# espressopp.external.transformations¶

Homogeneous Transformation Matrices and Quaternions.

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 2011.01.25

Requirements

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

1. Matrices and transformations. Ronald Goldman. In “Graphics Gems I”, pp 472-475. Morgan Kaufmann, 1990.
2. More matrices and transformations: shear and pseudo-perspective. Ronald Goldman. In “Graphics Gems II”, pp 320-323. Morgan Kaufmann, 1991.
3. Decomposing a matrix into simple transformations. Spencer Thomas. In “Graphics Gems II”, pp 320-323. Morgan Kaufmann, 1991.
4. Recovering the data from the transformation matrix. Ronald Goldman. In “Graphics Gems II”, pp 324-331. Morgan Kaufmann, 1991.
5. Euler angle conversion. Ken Shoemake. In “Graphics Gems IV”, pp 222-229. Morgan Kaufmann, 1994.
6. Arcball rotation control. Ken Shoemake. In “Graphics Gems IV”, pp 175-192. Morgan Kaufmann, 1994.
7. Representing attitude: Euler angles, unit quaternions, and rotation vectors. James Diebel. 2006.
8. A discussion of the solution for the best rotation to relate two sets of vectors. W Kabsch. Acta Cryst. 1978. A34, 827-828.
9. Closed-form solution of absolute orientation using unit quaternions. BKP Horn. J Opt Soc Am A. 1987. 4(4):629-642.
10. Quaternions. Ken Shoemake. http://www.sfu.ca/~jwa3/cmpt461/files/quatut.pdf
11. From quaternion to matrix and back. JMP van Waveren. 2005. http://www.intel.com/cd/ids/developer/asmo-na/eng/293748.htm
12. Uniform random rotations. Ken Shoemake. In “Graphics Gems III”, pp 124-132. Morgan Kaufmann, 1992.
13. Quaternion in molecular modeling. CFF Karney. J Mol Graph Mod, 25(5):595-604
14. New method for extracting the quaternion from a rotation matrix. Itzhack Y Bar-Itzhack, J Guid Contr Dynam. 2000. 23(6): 1085-1087.

Examples

>>> 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')
>>> numpy.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
>>> 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(numpy.random.rand(3))
>>> M = concatenate_matrices(T, R, Z, S)
>>> scale, shear, angles, trans, persp = decompose_matrix(M)
>>> numpy.allclose(scale, 1.23)
True
>>> numpy.allclose(trans, (1, 2, 3))
True
>>> numpy.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
>>> v0, v1 = random_vector(3), random_vector(3)
>>> M = rotation_matrix(angle_between_vectors(v0, v1), vector_product(v0, v1))
>>> v2 = numpy.dot(v0, M[:3,:3].T)
>>> numpy.allclose(unit_vector(v1), unit_vector(v2))
True

class espressopp.external.transformations.Arcball(initial=None)

Virtual Trackball Control.

>>> ball = Arcball()
>>> ball = Arcball(initial=numpy.identity(4))
>>> ball.place([320, 320], 320)
>>> ball.down([500, 250])
>>> ball.drag([475, 275])
>>> R = ball.matrix()
>>> numpy.allclose(numpy.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()
>>> numpy.allclose(numpy.sum(R), 0.2055924)
True
>>> ball.next()

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.

center
: sequence[2]
Window coordinates of trackball center.
: float
Radius of trackball in window coordinates.
setaxes(*axes)

Set axes to constrain rotations.

setconstrain(constrain)

Set state of constrain to axis mode.

espressopp.external.transformations.angle_between_vectors(v0, v1, directed=True, axis=0)

Return angle between vectors.

If directed is False, the input vectors are interpreted as undirected axes, i.e. the maximum angle is pi/2.

>>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3])
>>> numpy.allclose(a, math.pi)
True
>>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3], directed=False)
>>> numpy.allclose(a, 0)
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> a = angle_between_vectors(v0, v1)
>>> numpy.allclose(a, [0., 1.5708, 1.5708, 0.95532])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> a = angle_between_vectors(v0, v1, axis=1)
>>> numpy.allclose(a, [1.5708, 1.5708, 1.5708, 0.95532])
True

espressopp.external.transformations.arcball_constrain_to_axis(point, axis)

Return sphere point perpendicular to axis.

espressopp.external.transformations.arcball_map_to_sphere(point, center, radius)

Return unit sphere coordinates from window coordinates.

espressopp.external.transformations.arcball_nearest_axis(point, axes)

Return axis, which arc is nearest to point.

espressopp.external.transformations.clip_matrix(left, right, bottom, top, near, far, perspective=False)

Return matrix to obtain normalized device coordinates from frustrum.

The frustrum bounds are axis-aligned along x (left, right), y (bottom, top) and z (near, far).

Normalized device coordinates are in range [-1, 1] if coordinates are inside the frustrum.

If perspective is True the frustrum is a truncated pyramid with the perspective point at origin and direction along z axis, otherwise an orthographic canonical view volume (a box).

Homogeneous coordinates transformed by the perspective clip matrix need to be dehomogenized (devided by w coordinate).

>>> frustrum = numpy.random.rand(6)
>>> frustrum[1] += frustrum[0]
>>> frustrum[3] += frustrum[2]
>>> frustrum[5] += frustrum[4]
>>> M = clip_matrix(perspective=False, *frustrum)
>>> numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1.0])
array([-1., -1., -1.,  1.])
>>> numpy.dot(M, [frustrum[1], frustrum[3], frustrum[5], 1.0])
array([ 1.,  1.,  1.,  1.])
>>> M = clip_matrix(perspective=True, *frustrum)
>>> v = numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1.0])
>>> v / v[3]
array([-1., -1., -1.,  1.])
>>> v = numpy.dot(M, [frustrum[1], frustrum[3], frustrum[4], 1.0])
>>> v / v[3]
array([ 1.,  1., -1.,  1.])

espressopp.external.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
>>> scale = numpy.random.random(3) - 0.5
>>> shear = numpy.random.random(3) - 0.5
>>> angles = (numpy.random.random(3) - 0.5) * (2*math.pi)
>>> trans = numpy.random.random(3) - 0.5
>>> persp = numpy.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

espressopp.external.transformations.concatenate_matrices(*matrices)

Return concatenation of series of transformation matrices.

>>> M = numpy.random.rand(16).reshape((4, 4)) - 0.5
>>> numpy.allclose(M, concatenate_matrices(M))
True
>>> numpy.allclose(numpy.dot(M, M.T), concatenate_matrices(M, M.T))
True

espressopp.external.transformations.decompose_matrix(matrix)

Return sequence of transformations from transformation matrix.

matrix
: array_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.

>>> T0 = translation_matrix((1, 2, 3))
>>> scale, shear, angles, trans, persp = decompose_matrix(T0)
>>> T1 = translation_matrix(trans)
>>> numpy.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)
>>> numpy.allclose(R0, R1)
True

espressopp.external.transformations.euler_from_matrix(matrix, axes='sxyz')

Return Euler angles from rotation matrix for specified axis sequence.

axes : One of 24 axis sequences as string or encoded tuple

Note that many Euler angle triplets can describe one matrix.

>>> R0 = euler_matrix(1, 2, 3, 'syxz')
>>> al, be, ga = euler_from_matrix(R0, 'syxz')
>>> R1 = euler_matrix(al, be, ga, 'syxz')
>>> numpy.allclose(R0, R1)
True
>>> angles = (4.0*math.pi) * (numpy.random.random(3) - 0.5)
>>> for axes in _AXES2TUPLE.keys():
...    R0 = euler_matrix(axes=axes, *angles)
...    R1 = euler_matrix(axes=axes, *euler_from_matrix(R0, axes))
...    if not numpy.allclose(R0, R1): print(axes, "failed")

espressopp.external.transformations.euler_from_quaternion(quaternion, axes='sxyz')

Return Euler angles from quaternion for specified axis sequence.

>>> angles = euler_from_quaternion([0.99810947, 0.06146124, 0, 0])
>>> numpy.allclose(angles, [0.123, 0, 0])
True

espressopp.external.transformations.euler_matrix(ai, aj, ak, axes='sxyz')

Return homogeneous rotation matrix from Euler angles and axis sequence.

ai, aj, ak : Euler’s roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple

>>> R = euler_matrix(1, 2, 3, 'syxz')
>>> numpy.allclose(numpy.sum(R[0]), -1.34786452)
True
>>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1))
>>> numpy.allclose(numpy.sum(R[0]), -0.383436184)
True
>>> ai, aj, ak = (4.0*math.pi) * (numpy.random.random(3) - 0.5)
>>> for axes in _AXES2TUPLE.keys():
...    R = euler_matrix(ai, aj, ak, axes)
>>> for axes in _TUPLE2AXES.keys():
...    R = euler_matrix(ai, aj, ak, axes)

espressopp.external.transformations.identity_matrix()

Return 4x4 identity/unit matrix.

>>> I = identity_matrix()
>>> numpy.allclose(I, numpy.dot(I, I))
True
>>> numpy.sum(I), numpy.trace(I)
(4.0, 4.0)
>>> numpy.allclose(I, numpy.identity(4, dtype=numpy.float64))
True

espressopp.external.transformations.inverse_matrix(matrix)

Return inverse of square transformation matrix.

>>> M0 = random_rotation_matrix()
>>> M1 = inverse_matrix(M0.T)
>>> numpy.allclose(M1, numpy.linalg.inv(M0.T))
True
>>> for size in xrange(1, 7):
...     M0 = numpy.random.rand(size, size)
...     M1 = inverse_matrix(M0)
...     if not numpy.allclose(M1, numpy.linalg.inv(M0)): print(size)

espressopp.external.transformations.is_same_transform(matrix0, matrix1)

Return True if two matrices perform same transformation.

>>> is_same_transform(numpy.identity(4), numpy.identity(4))
True
>>> is_same_transform(numpy.identity(4), random_rotation_matrix())
False

espressopp.external.transformations.orthogonalization_matrix(lengths, angles)

Return orthogonalization matrix for crystallographic cell coordinates.

Angles are expected in degrees.

The de-orthogonalization matrix is the inverse.

>>> O = orthogonalization_matrix((10., 10., 10.), (90., 90., 90.))
>>> numpy.allclose(O[:3, :3], numpy.identity(3, float) * 10)
True
>>> O = orthogonalization_matrix([9.8, 12.0, 15.5], [87.2, 80.7, 69.7])
>>> numpy.allclose(numpy.sum(O), 43.063229)
True

espressopp.external.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.

>>> point = numpy.random.random(3) - 0.5
>>> normal = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> persp = numpy.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

espressopp.external.transformations.projection_matrix(point, normal, direction=None, perspective=None, pseudo=False)

Return matrix to project onto plane defined by point and normal.

Using either perspective point, projection direction, or none of both.

If pseudo is True, perspective projections will preserve relative depth such that Perspective = dot(Orthogonal, PseudoPerspective).

>>> P = projection_matrix((0, 0, 0), (1, 0, 0))
>>> numpy.allclose(P[1:, 1:], numpy.identity(4)[1:, 1:])
True
>>> point = numpy.random.random(3) - 0.5
>>> normal = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> persp = numpy.random.random(3) - 0.5
>>> P0 = projection_matrix(point, normal)
>>> P1 = projection_matrix(point, normal, direction=direct)
>>> P2 = projection_matrix(point, normal, perspective=persp)
>>> P3 = projection_matrix(point, normal, perspective=persp, pseudo=True)
>>> is_same_transform(P2, numpy.dot(P0, P3))
True
>>> P = projection_matrix((3, 0, 0), (1, 1, 0), (1, 0, 0))
>>> v0 = (numpy.random.rand(4, 5) - 0.5) * 20.0
>>> v0[3] = 1.0
>>> v1 = numpy.dot(P, v0)
>>> numpy.allclose(v1[1], v0[1])
True
>>> numpy.allclose(v1[0], 3.0-v1[1])
True

espressopp.external.transformations.quaternion_about_axis(angle, axis)

Return quaternion for rotation about axis.

>>> q = quaternion_about_axis(0.123, (1, 0, 0))
>>> numpy.allclose(q, [0.99810947, 0.06146124, 0, 0])
True

espressopp.external.transformations.quaternion_conjugate(quaternion)

Return conjugate of quaternion.

>>> q0 = random_quaternion()
>>> q1 = quaternion_conjugate(q0)
>>> q1[0] == q0[0] and all(q1[1:] == -q0[1:])
True

espressopp.external.transformations.quaternion_from_euler(ai, aj, ak, axes='sxyz')

Return quaternion from Euler angles and axis sequence.

ai, aj, ak : Euler’s roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple

>>> q = quaternion_from_euler(1, 2, 3, 'ryxz')
>>> numpy.allclose(q, [0.435953, 0.310622, -0.718287, 0.444435])
True

espressopp.external.transformations.quaternion_from_matrix(matrix, isprecise=False)

Return quaternion from rotation matrix.

If isprecise=True, the input matrix is assumed to be a precise rotation matrix and a faster algorithm is used.

>>> q = quaternion_from_matrix(identity_matrix(), True)
>>> numpy.allclose(q, [1., 0., 0., 0.])
True
>>> q = quaternion_from_matrix(numpy.diag([1., -1., -1., 1.]))
>>> numpy.allclose(q, [0, 1, 0, 0]) or numpy.allclose(q, [0, -1, 0, 0])
True
>>> R = rotation_matrix(0.123, (1, 2, 3))
>>> q = quaternion_from_matrix(R, True)
>>> numpy.allclose(q, [0.9981095, 0.0164262, 0.0328524, 0.0492786])
True
>>> R = [[-0.545, 0.797, 0.260, 0], [0.733, 0.603, -0.313, 0],
...      [-0.407, 0.021, -0.913, 0], [0, 0, 0, 1]]
>>> q = quaternion_from_matrix(R)
>>> numpy.allclose(q, [0.19069, 0.43736, 0.87485, -0.083611])
True
>>> R = [[0.395, 0.362, 0.843, 0], [-0.626, 0.796, -0.056, 0],
...      [-0.677, -0.498, 0.529, 0], [0, 0, 0, 1]]
>>> q = quaternion_from_matrix(R)
>>> numpy.allclose(q, [0.82336615, -0.13610694, 0.46344705, -0.29792603])
True
>>> R = random_rotation_matrix()
>>> q = quaternion_from_matrix(R)
>>> is_same_transform(R, quaternion_matrix(q))
True

espressopp.external.transformations.quaternion_imag(quaternion)

Return imaginary part of quaternion.

>>> quaternion_imag([3.0, 0.0, 1.0, 2.0])
[0.0, 1.0, 2.0]

espressopp.external.transformations.quaternion_inverse(quaternion)

Return inverse of quaternion.

>>> q0 = random_quaternion()
>>> q1 = quaternion_inverse(q0)
>>> numpy.allclose(quaternion_multiply(q0, q1), [1, 0, 0, 0])
True

espressopp.external.transformations.quaternion_matrix(quaternion)

Return homogeneous rotation matrix from quaternion.

>>> M = quaternion_matrix([0.99810947, 0.06146124, 0, 0])
>>> numpy.allclose(M, rotation_matrix(0.123, (1, 0, 0)))
True
>>> M = quaternion_matrix([1, 0, 0, 0])
>>> numpy.allclose(M, identity_matrix())
True
>>> M = quaternion_matrix([0, 1, 0, 0])
>>> numpy.allclose(M, numpy.diag([1, -1, -1, 1]))
True

espressopp.external.transformations.quaternion_multiply(quaternion1, quaternion0)

Return multiplication of two quaternions.

>>> q = quaternion_multiply([4, 1, -2, 3], [8, -5, 6, 7])
>>> numpy.allclose(q, [28, -44, -14, 48])
True

espressopp.external.transformations.quaternion_real(quaternion)

Return real part of quaternion.

>>> quaternion_real([3.0, 0.0, 1.0, 2.0])
3.0

espressopp.external.transformations.quaternion_slerp(quat0, quat1, fraction, spin=0, shortestpath=True)

Return spherical linear interpolation between two quaternions.

>>> q0 = random_quaternion()
>>> q1 = random_quaternion()
>>> q = quaternion_slerp(q0, q1, 0.0)
>>> numpy.allclose(q, q0)
True
>>> q = quaternion_slerp(q0, q1, 1.0, 1)
>>> numpy.allclose(q, q1)
True
>>> q = quaternion_slerp(q0, q1, 0.5)
>>> angle = math.acos(numpy.dot(q0, q))
>>> numpy.allclose(2.0, math.acos(numpy.dot(q0, q1)) / angle) or         numpy.allclose(2.0, math.acos(-numpy.dot(q0, q1)) / angle)
True

espressopp.external.transformations.random_quaternion(rand=None)

Return uniform random unit quaternion.

rand: array like or None
Three independent random variables that are uniformly distributed between 0 and 1.
>>> q = random_quaternion()
>>> numpy.allclose(1.0, vector_norm(q))
True
>>> q = random_quaternion(numpy.random.random(3))
>>> len(q.shape), q.shape[0]==4
(1, True)

espressopp.external.transformations.random_rotation_matrix(rand=None)

Return uniform random rotation matrix.

rnd: array like
Three independent random variables that are uniformly distributed between 0 and 1 for each returned quaternion.
>>> R = random_rotation_matrix()
>>> numpy.allclose(numpy.dot(R.T, R), numpy.identity(4))
True

espressopp.external.transformations.random_vector(size)

Return array of random doubles in the half-open interval [0.0, 1.0).

>>> v = random_vector(10000)
>>> numpy.all(v >= 0.0) and numpy.all(v < 1.0)
True
>>> v0 = random_vector(10)
>>> v1 = random_vector(10)
>>> numpy.any(v0 == v1)
False

espressopp.external.transformations.reflection_from_matrix(matrix)

Return mirror plane point and normal vector from reflection matrix.

>>> v0 = numpy.random.random(3) - 0.5
>>> v1 = numpy.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

espressopp.external.transformations.reflection_matrix(point, normal)

Return matrix to mirror at plane defined by point and normal vector.

>>> v0 = numpy.random.random(4) - 0.5
>>> v0[3] = 1.0
>>> v1 = numpy.random.random(3) - 0.5
>>> R = reflection_matrix(v0, v1)
>>> numpy.allclose(2., numpy.trace(R))
True
>>> numpy.allclose(v0, numpy.dot(R, v0))
True
>>> v2 = v0.copy()
>>> v2[:3] += v1
>>> v3 = v0.copy()
>>> v2[:3] -= v1
>>> numpy.allclose(v2, numpy.dot(R, v3))
True

espressopp.external.transformations.rotation_from_matrix(matrix)

Return rotation angle and axis from rotation matrix.

>>> angle = (random.random() - 0.5) * (2*math.pi)
>>> direc = numpy.random.random(3) - 0.5
>>> point = numpy.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

espressopp.external.transformations.rotation_matrix(angle, direction, point=None)

Return matrix to rotate about axis defined by point and direction.

>>> R = rotation_matrix(math.pi/2.0, [0, 0, 1], [1, 0, 0])
>>> numpy.allclose(numpy.dot(R, [0, 0, 0, 1]), [ 1., -1.,  0.,  1.])
True
>>> angle = (random.random() - 0.5) * (2*math.pi)
>>> direc = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> R0 = rotation_matrix(angle, direc, point)
>>> R1 = rotation_matrix(angle-2*math.pi, direc, point)
>>> is_same_transform(R0, R1)
True
>>> R0 = rotation_matrix(angle, direc, point)
>>> R1 = rotation_matrix(-angle, -direc, point)
>>> is_same_transform(R0, R1)
True
>>> I = numpy.identity(4, numpy.float64)
>>> numpy.allclose(I, rotation_matrix(math.pi*2, direc))
True
>>> numpy.allclose(2., numpy.trace(rotation_matrix(math.pi/2,
...                                                direc, point)))
True

espressopp.external.transformations.scale_from_matrix(matrix)

Return scaling factor, origin and direction from scaling matrix.

>>> factor = random.random() * 10 - 5
>>> origin = numpy.random.random(3) - 0.5
>>> direct = numpy.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

espressopp.external.transformations.scale_matrix(factor, origin=None, direction=None)

Return matrix to scale by factor around origin in direction.

Use factor -1 for point symmetry.

>>> v = (numpy.random.rand(4, 5) - 0.5) * 20.0
>>> v[3] = 1.0
>>> S = scale_matrix(-1.234)
>>> numpy.allclose(numpy.dot(S, v)[:3], -1.234*v[:3])
True
>>> factor = random.random() * 10 - 5
>>> origin = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> S = scale_matrix(factor, origin)
>>> S = scale_matrix(factor, origin, direct)

espressopp.external.transformations.shear_from_matrix(matrix)

Return shear angle, direction and plane from shear matrix.

>>> angle = (random.random() - 0.5) * 4*math.pi
>>> direct = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> normal = numpy.cross(direct, numpy.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

espressopp.external.transformations.shear_matrix(angle, direction, point, normal)

Return matrix to shear by angle along direction vector on shear plane.

The shear plane is defined by a point and normal vector. The direction vector must be orthogonal to the plane’s normal vector.

A point P is transformed by the shear matrix into P” such that the vector P-P” is parallel to the direction vector and its extent is given by the angle of P-P’-P”, where P’ is the orthogonal projection of P onto the shear plane.

>>> angle = (random.random() - 0.5) * 4*math.pi
>>> direct = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> normal = numpy.cross(direct, numpy.random.random(3))
>>> S = shear_matrix(angle, direct, point, normal)
>>> numpy.allclose(1.0, numpy.linalg.det(S))
True

espressopp.external.transformations.superimposition_matrix(v0, v1, scaling=False, usesvd=True)

Return matrix to transform given vector set into second vector set.

v0 and v1 are shape (3, *) or (4, *) arrays of at least 3 vectors.

If usesvd is True, the weighted sum of squared deviations (RMSD) is minimized according to the algorithm by W. Kabsch [8]. Otherwise the quaternion based algorithm by B. Horn [9] is used (slower when using this Python implementation).

The returned matrix performs rotation, translation and uniform scaling (if specified).

>>> v0 = numpy.random.rand(3, 10)
>>> M = superimposition_matrix(v0, v0)
>>> numpy.allclose(M, numpy.identity(4))
True
>>> R = random_rotation_matrix(numpy.random.random(3))
>>> v0 = ((1,0,0), (0,1,0), (0,0,1), (1,1,1))
>>> v1 = numpy.dot(R, v0)
>>> M = superimposition_matrix(v0, v1)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20.0
>>> v0[3] = 1.0
>>> v1 = numpy.dot(R, v0)
>>> M = superimposition_matrix(v0, v1)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> S = scale_matrix(random.random())
>>> T = translation_matrix(numpy.random.random(3)-0.5)
>>> M = concatenate_matrices(T, R, S)
>>> v1 = numpy.dot(M, v0)
>>> v0[:3] += numpy.random.normal(0.0, 1e-9, 300).reshape(3, -1)
>>> M = superimposition_matrix(v0, v1, scaling=True)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> M = superimposition_matrix(v0, v1, scaling=True, usesvd=False)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> v = numpy.empty((4, 100, 3), dtype=numpy.float64)
>>> v[:, :, 0] = v0
>>> M = superimposition_matrix(v0, v1, scaling=True, usesvd=False)
>>> numpy.allclose(v1, numpy.dot(M, v[:, :, 0]))
True

espressopp.external.transformations.translation_from_matrix(matrix)

Return translation vector from translation matrix.

>>> v0 = numpy.random.random(3) - 0.5
>>> v1 = translation_from_matrix(translation_matrix(v0))
>>> numpy.allclose(v0, v1)
True

espressopp.external.transformations.translation_matrix(direction)

Return matrix to translate by direction vector.

>>> v = numpy.random.random(3) - 0.5
>>> numpy.allclose(v, translation_matrix(v)[:3, 3])
True

espressopp.external.transformations.unit_vector(data, axis=None, out=None)

Return ndarray normalized by length, i.e. eucledian norm, along axis.

>>> v0 = numpy.random.random(3)
>>> v1 = unit_vector(v0)
>>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0))
True
>>> v0 = numpy.random.rand(5, 4, 3)
>>> v1 = unit_vector(v0, axis=-1)
>>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=2)), 2)
>>> numpy.allclose(v1, v2)
True
>>> v1 = unit_vector(v0, axis=1)
>>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=1)), 1)
>>> numpy.allclose(v1, v2)
True
>>> v1 = numpy.empty((5, 4, 3), dtype=numpy.float64)
>>> unit_vector(v0, axis=1, out=v1)
>>> numpy.allclose(v1, v2)
True
>>> list(unit_vector([]))
[]
>>> list(unit_vector([1.0]))
[1.0]

espressopp.external.transformations.vector_norm(data, axis=None, out=None)

Return length, i.e. eucledian norm, of ndarray along axis.

>>> v = numpy.random.random(3)
>>> n = vector_norm(v)
>>> numpy.allclose(n, numpy.linalg.norm(v))
True
>>> v = numpy.random.rand(6, 5, 3)
>>> n = vector_norm(v, axis=-1)
>>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=2)))
True
>>> n = vector_norm(v, axis=1)
>>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1)))
True
>>> v = numpy.random.rand(5, 4, 3)
>>> n = numpy.empty((5, 3), dtype=numpy.float64)
>>> vector_norm(v, axis=1, out=n)
>>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1)))
True
>>> vector_norm([])
0.0
>>> vector_norm([1.0])
1.0

espressopp.external.transformations.vector_product(v0, v1, axis=0)

Return vector perpendicular to vectors.

>>> v = vector_product([2, 0, 0], [0, 3, 0])
>>> numpy.allclose(v, [0, 0, 6])
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> v = vector_product(v0, v1)
>>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> v = vector_product(v0, v1, axis=1)
>>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]])
True