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affine

esta.general.affine

Linear algebra operations and helpers.

Taken from the link (scikit-ued: https://github.com/LaurentRDC/scikit-ued/)

Inspired by Christoph Gohlke's transformation.py (link: http://www.lfd.uci.edu/~gohlke/ )

affine_map(array)

Extends 3x3 transform matrices to 4x4, i.e. general affine transforms.

Parameters:

  • array (ndarray, shape {(3,3), (4,4)}) –

    Transformation matrix. If shape = (4,4), returned intact.

Returns:

  • extended ( (ndarray, shape(4, 4)) ) –

    Extended array

    Raises

    ValueError : If the transformation matrix is neither 3x3 or 4x4

transform(matrix, array) 

Applies a matrix transform on an array.

Parameters:

  • matrix (ndarray, shape {(3,3), (4,4)}) –

    Transformation matrix.

  • array (ndarray, shape {(3,), (3,3), (4,4)}) –

    Array to be transformed. Either a 1x3 vector, or a transformation matrix in 3x3 or 4x4 shape.

Returns:

  • transformed ( ndarray ) –

    Transformed array, either a 1D vector or a 4x4 transformation matrix

    Raises

    ValueError : If the transformation matrix is neither 3x3 or 4x4

translation_matrix(direction)

Return matrix to translate by direction vector.

Parameters:

  • direction ((array_like, shape(3))) –

Returns:

  • translation ( `~numpy.ndarray`, shape (4,4) ) –

    4x4 translation matrix.

change_of_basis(basis1, basis2=(e1, e2, e3))

Returns the matrix that goes from one basis to the other.

Parameters:

  • basis1 (list of array_like, shape (3,)) – 
    First basis
  • basis2 (list of array_like, shape (3,), default: (e1, e2, e3) ) – 
    Second basis. By default, this is the standard basis

Returns:

  • cob ( `~numpy.ndarray`, shape (3,3) ) –

    Change-of-basis matrix that, applied to basis, will return basis2.

is_basis(basis)

Returns true if the set of vectors forms a basis. This is done by checking whether basis vectors are independent via an eigenvalue calculation.

Parameters:

  • basis (list of array-like, shape (3,)) –

Returns:

  • out ( bool ) –

    Whether or not the basis is valid.

is_rotation_matrix(matrix)

Checks whether a matrix is orthogonal with unit determinant (1 or -1), properties of rotation matrices.

Parameters:

  • matrix (ndarray, shape {(3,3), (4,4)}) –

    Rotation matrix candidate. If (4,4) matrix is provided, only the top-left block matrix of (3,) is checked

Returns:

  • result ( bool ) –

    If True, input could be a rotation matrix.

rotation_matrix(angle, axis=(0, 0, 1))

Return matrix to rotate about axis defined by direction around the origin [0,0,0]. To combine rotation and translations, see http://www.euclideanspace.com/maths/geometry/affine/matrix4x4/index.htm

Parameters:

  • angle (float) – 
    Rotation angle [rad]
  • axis (array-like of length 3, default: (0, 0, 1) ) – 
    Axis about which to rotate

Returns:

  • matrix ( `~numpy.ndarray`, shape (3,3) ) –

    Rotation matrix.

See also

translation_rotation_matrix

translation_rotation_matrix(angle, axis, translation)

Returns a 4x4 matrix that includes a rotation and a translation.

Parameters:

  • angle (float) – 
    Rotation angle [rad]
  • axis (array-like of length 3) – 
    Axis about which to rotate
  • translation ((array_like, shape(3))) – 
    Translation vector

Returns:

  • matrix ( `~numpy.ndarray`, shape (4,4) ) –

    Affine transform matrix.

change_basis_mesh(xx, yy, zz, basis1, basis2)

Changes the basis of meshgrid arrays.

Parameters:

  • xx (ndarrays) –

    Arrays of equal shape, such as produced by numpy.meshgrid.

  • yy (ndarrays) –

    Arrays of equal shape, such as produced by numpy.meshgrid.

  • zz (ndarrays) –

    Arrays of equal shape, such as produced by numpy.meshgrid.

  • basis1 (list of ndarrays, shape(3,)) –

    Basis of the mesh

  • basis2 (list of ndarrays, shape(3,)) –

    Basis in which to express the mesh

Returns:

  • XX, YY, ZZ : `~numpy.ndarray` –

minimum_image_distance(xx, yy, zz, lattice)

Returns a periodic array according to the minimum image convention.

Parameters:

  • xx (ndarrays) – 
    Arrays of equal shape, such as produced by numpy.meshgrid.
  • yy (ndarrays) – 
    Arrays of equal shape, such as produced by numpy.meshgrid.
  • zz (ndarrays) – 
    Arrays of equal shape, such as produced by numpy.meshgrid.
  • lattice (list of ndarrays, shape(3,)) – 
    Basis of the mesh

Returns:

  • r ( `~numpy.ndarray` ) –

    Minimum image distance over the lattice