# 14.2.8. Homogeneous Transformation Matrices and Quaternions — `MDAnalysis.lib.transformations`

A library for calculating 4x4 matrices for translating, rotating, reflecting, scaling, shearing, projecting, orthogonalizing, and superimposing arrays of 3D homogeneous coordinates as well as for converting between rotation matrices, Euler angles, and quaternions. Also includes an Arcball control object and functions to decompose transformation matrices.

Authors:

  Christoph Gohlke, Laboratory for Fluorescence Dynamics, University of California, Irvine

Version:

  2010.05.10

Licence:

  BSD 3-clause

## 14.2.8.1. Requirements

- Python 2.6 or 3.1

- Numpy 1.4

- transformations.c 2010.04.10 (optional implementation of some functions in C)

Notes

The API is not stable yet and is expected to change between revisions.

This Python code is not optimized for speed. Refer to the transformations.c module for a faster implementation of some functions.

Documentation in HTML format can be generated with epydoc.

Matrices (M) can be inverted using `numpy.linalg.inv(M)`, concatenated using `numpy.dot(M0, M1)`, or used to transform homogeneous coordinates (v) using `numpy.dot(M, v)` for shape `(4, *)` “point of arrays”, respectively `numpy.dot(v, M.T)` for shape `(*, 4)` “array of points”.

Use the transpose of transformation matrices for OpenGL `glMultMatrixd()`.

Calculations are carried out with `numpy.float64` precision.

Vector, point, quaternion, and matrix function arguments are expected to be “array like”, i.e. tuple, list, or numpy arrays.

Return types are numpy arrays unless specified otherwise.

Angles are in radians unless specified otherwise.

Quaternions w+ix+jy+kz are represented as `[w, x, y, z]`.

A triple of Euler angles can be applied/interpreted in 24 ways, which can be specified using a 4 character string or encoded 4-tuple:

> - Axes 4-string: e.g. ‘sxyz’ or ‘ryxy’

>   - first character : rotations are applied to ‘s’tatic or ‘r’otating frame

>   - remaining characters : successive rotation axis ‘x’, ‘y’, or ‘z’

> - Axes 4-tuple: e.g. (0, 0, 0, 0) or (1, 1, 1, 1)

>   - inner axis: code of axis (‘x’:0, ‘y’:1, ‘z’:2) of rightmost matrix.

>   - parity : even (0) if inner axis ‘x’ is followed by ‘y’, ‘y’ is followed by ‘z’, or ‘z’ is followed by ‘x’. Otherwise odd (1).

>   - repetition : first and last axis are same (1) or different (0).

>   - frame : rotations are applied to static (0) or rotating (1) frame.

References

Examples

    >>> from MDAnalysis.lib.transformations import *
    >>> import numpy as np
    >>> alpha, beta, gamma = 0.123, -1.234, 2.345
    >>> origin, xaxis, yaxis, zaxis = (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)
    >>> I = identity_matrix()
    >>> Rx = rotation_matrix(alpha, xaxis)
    >>> Ry = rotation_matrix(beta, yaxis)
    >>> Rz = rotation_matrix(gamma, zaxis)
    >>> R = concatenate_matrices(Rx, Ry, Rz)
    >>> euler = euler_from_matrix(R, 'rxyz')
    >>> np.allclose([alpha, beta, gamma], euler)
    True
    >>> Re = euler_matrix(alpha, beta, gamma, 'rxyz')
    >>> is_same_transform(R, Re)
    True
    >>> al, be, ga = euler_from_matrix(Re, 'rxyz')
    >>> is_same_transform(Re, euler_matrix(al, be, ga, 'rxyz'))
    True
    >>> qx = quaternion_about_axis(alpha, xaxis)
    >>> qy = quaternion_about_axis(beta, yaxis)
    >>> qz = quaternion_about_axis(gamma, zaxis)
    >>> q = quaternion_multiply(qx, qy)
    >>> q = quaternion_multiply(q, qz)
    >>> Rq = quaternion_matrix(q)
    >>> is_same_transform(R, Rq)
    True
    >>> S = scale_matrix(1.23, origin)
    >>> T = translation_matrix((1, 2, 3))
    >>> Z = shear_matrix(beta, xaxis, origin, zaxis)
    >>> R = random_rotation_matrix(np.random.rand(3))
    >>> M = concatenate_matrices(T, R, Z, S)
    >>> scale, shear, angles, trans, persp = decompose_matrix(M)
    >>> np.allclose(scale, 1.23)
    True
    >>> np.allclose(trans, (1, 2, 3))
    True
    >>> np.allclose(shear, (0, math.tan(beta), 0))
    True
    >>> is_same_transform(R, euler_matrix(axes='sxyz', *angles))
    True
    >>> M1 = compose_matrix(scale, shear, angles, trans, persp)
    >>> is_same_transform(M, M1)
    True

## 14.2.8.2. Functions

Changed in version 0.11.0: Transformations library moved from MDAnalysis.core.transformations to MDAnalysis.lib.transformations

class MDAnalysis.lib.transformations.Arcball(initial=None)

  Virtual Trackball Control.

      >>> from MDAnalysis.lib.transformations import Arcball
      >>> import numpy as np
      >>> ball = Arcball()
      >>> ball = Arcball(initial=np.identity(4))
      >>> ball.place([320, 320], 320)
      >>> ball.down([500, 250])
      >>> ball.drag([475, 275])
      >>> R = ball.matrix()
      >>> np.allclose(np.sum(R), 3.90583455)
      True
      >>> ball = Arcball(initial=[1, 0, 0, 0])
      >>> ball.place([320, 320], 320)
      >>> ball.setaxes([1,1,0], [-1, 1, 0])
      >>> ball.setconstrain(True)
      >>> ball.down([400, 200])
      >>> ball.drag([200, 400])
      >>> R = ball.matrix()
      >>> np.allclose(np.sum(R), 0.2055924)
      True
      >>> ball.next()

  Initialize virtual trackball control.

  initial : quaternion or rotation matrix

  down(point)

    Set initial cursor window coordinates and pick constrain-axis.

  drag(point)

    Update current cursor window coordinates.

  getconstrain()

    Return state of constrain to axis mode.

  matrix()

    Return homogeneous rotation matrix.

  next(acceleration=0.0)

    Continue rotation in direction of last drag.

  place(center, radius)

    Place Arcball, e.g. when window size changes.

    centersequence[2]

      Window coordinates of trackball center.

    radiusfloat

      Radius of trackball in window coordinates.

  setaxes(*axes)

    Set axes to constrain rotations.

  setconstrain(constrain)

    Set state of constrain to axis mode.

MDAnalysis.lib.transformations.arcball_nearest_axis(point, axes)

  Return axis, which arc is nearest to point.

MDAnalysis.lib.transformations.compose_matrix(scale=None, shear=None, angles=None, translate=None, perspective=None)

  Return transformation matrix from sequence of transformations.

  This is the inverse of the decompose_matrix function.

  Sequence of transformations:

    scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix

      >>> from MDAnalysis.lib.transformations import (compose_matrix,
      ...     decompose_matrix, is_same_transform)
      >>> import math
      >>> import numpy as np
      >>> scale = np.random.random(3) - 0.5
      >>> shear = np.random.random(3) - 0.5
      >>> angles = (np.random.random(3) - 0.5) * (2*math.pi)
      >>> trans = np.random.random(3) - 0.5
      >>> persp = np.random.random(4) - 0.5
      >>> M0 = compose_matrix(scale, shear, angles, trans, persp)
      >>> result = decompose_matrix(M0)
      >>> M1 = compose_matrix(*result)
      >>> is_same_transform(M0, M1)
      True

MDAnalysis.lib.transformations.concatenate_matrices(*matrices)

  Return concatenation of series of transformation matrices.

      >>> from MDAnalysis.lib.transformations import concatenate_matrices
      >>> import numpy as np
      >>> M = np.random.rand(16).reshape((4, 4)) - 0.5
      >>> np.allclose(M, concatenate_matrices(M))
      True
      >>> np.allclose(np.dot(M, M.T), concatenate_matrices(M, M.T))
      True

MDAnalysis.lib.transformations.decompose_matrix(matrix)

  Return sequence of transformations from transformation matrix.

  matrixarray_like

    Non-degenerative homogeneous transformation matrix

  Return tuple of:

    scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix

  Raise ValueError if matrix is of wrong type or degenerative.

      >>> from MDAnalysis.lib.transformations import (translation_matrix,
      ...     decompose_matrix, scale_matrix, euler_matrix)
      >>> import numpy as np
      >>> T0 = translation_matrix((1, 2, 3))
      >>> scale, shear, angles, trans, persp = decompose_matrix(T0)
      >>> T1 = translation_matrix(trans)
      >>> np.allclose(T0, T1)
      True
      >>> S = scale_matrix(0.123)
      >>> scale, shear, angles, trans, persp = decompose_matrix(S)
      >>> scale[0]
      0.123
      >>> R0 = euler_matrix(1, 2, 3)
      >>> scale, shear, angles, trans, persp = decompose_matrix(R0)
      >>> R1 = euler_matrix(*angles)
      >>> np.allclose(R0, R1)
      True

MDAnalysis.lib.transformations.euler_from_quaternion(quaternion, axes='sxyz')

  Return Euler angles from quaternion for specified axis sequence.

      >>> from MDAnalysis.lib.transformations import euler_from_quaternion
      >>> import numpy as np
      >>> angles = euler_from_quaternion([0.99810947, 0.06146124, 0, 0])
      >>> np.allclose(angles, [0.123, 0, 0])
      True

MDAnalysis.lib.transformations.projection_from_matrix(matrix, pseudo=False)

  Return projection plane and perspective point from projection matrix.

  Return values are same as arguments for projection_matrix function: point, normal, direction, perspective, and pseudo.

      >>> from MDAnalysis.lib.transformations import (projection_matrix,
      ...     projection_from_matrix, is_same_transform)
      >>> import numpy as np
      >>> point = np.random.random(3) - 0.5
      >>> normal = np.random.random(3) - 0.5
      >>> direct = np.random.random(3) - 0.5
      >>> persp = np.random.random(3) - 0.5
      >>> P0 = projection_matrix(point, normal)
      >>> result = projection_from_matrix(P0)
      >>> P1 = projection_matrix(*result)
      >>> is_same_transform(P0, P1)
      True
      >>> P0 = projection_matrix(point, normal, direct)
      >>> result = projection_from_matrix(P0)
      >>> P1 = projection_matrix(*result)
      >>> is_same_transform(P0, P1)
      True
      >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=False)
      >>> result = projection_from_matrix(P0, pseudo=False)
      >>> P1 = projection_matrix(*result)
      >>> is_same_transform(P0, P1)
      True
      >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=True)
      >>> result = projection_from_matrix(P0, pseudo=True)
      >>> P1 = projection_matrix(*result)
      >>> is_same_transform(P0, P1)
      True

MDAnalysis.lib.transformations.quaternion_imag(quaternion)

  Return imaginary part of quaternion.

      >>> from MDAnalysis.lib.transformations import quaternion_imag
      >>> quaternion_imag([3.0, 0.0, 1.0, 2.0])
      [0.0, 1.0, 2.0]

MDAnalysis.lib.transformations.quaternion_real(quaternion)

  Return real part of quaternion.

      >>> from MDAnalysis.lib.transformations import quaternion_real
      >>> quaternion_real([3.0, 0.0, 1.0, 2.0])
      3.0

MDAnalysis.lib.transformations.reflection_from_matrix(matrix)

  Return mirror plane point and normal vector from reflection matrix.

      >>> from MDAnalysis.lib.transformations import (reflection_matrix,
      ...     reflection_from_matrix, is_same_transform)
      >>> import numpy as np
      >>> v0 = np.random.random(3) - 0.5
      >>> v1 = np.random.random(3) - 0.5
      >>> M0 = reflection_matrix(v0, v1)
      >>> point, normal = reflection_from_matrix(M0)
      >>> M1 = reflection_matrix(point, normal)
      >>> is_same_transform(M0, M1)
      True

MDAnalysis.lib.transformations.rotation_from_matrix(matrix)

  Return rotation angle and axis from rotation matrix.

      >>> from MDAnalysis.lib.transformations import (rotation_matrix,
      ...     is_same_transform, rotation_from_matrix)
      >>> import random, math
      >>> import numpy as np
      >>> angle = (random.random() - 0.5) * (2*math.pi)
      >>> direc = np.random.random(3) - 0.5
      >>> point = np.random.random(3) - 0.5
      >>> R0 = rotation_matrix(angle, direc, point)
      >>> angle, direc, point = rotation_from_matrix(R0)
      >>> R1 = rotation_matrix(angle, direc, point)
      >>> is_same_transform(R0, R1)
      True

MDAnalysis.lib.transformations.rotaxis(a, b)

  Return the rotation axis to rotate vector a into b.

  Parameters:

    - a (array_like) – two vectors

    - b (array_like) – two vectors

  Returns:

    c – vector to rotate a into b

  Return type:

    np.ndarray

  Note

  If a == b this will always return [1, 0, 0]

MDAnalysis.lib.transformations.scale_from_matrix(matrix)

  Return scaling factor, origin and direction from scaling matrix.

      >>> from MDAnalysis.lib.transformations import (scale_matrix,
      ...      scale_from_matrix, is_same_transform)
      >>> import random
      >>> import numpy as np
      >>> factor = random.random() * 10 - 5
      >>> origin = np.random.random(3) - 0.5
      >>> direct = np.random.random(3) - 0.5
      >>> S0 = scale_matrix(factor, origin)
      >>> factor, origin, direction = scale_from_matrix(S0)
      >>> S1 = scale_matrix(factor, origin, direction)
      >>> is_same_transform(S0, S1)
      True
      >>> S0 = scale_matrix(factor, origin, direct)
      >>> factor, origin, direction = scale_from_matrix(S0)
      >>> S1 = scale_matrix(factor, origin, direction)
      >>> is_same_transform(S0, S1)
      True

MDAnalysis.lib.transformations.shear_from_matrix(matrix)

  Return shear angle, direction and plane from shear matrix.

      >>> from MDAnalysis.lib.transformations import (shear_matrix,
      ...     shear_from_matrix, is_same_transform)
      >>> import random, math
      >>> import numpy as np
      >>> angle = (random.random() - 0.5) * 4*math.pi
      >>> direct = np.random.random(3) - 0.5
      >>> point = np.random.random(3) - 0.5
      >>> normal = np.cross(direct, np.random.random(3))
      >>> S0 = shear_matrix(angle, direct, point, normal)
      >>> angle, direct, point, normal = shear_from_matrix(S0)
      >>> S1 = shear_matrix(angle, direct, point, normal)
      >>> is_same_transform(S0, S1)
      True

MDAnalysis.lib.transformations.translation_from_matrix(matrix)

  Return translation vector from translation matrix.

      >>> from MDAnalysis.lib.transformations import (translation_matrix,
      ...     translation_from_matrix)
      >>> import numpy as np
      >>> v0 = np.random.random(3) - 0.5
      >>> v1 = translation_from_matrix(translation_matrix(v0))
      >>> np.allclose(v0, v1)
      True