ipymd.shared package

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Submodules

ipymd.shared.colors module

Created on Wed Jun 29 01:56:51 2016

@author: cjs14

ipymd.shared.colors.any_to_rgb(color)

If color is an rgb tuple return it, if it is a string, parse it and return the respective rgb tuple.

ipymd.shared.colors.available_colors()

ipymd.shared.colors.get(name)

Given a string color, return the color as a tuple (r, g, b, a) where each value is between 0 and 255.

As for the color name follow the HTML color names <http://www.w3schools.com/tags/ref_colornames.asp> in lowescore style eg. forest_green.

ipymd.shared.colors.hsl_to_rgb(arr)

Converts HSL color array to RGB array

H = [0..360] S = [0..1] l = [0..1]

http://en.wikipedia.org/wiki/HSL_and_HSV#From_HSL

Returns R,G,B in [0..255]

ipymd.shared.colors.html_to_rgb(colorstring)

convert #RRGGBB to an (R, G, B) tuple

ipymd.shared.colors.mix(a, b, ratio=0.5)

ipymd.shared.colors.parse_color(color)

Return the RGB 0-255 representation of the current string passed.

It first tries to match the string with DVI color names.

ipymd.shared.colors.rgb_to_hsl(a)

ipymd.shared.colors.rgb_to_hsl_hsv(a, isHSV=True)

Converts RGB image data to HSV or HSL. :param a: 3D array. Retval of numpy.asarray(Image.open(...), int) :param isHSV: True = HSV, False = HSL :return: H,S,L or H,S,V array

ipymd.shared.colors.rgb_to_hsv(a)

ipymd.shared.transformations module

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 Version:2012.10.14

Requirements

• CPython 2.7 or 3.2
• Numpy 1.6
• transformations.c 2012.01.01 (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), be concatenated using numpy.dot(M0, M1), or transform homogeneous coordinate arrays (v) using numpy.dot(M, v) for shape (4, *) column vectors, respectively numpy.dot(v, M.T) for shape (*, 4) row vectors (“array of points”).

This module follows the “column vectors on the right” and “row major storage” (C contiguous) conventions. The translation components are in the right column of the transformation matrix, i.e. M[:3, 3]. The transpose of the transformation matrices may have to be used to interface with other graphics systems, e.g. with OpenGL’s glMultMatrixd(). See also [16].

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.
15. Multiple View Geometry in Computer Vision. Hartley and Zissermann. Cambridge University Press; 2nd Ed. 2004. Chapter 4, Algorithm 4.7, p 130.
16. Column Vectors vs. Row Vectors. http://steve.hollasch.net/cgindex/math/matrix/column-vec.html

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
>>> 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(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 ipymd.shared.transformations.Arcball(initial=None)

Bases: object

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.

center : sequence[2]

Window coordinates of trackball center.

radius : float

Radius of trackball in window coordinates.

setaxes(*axes)

Set axes to constrain rotations.

setconstrain(constrain)

Set state of constrain to axis mode.

ipymd.shared.transformations.affine_matrix_from_points(v0, v1, shear=True, scale=True, usesvd=True)

Return affine transform matrix to register two point sets.

v0 and v1 are shape (ndims, *) arrays of at least ndims non-homogeneous coordinates, where ndims is the dimensionality of the coordinate space.

If shear is False, a similarity transformation matrix is returned. If also scale is False, a rigid/Eucledian transformation matrix is returned.

By default the algorithm by Hartley and Zissermann [15] is used. If usesvd is True, similarity and Eucledian transformation matrices are calculated by minimizing the weighted sum of squared deviations (RMSD) according to the algorithm by Kabsch [8]. Otherwise, and if ndims is 3, the quaternion based algorithm by Horn [9] is used, which is slower when using this Python implementation.

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

>>> v0 = [[0, 1031, 1031, 0], [0, 0, 1600, 1600]]
>>> v1 = [[675, 826, 826, 677], [55, 52, 281, 277]]
>>> affine_matrix_from_points(v0, v1)
array([[   0.14549,    0.00062,  675.50008],
       [   0.00048,    0.14094,   53.24971],
       [   0.     ,    0.     ,    1.     ]])
>>> T = translation_matrix(numpy.random.random(3)-0.5)
>>> R = random_rotation_matrix(numpy.random.random(3))
>>> S = scale_matrix(random.random())
>>> M = concatenate_matrices(T, R, S)
>>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = numpy.dot(M, v0)
>>> v0[:3] += numpy.random.normal(0, 1e-8, 300).reshape(3, -1)
>>> M = affine_matrix_from_points(v0[:3], v1[:3])
>>> numpy.allclose(v1, numpy.dot(M, v0))
True

More examples in superimposition_matrix()

ipymd.shared.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

ipymd.shared.transformations.arcball_constrain_to_axis(point, axis)

Return sphere point perpendicular to axis.

ipymd.shared.transformations.arcball_map_to_sphere(point, center, radius)

Return unit sphere coordinates from window coordinates.

ipymd.shared.transformations.arcball_nearest_axis(point, axes)

Return axis, which arc is nearest to point.

ipymd.shared.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 (divided 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])
array([-1., -1., -1.,  1.])
>>> numpy.dot(M, [frustrum[1], frustrum[3], frustrum[5], 1])
array([ 1.,  1.,  1.,  1.])
>>> M = clip_matrix(perspective=True, *frustrum)
>>> v = numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1])
>>> v / v[3]
array([-1., -1., -1.,  1.])
>>> v = numpy.dot(M, [frustrum[1], frustrum[3], frustrum[4], 1])
>>> v / v[3]
array([ 1.,  1., -1.,  1.])

ipymd.shared.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

ipymd.shared.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

ipymd.shared.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

ipymd.shared.transformations.distance(x1, x2)

Distance between two points in space

ipymd.shared.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*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")

ipymd.shared.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

ipymd.shared.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*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)

ipymd.shared.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))
True

ipymd.shared.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 range(1, 7):
...     M0 = numpy.random.rand(size, size)
...     M1 = inverse_matrix(M0)
...     if not numpy.allclose(M1, numpy.linalg.inv(M0)): print(size)

ipymd.shared.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

ipymd.shared.transformations.normalized(x)

Return the x vector normalized

ipymd.shared.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

ipymd.shared.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

ipymd.shared.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
>>> v0[3] = 1
>>> v1 = numpy.dot(P, v0)
>>> numpy.allclose(v1[1], v0[1])
True
>>> numpy.allclose(v1[0], 3-v1[1])
True

ipymd.shared.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

ipymd.shared.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

ipymd.shared.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

ipymd.shared.transformations.quaternion_from_matrix(matrix, isprecise=False)

Return quaternion from rotation matrix.

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

>>> q = quaternion_from_matrix(numpy.identity(4), 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

ipymd.shared.transformations.quaternion_imag(quaternion)

Return imaginary part of quaternion.

>>> quaternion_imag([3, 0, 1, 2])
array([ 0.,  1.,  2.])

ipymd.shared.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

ipymd.shared.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, numpy.identity(4))
True
>>> M = quaternion_matrix([0, 1, 0, 0])
>>> numpy.allclose(M, numpy.diag([1, -1, -1, 1]))
True

ipymd.shared.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

ipymd.shared.transformations.quaternion_real(quaternion)

Return real part of quaternion.

>>> quaternion_real([3, 0, 1, 2])
3.0

ipymd.shared.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)
>>> numpy.allclose(q, q0)
True
>>> q = quaternion_slerp(q0, q1, 1, 1)
>>> numpy.allclose(q, q1)
True
>>> q = quaternion_slerp(q0, q1, 0.5)
>>> angle = math.acos(numpy.dot(q0, q))
>>> numpy.allclose(2, math.acos(numpy.dot(q0, q1)) / angle) or         numpy.allclose(2, math.acos(-numpy.dot(q0, q1)) / angle)
True

ipymd.shared.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, vector_norm(q))
True
>>> q = random_quaternion(numpy.random.random(3))
>>> len(q.shape), q.shape[0]==4
(1, True)

ipymd.shared.transformations.random_rotation_matrix(rand=None)

Return uniform random rotation matrix.

rand: 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

ipymd.shared.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) and numpy.all(v < 1)
True
>>> v0 = random_vector(10)
>>> v1 = random_vector(10)
>>> numpy.any(v0 == v1)
False

ipymd.shared.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

ipymd.shared.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.
>>> 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

ipymd.shared.transformations.rotate_vectors(vector, axis, theta)

rotate the vector v clockwise about the given axis vector by theta degrees.

e.g. rotate([0,1,0],[0,0,1],90) -> [1,0,0]

vector : iterable or list of iterables

vector to rotate [x,y,z] or [[x1,y1,z1],[x2,y2,z2]]

axis : iterable

axis to rotate around [x0,y0,z0]

theta : float

rotation angle in degrees

ipymd.shared.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

ipymd.shared.transformations.rotation_matrix(angle, direction)

Create a rotation matrix corresponding to the rotation around a general axis by a specified angle.

R = dd^T + cos(a) (I - dd^T) + sin(a) skew(d)

Parameters:

• angle – float a
• direction – array d

ipymd.shared.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

ipymd.shared.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
>>> v[3] = 1
>>> 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)

ipymd.shared.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

ipymd.shared.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, numpy.linalg.det(S))
True

ipymd.shared.transformations.simple_clip_matrix(scale, znear, zfar, aspectratio=1.0)

Given the parameters for a frustum returns a 4x4 perspective projection matrix

Parameters:

• scale (float) –
• znear,zfar (float) – near/far plane z, float

Return: a 4x4 perspective matrix

ipymd.shared.transformations.superimposition_matrix(v0, v1, scale=False, usesvd=True)

Return matrix to transform given 3D point set into second point set.

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

The parameters scale and usesvd are explained in the more general affine_matrix_from_points function.

The returned matrix is a similarity or Eucledian transformation matrix. This function has a fast C implementation in transformations.c.

>>> 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
>>> v0[3] = 1
>>> 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, 1e-9, 300).reshape(3, -1)
>>> M = superimposition_matrix(v0, v1, scale=True)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> v = numpy.empty((4, 100, 3))
>>> v[:, :, 0] = v0
>>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False)
>>> numpy.allclose(v1, numpy.dot(M, v[:, :, 0]))
True

ipymd.shared.transformations.transform_from_crytal(coords, a, b, c, origin=[0, 0, 0])

transform from crystal fractional coordinates to cartesian

coords : numpy.array((N,3))

a : numpy.array(3)

b : numpy.array(3)

c : numpy.array(3)

origin : numpy.array(3)

Notes

From https://en.wikipedia.org/wiki/Fractional_coordinates

\[\begin{split}\begin{bmatrix}x\\y\\z\\\end{bmatrix}= \begin{bmatrix}a&b\cos(\gamma )&c\cos(\beta )\\0&b\sin(\gamma )&c{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}\\0&0&c{\frac {v}{\sin(\gamma )}}\\\end{bmatrix} \begin{bmatrix}x_{frac}\\y_{frac}\\z_{frac}\\\end{bmatrix}\end{split}\]

such that v is the volume of a unit parallelepiped defined as:

\[v={\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}\]

ipymd.shared.transformations.transform_to_crystal(coords, a, b, c, origin=[0, 0, 0])

transform from cartesian to crystal fractional coordinates

coords : numpy.array((N,3)) a : numpy.array(3) b : numpy.array(3) c : numpy.array(3) origin : numpy.array(3)

Notes

From https://en.wikipedia.org/wiki/Fractional_coordinates

\[\begin{split}\begin{bmatrix}x_{frac}\\y_{frac}\\z_{frac}\\\end{bmatrix}= \begin{bmatrix}{ \frac {1}{a}}&-{\frac {\cos(\gamma )}{a\sin(\gamma )}}&{\frac {\cos(\alpha )\cos(\gamma )-\cos(\beta )}{av\sin(\gamma )}}\\ 0&{\frac {1}{b\sin(\gamma )}}&{\frac {\cos(\beta )\cos(\gamma )-\cos(\alpha )}{bv\sin(\gamma )}}\\ 0&0&{\frac {\sin(\gamma )}{cv}}\\\end{bmatrix} \begin{bmatrix}x\\y\\z\\\end{bmatrix}\end{split}\]

such that v is the volume of a unit parallelepiped defined as:

\[v={\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}\]

ipymd.shared.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

ipymd.shared.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

ipymd.shared.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))
>>> unit_vector(v0, axis=1, out=v1)
>>> numpy.allclose(v1, v2)
True
>>> list(unit_vector([]))
[]
>>> list(unit_vector([1]))
[1.0]

ipymd.shared.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))
>>> 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])
1.0

ipymd.shared.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

Module contents

ipymd.shared.atom_data()

return a dataframe of atomic data

ipymd.shared.get_data_path(data, check_exists=False, module=<module 'ipymd.test_data' from '/home/docs/checkouts/readthedocs.org/user_builds/ipymd/checkouts/latest/ipymd/test_data/__init__.pyc'>)

return a directory path to data within a module

data : str or list of str

file name or list of sub-directories and file name (e.g. [‘lammps’,’data.txt’])