶: Perform rotation chaining / rotation multiplication using the theta.u rotation representation. See: this answer for some details about the maths.

__ne__(*args, **kwargs)¶

Overloaded function.

__ne__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Not equal to comparison operator of a 2D array.

__ne__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Not equal to comparison operator of a 2D array.

__ne__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Not equal to comparison operator of a 2D array.

buildFrom(*args, **kwargs)¶

Overloaded function.

buildFrom(self: visp._visp.core.ThetaUVector, M: visp._visp.core.HomogeneousMatrix) -> visp._visp.core.ThetaUVector

Converts an homogeneous matrix into a \(\theta {\bf u}\) vector.

buildFrom(self: visp._visp.core.ThetaUVector, p: visp._visp.core.PoseVector) -> visp._visp.core.ThetaUVector

Converts a pose vector into a \(\theta {\bf u}\) vector copying the \(\theta {\bf u}\) values contained in the pose vector.

buildFrom(self: visp._visp.core.ThetaUVector, R: visp._visp.core.RotationMatrix) -> visp._visp.core.ThetaUVector

Converts a rotation matrix into a \(\theta {\bf u}\) vector.

buildFrom(self: visp._visp.core.ThetaUVector, rzyx: visp._visp.core.RzyxVector) -> visp._visp.core.ThetaUVector

Build a \(\theta {\bf u}\) vector from an Euler z-y-x representation vector.

buildFrom(self: visp._visp.core.ThetaUVector, zyz: visp._visp.core.RzyzVector) -> visp._visp.core.ThetaUVector

Build a \(\theta {\bf u}\) vector from an Euler z-y-z representation vector.

buildFrom(self: visp._visp.core.ThetaUVector, xyz: visp._visp.core.RxyzVector) -> visp._visp.core.ThetaUVector

Build a \(\theta {\bf u}\) vector from an Euler x-y-z representation vector.

buildFrom(self: visp._visp.core.ThetaUVector, q: visp._visp.core.QuaternionVector) -> visp._visp.core.ThetaUVector

Build a \(\theta {\bf u}\) vector from a quaternion representation vector.

buildFrom(self: visp._visp.core.ThetaUVector, tu: visp._visp.core.ColVector) -> visp._visp.core.ThetaUVector

Build a \(\theta {\bf u}\) vector from a 3-dim vector.

buildFrom(self: visp._visp.core.ThetaUVector, tu: list[float]) -> visp._visp.core.ThetaUVector

Build a \(\theta {\bf u}\) vector from a 3-dim vectors.

buildFrom(self: visp._visp.core.ThetaUVector, tux: float, tuy: float, tuz: float) -> None

Build a \(\theta {\bf u}\) vector from 3 angles in radian.

static conv2(*args, **kwargs)¶

Overloaded function.

conv2(M: visp._visp.core.ArrayDouble2D, kernel: visp._visp.core.ArrayDouble2D, mode: str) -> visp._visp.core.ArrayDouble2D

Perform a 2D convolution similar to Matlab conv2 function: \(M \star kernel\) .

Note

This is a very basic implementation that does not use FFT.

Parameters:

M: First matrix.
kernel: Second matrix.
mode: Convolution mode: “full” (default), “same”, “valid”.

conv2(M: visp._visp.core.ArrayDouble2D, kernel: visp._visp.core.ArrayDouble2D, res: visp._visp.core.ArrayDouble2D, mode: str) -> None

Perform a 2D convolution similar to Matlab conv2 function: \(M \star kernel\) .

Note

This is a very basic implementation that does not use FFT.

Parameters:

M: First array.
kernel: Second array.
res: Result.
mode: Convolution mode: “full” (default), “same”, “valid”.

extract(self) → tuple[float, visp._visp.core.ColVector]¶

Extract the rotation angle \(\theta\) and the unit vector \(\bf u\) from the \(\theta {\bf u}\) representation.

The following example shows how to use this function:

#include <visp3/core/vpThetaUVector.h>

int main()
{
  vpHomogeneousMatrix M(0, 0, 1., vpMath::rad(10), vpMath::rad(20), vpMath::rad(30));

  double theta;
  vpColVector u;
  M.getRotationMatrix().getThetaUVector().extract(theta, u);
  std::cout << "theta: " << theta << std::endl;
  std::cout << "u    : " << u.t() << std::endl;
}

Note

See getTheta() , getU()

Returns:

A tuple containing:

theta: Rotation angle \(\theta\) in rad.
u: 3-dim unit vector \({\bf u} = (u_{x},u_{y},u_{z})^{\top}\) representing the rotation axis.

getCols(self) → int¶: Return the number of columns of the 2D array.

Note

See getRows() , size()

getMaxValue(self) → float¶: Return the array max value.

getMinValue(self) → float¶: Return the array min value.

getRows(self) → int¶: Return the number of rows of the 2D array.

Note

See getCols() , size()

getTheta(self) → float¶

Get the rotation angle \(\theta\) from the \(\theta {\bf u}\) representation.

The following example shows how to use this function:

#include <visp3/core/vpThetaUVector.h>

int main()
{
  vpHomogeneousMatrix M(0, 0, 1., vpMath::rad(10), vpMath::rad(20), vpMath::rad(30));

  std::cout << "theta: " << M.getRotationMatrix().getThetaUVector().getTheta() << std::endl;
  std::cout << "u    : " << M.getRotationMatrix().getThetaUVector().getU().t() << std::endl;
}

Note

See getTheta() , extract()

Returns:: Rotation angle \(\theta\) in rad.

getU(self) → visp._visp.core.ColVector¶

Get the unit vector \(\bf u\) from the \(\theta {\bf u}\) representation.

The following example shows how to use this function:

#include <visp3/core/vpThetaUVector.h>

int main()
{
  vpHomogeneousMatrix M(0, 0, 1., vpMath::rad(10), vpMath::rad(20), pMath::rad(30));

  std::cout << "theta: " << M.getRotationMatrix().getThetaUVector().getTheta() << std::endl;
  std::cout << "u    : " << M.getRotationMatrix().getThetaUVector().getU().t() << std::endl;
}

Note

See getTheta() , extract()

Returns:: 3-dim unit vector \({\bf u} = (u_{x},u_{y},u_{z})^{\top}\) representing the rotation axis.

hadamard(self, m: visp._visp.core.ArrayDouble2D) → visp._visp.core.ArrayDouble2D¶

Parameters:

m: visp._visp.core.ArrayDouble2D¶: Second matrix;

Returns:

m1.hadamard(m2) The Hadamard product : \(m1 \circ m2 = (m1 \circ m2)_{i,j} = (m1)_{i,j} (m2)_{i,j}\)

insert(*args, **kwargs)¶

Overloaded function.

insert(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D, r: int, c: int) -> None

Insert array A at the given position in the current array.

Warning

Throw vpException::dimensionError if the dimensions of the matrices do not allow the operation.

Parameters:

A: The array to insert.
r: The index of the row to begin to insert data.
c: The index of the column to begin to insert data.

insert(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D, B: visp._visp.core.ArrayDouble2D, r: int, c: int) -> visp._visp.core.ArrayDouble2D

Insert array B in array A at the given position.

Warning

Throw exception if the sizes of the arrays do not allow the insertion.

Parameters:

A: Main array.
B: Array to insert.
r: Index of the row where to add the array.
c: Index of the column where to add the array.

Returns:

Array with B insert in A.

static insertStatic(A: visp._visp.core.ArrayDouble2D, B: visp._visp.core.ArrayDouble2D, C: visp._visp.core.ArrayDouble2D, r: int, c: int) → None¶

Insert array B in array A at the given position.

Warning

Throw exception if the sizes of the arrays do not allow the insertion.

Parameters:

A: visp._visp.core.ArrayDouble2D¶: Main array.
B: visp._visp.core.ArrayDouble2D¶: Array to insert.
C: visp._visp.core.ArrayDouble2D¶: Result array.
r: int¶: Index of the row where to insert array B.
c: int¶: Index of the column where to insert array B.

numpy(self) → numpy.ndarray[numpy.float64]¶: Numpy view of the underlying array data. This numpy view can be used to directly modify the array.

reshape(self, nrows: int, ncols: int) → None¶

resize(self, nrows: int, ncols: int, flagNullify: bool = true, recopy_: bool = true) → None¶

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

Parameters:

nrows: int¶: number of rows.
ncols: int¶: number of column.
flagNullify: bool = true¶: 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_: bool = true¶: if true, will perform an explicit recopy of the old data.

static save(*args, **kwargs)¶

Overloaded function.

save(filename: str, A: visp._visp.core.ArrayDouble2D, binary: bool = false, header: str = ) -> bool

Save a matrix to a file.

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

Note

See load()

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.

save(filename: str, A: visp._visp.core.ArrayDouble2D, binary: bool = false, header: str = ) -> bool

Save a matrix to a file.

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

Note

See load()

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.

save(filename: str, A: visp._visp.core.ArrayDouble2D, binary: bool = false, header: str = ) -> bool

Save a matrix to a file.

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

Note

See load()

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.

static saveYAML(filename: str, A: visp._visp.core.ArrayDouble2D, header: str =) → bool¶

Save an array in a YAML-formatted file.

Here is an example of outputs.

vpArray2D<double> M(3,4);
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
data:
  - [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
data:
    - [0, 0, 0, 0]
    - [0, 0, 0, 0]
    - [0, 0, 0, 0]

Note

See loadYAML()

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.

size(self) → int¶: Return the number of elements of the 2D array.

sumSquare(self) → float¶

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

Returns:: The value

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

t(*args, **kwargs)¶

Overloaded function.

t(self: visp._visp.core.RotationVector) -> visp._visp.core.RowVector

Return the transpose of the rotation vector.

t(self: visp._visp.core.ArrayDouble2D) -> visp._visp.core.ArrayDouble2D

Compute the transpose of the array.

Returns:: vpArray2D<Type> C = A^T

toStdVector(self) → list[float]¶

Returns:: The corresponding std::vector<double>.

__hash__ = None¶