[pyassimp] Added missing transformations.py library

port/PyAssimp/scripts/transformations.py

@@ -0,0 +1,1705 @@
+# -*- coding: utf-8 -*-
+# transformations.py
+
+# Copyright (c) 2006, Christoph Gohlke
+# Copyright (c) 2006-2009, The Regents of the University of California
+# All rights reserved.
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions are met:
+#
+# * Redistributions of source code must retain the above copyright
+#   notice, this list of conditions and the following disclaimer.
+# * Redistributions in binary form must reproduce the above copyright
+#   notice, this list of conditions and the following disclaimer in the
+#   documentation and/or other materials provided with the distribution.
+# * Neither the name of the copyright holders nor the names of any
+#   contributors may be used to endorse or promote products derived
+#   from this software without specific prior written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED.  IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+# POSSIBILITY OF SUCH DAMAGE.
+
+"""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 <http://www.lfd.uci.edu/~gohlke/>`__,
+  Laboratory for Fluorescence Dynamics, University of California, Irvine
+
+:Version: 20090418
+
+Requirements
+------------
+
+* `Python 2.6 <http://www.python.org>`__
+* `Numpy 1.3 <http://numpy.scipy.org>`__
+* `transformations.c 20090418 <http://www.lfd.uci.edu/~gohlke/>`__
+  (optional implementation of some functions in C)
+
+Notes
+-----
+
+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".
+
+Calculations are carried out with numpy.float64 precision.
+
+This Python implementation is not optimized for speed.
+
+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 ix+jy+kz+w are represented as [x, y, z, w].
+
+Use the transpose of transformation matrices for OpenGL glMultMatrixd().
+
+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.
+
+
+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
+
+"""
+
+from __future__ import division
+
+import warnings
+import math
+
+import numpy
+
+# Documentation in HTML format can be generated with Epydoc
+__docformat__ = "restructuredtext en"
+
+
+def 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
+
+    """
+    return numpy.identity(4, dtype=numpy.float64)
+
+
+def 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
+
+    """
+    M = numpy.identity(4)
+    M[:3, 3] = direction[:3]
+    return M
+
+
+def 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
+
+    """
+    return numpy.array(matrix, copy=False)[:3, 3].copy()
+
+
+def 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
+
+    """
+    normal = unit_vector(normal[:3])
+    M = numpy.identity(4)
+    M[:3, :3] -= 2.0 * numpy.outer(normal, normal)
+    M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal
+    return M
+
+
+def 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
+
+    """
+    M = numpy.array(matrix, dtype=numpy.float64, copy=False)
+    # normal: unit eigenvector corresponding to eigenvalue -1
+    l, V = numpy.linalg.eig(M[:3, :3])
+    i = numpy.where(abs(numpy.real(l) + 1.0) < 1e-8)[0]
+    if not len(i):
+        raise ValueError("no unit eigenvector corresponding to eigenvalue -1")
+    normal = numpy.real(V[:, i[0]]).squeeze()
+    # point: any unit eigenvector corresponding to eigenvalue 1
+    l, V = numpy.linalg.eig(M)
+    i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
+    if not len(i):
+        raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
+    point = numpy.real(V[:, i[-1]]).squeeze()
+    point /= point[3]
+    return point, normal
+
+
+def rotation_matrix(angle, direction, point=None):
+    """Return matrix to rotate about axis defined by point and direction.
+
+    >>> 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
+
+    """
+    sina = math.sin(angle)
+    cosa = math.cos(angle)
+    direction = unit_vector(direction[:3])
+    # rotation matrix around unit vector
+    R = numpy.array(((cosa, 0.0,  0.0),
+                     (0.0,  cosa, 0.0),
+                     (0.0,  0.0,  cosa)), dtype=numpy.float64)
+    R += numpy.outer(direction, direction) * (1.0 - cosa)
+    direction *= sina
+    R += numpy.array((( 0.0,         -direction[2],  direction[1]),
+                      ( direction[2], 0.0,          -direction[0]),
+                      (-direction[1], direction[0],  0.0)),
+                     dtype=numpy.float64)
+    M = numpy.identity(4)
+    M[:3, :3] = R
+    if point is not None:
+        # rotation not around origin
+        point = numpy.array(point[:3], dtype=numpy.float64, copy=False)
+        M[:3, 3] = point - numpy.dot(R, point)
+    return M
+
+
+def 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
+
+    """
+    R = numpy.array(matrix, dtype=numpy.float64, copy=False)
+    R33 = R[:3, :3]
+    # direction: unit eigenvector of R33 corresponding to eigenvalue of 1
+    l, W = numpy.linalg.eig(R33.T)
+    i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
+    if not len(i):
+        raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
+    direction = numpy.real(W[:, i[-1]]).squeeze()
+    # point: unit eigenvector of R33 corresponding to eigenvalue of 1
+    l, Q = numpy.linalg.eig(R)
+    i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
+    if not len(i):
+        raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
+    point = numpy.real(Q[:, i[-1]]).squeeze()
+    point /= point[3]
+    # rotation angle depending on direction
+    cosa = (numpy.trace(R33) - 1.0) / 2.0
+    if abs(direction[2]) > 1e-8:
+        sina = (R[1, 0] + (cosa-1.0)*direction[0]*direction[1]) / direction[2]
+    elif abs(direction[1]) > 1e-8:
+        sina = (R[0, 2] + (cosa-1.0)*direction[0]*direction[2]) / direction[1]
+    else:
+        sina = (R[2, 1] + (cosa-1.0)*direction[1]*direction[2]) / direction[0]
+    angle = math.atan2(sina, cosa)
+    return angle, direction, point
+
+
+def 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)
+
+    """
+    if direction is None:
+        # uniform scaling
+        M = numpy.array(((factor, 0.0,    0.0,    0.0),
+                         (0.0,    factor, 0.0,    0.0),
+                         (0.0,    0.0,    factor, 0.0),
+                         (0.0,    0.0,    0.0,    1.0)), dtype=numpy.float64)
+        if origin is not None:
+            M[:3, 3] = origin[:3]
+            M[:3, 3] *= 1.0 - factor
+    else:
+        # nonuniform scaling
+        direction = unit_vector(direction[:3])
+        factor = 1.0 - factor
+        M = numpy.identity(4)
+        M[:3, :3] -= factor * numpy.outer(direction, direction)
+        if origin is not None:
+            M[:3, 3] = (factor * numpy.dot(origin[:3], direction)) * direction
+    return M
+
+
+def 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
+
+    """
+    M = numpy.array(matrix, dtype=numpy.float64, copy=False)
+    M33 = M[:3, :3]
+    factor = numpy.trace(M33) - 2.0
+    try:
+        # direction: unit eigenvector corresponding to eigenvalue factor
+        l, V = numpy.linalg.eig(M33)
+        i = numpy.where(abs(numpy.real(l) - factor) < 1e-8)[0][0]
+        direction = numpy.real(V[:, i]).squeeze()
+        direction /= vector_norm(direction)
+    except IndexError:
+        # uniform scaling
+        factor = (factor + 2.0) / 3.0
+        direction = None
+    # origin: any eigenvector corresponding to eigenvalue 1
+    l, V = numpy.linalg.eig(M)
+    i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
+    if not len(i):
+        raise ValueError("no eigenvector corresponding to eigenvalue 1")
+    origin = numpy.real(V[:, i[-1]]).squeeze()
+    origin /= origin[3]
+    return factor, origin, direction
+
+
+def 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
+
+    """
+    M = numpy.identity(4)
+    point = numpy.array(point[:3], dtype=numpy.float64, copy=False)
+    normal = unit_vector(normal[:3])
+    if perspective is not None:
+        # perspective projection
+        perspective = numpy.array(perspective[:3], dtype=numpy.float64,
+                                  copy=False)
+        M[0, 0] = M[1, 1] = M[2, 2] = numpy.dot(perspective-point, normal)
+        M[:3, :3] -= numpy.outer(perspective, normal)
+        if pseudo:
+            # preserve relative depth
+            M[:3, :3] -= numpy.outer(normal, normal)
+            M[:3, 3] = numpy.dot(point, normal) * (perspective+normal)
+        else:
+            M[:3, 3] = numpy.dot(point, normal) * perspective
+        M[3, :3] = -normal
+        M[3, 3] = numpy.dot(perspective, normal)
+    elif direction is not None:
+        # parallel projection
+        direction = numpy.array(direction[:3], dtype=numpy.float64, copy=False)
+        scale = numpy.dot(direction, normal)
+        M[:3, :3] -= numpy.outer(direction, normal) / scale
+        M[:3, 3] = direction * (numpy.dot(point, normal) / scale)
+    else:
+        # orthogonal projection
+        M[:3, :3] -= numpy.outer(normal, normal)
+        M[:3, 3] = numpy.dot(point, normal) * normal
+    return M
+
+
+def 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
+
+    """
+    M = numpy.array(matrix, dtype=numpy.float64, copy=False)
+    M33 = M[:3, :3]
+    l, V = numpy.linalg.eig(M)
+    i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
+    if not pseudo and len(i):
+        # point: any eigenvector corresponding to eigenvalue 1
+        point = numpy.real(V[:, i[-1]]).squeeze()
+        point /= point[3]
+        # direction: unit eigenvector corresponding to eigenvalue 0
+        l, V = numpy.linalg.eig(M33)
+        i = numpy.where(abs(numpy.real(l)) < 1e-8)[0]
+        if not len(i):
+            raise ValueError("no eigenvector corresponding to eigenvalue 0")
+        direction = numpy.real(V[:, i[0]]).squeeze()
+        direction /= vector_norm(direction)
+        # normal: unit eigenvector of M33.T corresponding to eigenvalue 0
+        l, V = numpy.linalg.eig(M33.T)
+        i = numpy.where(abs(numpy.real(l)) < 1e-8)[0]
+        if len(i):
+            # parallel projection
+            normal = numpy.real(V[:, i[0]]).squeeze()
+            normal /= vector_norm(normal)
+            return point, normal, direction, None, False
+        else:
+            # orthogonal projection, where normal equals direction vector
+            return point, direction, None, None, False
+    else:
+        # perspective projection
+        i = numpy.where(abs(numpy.real(l)) > 1e-8)[0]
+        if not len(i):
+            raise ValueError(
+                "no eigenvector not corresponding to eigenvalue 0")
+        point = numpy.real(V[:, i[-1]]).squeeze()
+        point /= point[3]
+        normal = - M[3, :3]
+        perspective = M[:3, 3] / numpy.dot(point[:3], normal)
+        if pseudo:
+            perspective -= normal
+        return point, normal, None, perspective, pseudo
+
+
+def 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(*frustrum, perspective=False)
+    >>> 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(*frustrum, perspective=True)
+    >>> 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.])
+
+    """
+    if left >= right or bottom >= top or near >= far:
+        raise ValueError("invalid frustrum")
+    if perspective:
+        if near <= _EPS:
+            raise ValueError("invalid frustrum: near <= 0")
+        t = 2.0 * near
+        M = ((-t/(right-left), 0.0, (right+left)/(right-left), 0.0),
+             (0.0, -t/(top-bottom), (top+bottom)/(top-bottom), 0.0),
+             (0.0, 0.0, -(far+near)/(far-near), t*far/(far-near)),
+             (0.0, 0.0, -1.0, 0.0))
+    else:
+        M = ((2.0/(right-left), 0.0, 0.0, (right+left)/(left-right)),
+             (0.0, 2.0/(top-bottom), 0.0, (top+bottom)/(bottom-top)),
+             (0.0, 0.0, 2.0/(far-near), (far+near)/(near-far)),
+             (0.0, 0.0, 0.0, 1.0))
+    return numpy.array(M, dtype=numpy.float64)
+
+
+def 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
+
+    """
+    normal = unit_vector(normal[:3])
+    direction = unit_vector(direction[:3])
+    if abs(numpy.dot(normal, direction)) > 1e-6:
+        raise ValueError("direction and normal vectors are not orthogonal")
+    angle = math.tan(angle)
+    M = numpy.identity(4)
+    M[:3, :3] += angle * numpy.outer(direction, normal)
+    M[:3, 3] = -angle * numpy.dot(point[:3], normal) * direction
+    return M
+
+
+def 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
+
+    """
+    M = numpy.array(matrix, dtype=numpy.float64, copy=False)
+    M33 = M[:3, :3]
+    # normal: cross independent eigenvectors corresponding to the eigenvalue 1
+    l, V = numpy.linalg.eig(M33)
+    i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-4)[0]
+    if len(i) < 2:
+        raise ValueError("No two linear independent eigenvectors found %s" % l)
+    V = numpy.real(V[:, i]).squeeze().T
+    lenorm = -1.0
+    for i0, i1 in ((0, 1), (0, 2), (1, 2)):
+        n = numpy.cross(V[i0], V[i1])
+        l = vector_norm(n)
+        if l > lenorm:
+            lenorm = l
+            normal = n
+    normal /= lenorm
+    # direction and angle
+    direction = numpy.dot(M33 - numpy.identity(3), normal)
+    angle = vector_norm(direction)
+    direction /= angle
+    angle = math.atan(angle)
+    # point: eigenvector corresponding to eigenvalue 1
+    l, V = numpy.linalg.eig(M)
+    i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
+    if not len(i):
+        raise ValueError("no eigenvector corresponding to eigenvalue 1")
+    point = numpy.real(V[:, i[-1]]).squeeze()
+    point /= point[3]
+    return angle, direction, point, normal
+
+
+def 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
+
+    """
+    M = numpy.array(matrix, dtype=numpy.float64, copy=True).T
+    if abs(M[3, 3]) < _EPS:
+        raise ValueError("M[3, 3] is zero")
+    M /= M[3, 3]
+    P = M.copy()
+    P[:, 3] = 0, 0, 0, 1
+    if not numpy.linalg.det(P):
+        raise ValueError("Matrix is singular")
+
+    scale = numpy.zeros((3, ), dtype=numpy.float64)
+    shear = [0, 0, 0]
+    angles = [0, 0, 0]
+
+    if any(abs(M[:3, 3]) > _EPS):
+        perspective = numpy.dot(M[:, 3], numpy.linalg.inv(P.T))
+        M[:, 3] = 0, 0, 0, 1
+    else:
+        perspective = numpy.array((0, 0, 0, 1), dtype=numpy.float64)
+
+    translate = M[3, :3].copy()
+    M[3, :3] = 0
+
+    row = M[:3, :3].copy()
+    scale[0] = vector_norm(row[0])
+    row[0] /= scale[0]
+    shear[0] = numpy.dot(row[0], row[1])
+    row[1] -= row[0] * shear[0]
+    scale[1] = vector_norm(row[1])
+    row[1] /= scale[1]
+    shear[0] /= scale[1]
+    shear[1] = numpy.dot(row[0], row[2])
+    row[2] -= row[0] * shear[1]
+    shear[2] = numpy.dot(row[1], row[2])
+    row[2] -= row[1] * shear[2]
+    scale[2] = vector_norm(row[2])
+    row[2] /= scale[2]
+    shear[1:] /= scale[2]
+
+    if numpy.dot(row[0], numpy.cross(row[1], row[2])) < 0:
+        scale *= -1
+        row *= -1
+
+    angles[1] = math.asin(-row[0, 2])
+    if math.cos(angles[1]):
+        angles[0] = math.atan2(row[1, 2], row[2, 2])
+        angles[2] = math.atan2(row[0, 1], row[0, 0])
+    else:
+        #angles[0] = math.atan2(row[1, 0], row[1, 1])
+        angles[0] = math.atan2(-row[2, 1], row[1, 1])
+        angles[2] = 0.0
+
+    return scale, shear, angles, translate, perspective
+
+
+def 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
+
+    """
+    M = numpy.identity(4)
+    if perspective is not None:
+        P = numpy.identity(4)
+        P[3, :] = perspective[:4]
+        M = numpy.dot(M, P)
+    if translate is not None:
+        T = numpy.identity(4)
+        T[:3, 3] = translate[:3]
+        M = numpy.dot(M, T)
+    if angles is not None:
+        R = euler_matrix(angles[0], angles[1], angles[2], 'sxyz')
+        M = numpy.dot(M, R)
+    if shear is not None:
+        Z = numpy.identity(4)
+        Z[1, 2] = shear[2]
+        Z[0, 2] = shear[1]
+        Z[0, 1] = shear[0]
+        M = numpy.dot(M, Z)
+    if scale is not None:
+        S = numpy.identity(4)
+        S[0, 0] = scale[0]
+        S[1, 1] = scale[1]
+        S[2, 2] = scale[2]
+        M = numpy.dot(M, S)
+    M /= M[3, 3]
+    return M
+
+
+def 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
+
+    """
+    a, b, c = lengths
+    angles = numpy.radians(angles)
+    sina, sinb, _ = numpy.sin(angles)
+    cosa, cosb, cosg = numpy.cos(angles)
+    co = (cosa * cosb - cosg) / (sina * sinb)
+    return numpy.array((
+        ( a*sinb*math.sqrt(1.0-co*co),  0.0,    0.0, 0.0),
+        (-a*sinb*co,                    b*sina, 0.0, 0.0),
+        ( a*cosb,                       b*cosa, c,   0.0),
+        ( 0.0,                          0.0,    0.0, 1.0)),
+        dtype=numpy.float64)
+
+
+def 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
+
+    """
+    v0 = numpy.array(v0, dtype=numpy.float64, copy=False)[:3]
+    v1 = numpy.array(v1, dtype=numpy.float64, copy=False)[:3]
+
+    if v0.shape != v1.shape or v0.shape[1] < 3:
+        raise ValueError("Vector sets are of wrong shape or type.")
+
+    # move centroids to origin
+    t0 = numpy.mean(v0, axis=1)
+    t1 = numpy.mean(v1, axis=1)
+    v0 = v0 - t0.reshape(3, 1)
+    v1 = v1 - t1.reshape(3, 1)
+
+    if usesvd:
+        # Singular Value Decomposition of covariance matrix
+        u, s, vh = numpy.linalg.svd(numpy.dot(v1, v0.T))
+        # rotation matrix from SVD orthonormal bases
+        R = numpy.dot(u, vh)
+        if numpy.linalg.det(R) < 0.0:
+            # R does not constitute right handed system
+            R -= numpy.outer(u[:, 2], vh[2, :]*2.0)
+            s[-1] *= -1.0
+        # homogeneous transformation matrix
+        M = numpy.identity(4)
+        M[:3, :3] = R
+    else:
+        # compute symmetric matrix N
+        xx, yy, zz = numpy.sum(v0 * v1, axis=1)
+        xy, yz, zx = numpy.sum(v0 * numpy.roll(v1, -1, axis=0), axis=1)
+        xz, yx, zy = numpy.sum(v0 * numpy.roll(v1, -2, axis=0), axis=1)
+        N = ((xx+yy+zz, yz-zy,    zx-xz,    xy-yx),
+             (yz-zy,    xx-yy-zz, xy+yx,    zx+xz),
+             (zx-xz,    xy+yx,   -xx+yy-zz, yz+zy),
+             (xy-yx,    zx+xz,    yz+zy,   -xx-yy+zz))
+        # quaternion: eigenvector corresponding to most positive eigenvalue
+        l, V = numpy.linalg.eig(N)
+        q = V[:, numpy.argmax(l)]
+        q /= vector_norm(q) # unit quaternion
+        q = numpy.roll(q, -1) # move w component to end
+        # homogeneous transformation matrix
+        M = quaternion_matrix(q)
+
+    # scale: ratio of rms deviations from centroid
+    if scaling:
+        v0 *= v0
+        v1 *= v1
+        M[:3, :3] *= math.sqrt(numpy.sum(v1) / numpy.sum(v0))
+
+    # translation
+    M[:3, 3] = t1
+    T = numpy.identity(4)
+    T[:3, 3] = -t0
+    M = numpy.dot(M, T)
+    return M
+
+
+def 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)
+
+    """
+    try:
+        firstaxis, parity, repetition, frame = _AXES2TUPLE[axes]
+    except (AttributeError, KeyError):
+        _ = _TUPLE2AXES[axes]
+        firstaxis, parity, repetition, frame = axes
+
+    i = firstaxis
+    j = _NEXT_AXIS[i+parity]
+    k = _NEXT_AXIS[i-parity+1]
+
+    if frame:
+        ai, ak = ak, ai
+    if parity:
+        ai, aj, ak = -ai, -aj, -ak
+
+    si, sj, sk = math.sin(ai), math.sin(aj), math.sin(ak)
+    ci, cj, ck = math.cos(ai), math.cos(aj), math.cos(ak)
+    cc, cs = ci*ck, ci*sk
+    sc, ss = si*ck, si*sk
+
+    M = numpy.identity(4)
+    if repetition:
+        M[i, i] = cj
+        M[i, j] = sj*si
+        M[i, k] = sj*ci
+        M[j, i] = sj*sk
+        M[j, j] = -cj*ss+cc
+        M[j, k] = -cj*cs-sc
+        M[k, i] = -sj*ck
+        M[k, j] = cj*sc+cs
+        M[k, k] = cj*cc-ss
+    else:
+        M[i, i] = cj*ck
+        M[i, j] = sj*sc-cs
+        M[i, k] = sj*cc+ss
+        M[j, i] = cj*sk
+        M[j, j] = sj*ss+cc
+        M[j, k] = sj*cs-sc
+        M[k, i] = -sj
+        M[k, j] = cj*si
+        M[k, k] = cj*ci
+    return M
+
+
+def 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"
+
+    """
+    try:
+        firstaxis, parity, repetition, frame = _AXES2TUPLE[axes.lower()]
+    except (AttributeError, KeyError):
+        _ = _TUPLE2AXES[axes]
+        firstaxis, parity, repetition, frame = axes
+
+    i = firstaxis
+    j = _NEXT_AXIS[i+parity]
+    k = _NEXT_AXIS[i-parity+1]
+
+    M = numpy.array(matrix, dtype=numpy.float64, copy=False)[:3, :3]
+    if repetition:
+        sy = math.sqrt(M[i, j]*M[i, j] + M[i, k]*M[i, k])
+        if sy > _EPS:
+            ax = math.atan2( M[i, j],  M[i, k])
+            ay = math.atan2( sy,       M[i, i])
+            az = math.atan2( M[j, i], -M[k, i])
+        else:
+            ax = math.atan2(-M[j, k],  M[j, j])
+            ay = math.atan2( sy,       M[i, i])
+            az = 0.0
+    else:
+        cy = math.sqrt(M[i, i]*M[i, i] + M[j, i]*M[j, i])
+        if cy > _EPS:
+            ax = math.atan2( M[k, j],  M[k, k])
+            ay = math.atan2(-M[k, i],  cy)
+            az = math.atan2( M[j, i],  M[i, i])
+        else:
+            ax = math.atan2(-M[j, k],  M[j, j])
+            ay = math.atan2(-M[k, i],  cy)
+            az = 0.0
+
+    if parity:
+        ax, ay, az = -ax, -ay, -az
+    if frame:
+        ax, az = az, ax
+    return ax, ay, az
+
+
+def euler_from_quaternion(quaternion, axes='sxyz'):
+    """Return Euler angles from quaternion for specified axis sequence.
+
+    >>> angles = euler_from_quaternion([0.06146124, 0, 0, 0.99810947])
+    >>> numpy.allclose(angles, [0.123, 0, 0])
+    True
+
+    """
+    return euler_from_matrix(quaternion_matrix(quaternion), axes)
+
+
+def 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, [0.310622, -0.718287, 0.444435, 0.435953])
+    True
+
+    """
+    try:
+        firstaxis, parity, repetition, frame = _AXES2TUPLE[axes.lower()]
+    except (AttributeError, KeyError):
+        _ = _TUPLE2AXES[axes]
+        firstaxis, parity, repetition, frame = axes
+
+    i = firstaxis
+    j = _NEXT_AXIS[i+parity]
+    k = _NEXT_AXIS[i-parity+1]
+
+    if frame:
+        ai, ak = ak, ai
+    if parity:
+        aj = -aj
+
+    ai /= 2.0
+    aj /= 2.0
+    ak /= 2.0
+    ci = math.cos(ai)
+    si = math.sin(ai)
+    cj = math.cos(aj)
+    sj = math.sin(aj)
+    ck = math.cos(ak)
+    sk = math.sin(ak)
+    cc = ci*ck
+    cs = ci*sk
+    sc = si*ck
+    ss = si*sk
+
+    quaternion = numpy.empty((4, ), dtype=numpy.float64)
+    if repetition:
+        quaternion[i] = cj*(cs + sc)
+        quaternion[j] = sj*(cc + ss)
+        quaternion[k] = sj*(cs - sc)
+        quaternion[3] = cj*(cc - ss)
+    else:
+        quaternion[i] = cj*sc - sj*cs
+        quaternion[j] = cj*ss + sj*cc
+        quaternion[k] = cj*cs - sj*sc
+        quaternion[3] = cj*cc + sj*ss
+    if parity:
+        quaternion[j] *= -1
+
+    return quaternion
+
+
+def quaternion_about_axis(angle, axis):
+    """Return quaternion for rotation about axis.
+
+    >>> q = quaternion_about_axis(0.123, (1, 0, 0))
+    >>> numpy.allclose(q, [0.06146124, 0, 0, 0.99810947])
+    True
+
+    """
+    quaternion = numpy.zeros((4, ), dtype=numpy.float64)
+    quaternion[:3] = axis[:3]
+    qlen = vector_norm(quaternion)
+    if qlen > _EPS:
+        quaternion *= math.sin(angle/2.0) / qlen
+    quaternion[3] = math.cos(angle/2.0)
+    return quaternion
+
+
+def quaternion_matrix(quaternion):
+    """Return homogeneous rotation matrix from quaternion.
+
+    >>> R = quaternion_matrix([0.06146124, 0, 0, 0.99810947])
+    >>> numpy.allclose(R, rotation_matrix(0.123, (1, 0, 0)))
+    True
+
+    """
+    q = numpy.array(quaternion[:4], dtype=numpy.float64, copy=True)
+    nq = numpy.dot(q, q)
+    if nq < _EPS:
+        return numpy.identity(4)
+    q *= math.sqrt(2.0 / nq)
+    q = numpy.outer(q, q)
+    return numpy.array((
+        (1.0-q[1, 1]-q[2, 2],     q[0, 1]-q[2, 3],     q[0, 2]+q[1, 3], 0.0),
+        (    q[0, 1]+q[2, 3], 1.0-q[0, 0]-q[2, 2],     q[1, 2]-q[0, 3], 0.0),
+        (    q[0, 2]-q[1, 3],     q[1, 2]+q[0, 3], 1.0-q[0, 0]-q[1, 1], 0.0),
+        (                0.0,                 0.0,                 0.0, 1.0)
+        ), dtype=numpy.float64)
+
+
+def quaternion_from_matrix(matrix):
+    """Return quaternion from rotation matrix.
+
+    >>> R = rotation_matrix(0.123, (1, 2, 3))
+    >>> q = quaternion_from_matrix(R)
+    >>> numpy.allclose(q, [0.0164262, 0.0328524, 0.0492786, 0.9981095])
+    True
+
+    """
+    q = numpy.empty((4, ), dtype=numpy.float64)
+    M = numpy.array(matrix, dtype=numpy.float64, copy=False)[:4, :4]
+    t = numpy.trace(M)
+    if t > M[3, 3]:
+        q[3] = t
+        q[2] = M[1, 0] - M[0, 1]
+        q[1] = M[0, 2] - M[2, 0]
+        q[0] = M[2, 1] - M[1, 2]
+    else:
+        i, j, k = 0, 1, 2
+        if M[1, 1] > M[0, 0]:
+            i, j, k = 1, 2, 0
+        if M[2, 2] > M[i, i]:
+            i, j, k = 2, 0, 1
+        t = M[i, i] - (M[j, j] + M[k, k]) + M[3, 3]
+        q[i] = t
+        q[j] = M[i, j] + M[j, i]
+        q[k] = M[k, i] + M[i, k]
+        q[3] = M[k, j] - M[j, k]
+    q *= 0.5 / math.sqrt(t * M[3, 3])
+    return q
+
+
+def quaternion_multiply(quaternion1, quaternion0):
+    """Return multiplication of two quaternions.
+
+    >>> q = quaternion_multiply([1, -2, 3, 4], [-5, 6, 7, 8])
+    >>> numpy.allclose(q, [-44, -14, 48, 28])
+    True
+
+    """
+    x0, y0, z0, w0 = quaternion0
+    x1, y1, z1, w1 = quaternion1
+    return numpy.array((
+         x1*w0 + y1*z0 - z1*y0 + w1*x0,
+        -x1*z0 + y1*w0 + z1*x0 + w1*y0,
+         x1*y0 - y1*x0 + z1*w0 + w1*z0,
+        -x1*x0 - y1*y0 - z1*z0 + w1*w0), dtype=numpy.float64)
+
+
+def quaternion_conjugate(quaternion):
+    """Return conjugate of quaternion.
+
+    >>> q0 = random_quaternion()
+    >>> q1 = quaternion_conjugate(q0)
+    >>> q1[3] == q0[3] and all(q1[:3] == -q0[:3])
+    True
+
+    """
+    return numpy.array((-quaternion[0], -quaternion[1],
+                        -quaternion[2], quaternion[3]), dtype=numpy.float64)
+
+
+def quaternion_inverse(quaternion):
+    """Return inverse of quaternion.
+
+    >>> q0 = random_quaternion()
+    >>> q1 = quaternion_inverse(q0)
+    >>> numpy.allclose(quaternion_multiply(q0, q1), [0, 0, 0, 1])
+    True
+
+    """
+    return quaternion_conjugate(quaternion) / numpy.dot(quaternion, quaternion)
+
+
+def 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
+
+    """
+    q0 = unit_vector(quat0[:4])
+    q1 = unit_vector(quat1[:4])
+    if fraction == 0.0:
+        return q0
+    elif fraction == 1.0:
+        return q1
+    d = numpy.dot(q0, q1)
+    if abs(abs(d) - 1.0) < _EPS:
+        return q0
+    if shortestpath and d < 0.0:
+        # invert rotation
+        d = -d
+        q1 *= -1.0
+    angle = math.acos(d) + spin * math.pi
+    if abs(angle) < _EPS:
+        return q0
+    isin = 1.0 / math.sin(angle)
+    q0 *= math.sin((1.0 - fraction) * angle) * isin
+    q1 *= math.sin(fraction * angle) * isin
+    q0 += q1
+    return q0
+
+
+def 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))
+    >>> q.shape
+    (4,)
+
+    """
+    if rand is None:
+        rand = numpy.random.rand(3)
+    else:
+        assert len(rand) == 3
+    r1 = numpy.sqrt(1.0 - rand[0])
+    r2 = numpy.sqrt(rand[0])
+    pi2 = math.pi * 2.0
+    t1 = pi2 * rand[1]
+    t2 = pi2 * rand[2]
+    return numpy.array((numpy.sin(t1)*r1,
+                        numpy.cos(t1)*r1,
+                        numpy.sin(t2)*r2,
+                        numpy.cos(t2)*r2), dtype=numpy.float64)
+
+
+def 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
+
+    """
+    return quaternion_matrix(random_quaternion(rand))
+
+
+class Arcball(object):
+    """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=[0, 0, 0, 1])
+    >>> 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()
+
+    """
+
+    def __init__(self, initial=None):
+        """Initialize virtual trackball control.
+
+        initial : quaternion or rotation matrix
+
+        """
+        self._axis = None
+        self._axes = None
+        self._radius = 1.0
+        self._center = [0.0, 0.0]
+        self._vdown = numpy.array([0, 0, 1], dtype=numpy.float64)
+        self._constrain = False
+
+        if initial is None:
+            self._qdown = numpy.array([0, 0, 0, 1], dtype=numpy.float64)
+        else:
+            initial = numpy.array(initial, dtype=numpy.float64)
+            if initial.shape == (4, 4):
+                self._qdown = quaternion_from_matrix(initial)
+            elif initial.shape == (4, ):
+                initial /= vector_norm(initial)
+                self._qdown = initial
+            else:
+                raise ValueError("initial not a quaternion or matrix.")
+
+        self._qnow = self._qpre = self._qdown
+
+    def place(self, 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.
+
+        """
+        self._radius = float(radius)
+        self._center[0] = center[0]
+        self._center[1] = center[1]
+
+    def setaxes(self, *axes):
+        """Set axes to constrain rotations."""
+        if axes is None:
+            self._axes = None
+        else:
+            self._axes = [unit_vector(axis) for axis in axes]
+
+    def setconstrain(self, constrain):
+        """Set state of constrain to axis mode."""
+        self._constrain = constrain == True
+
+    def getconstrain(self):
+        """Return state of constrain to axis mode."""
+        return self._constrain
+
+    def down(self, point):
+        """Set initial cursor window coordinates and pick constrain-axis."""
+        self._vdown = arcball_map_to_sphere(point, self._center, self._radius)
+        self._qdown = self._qpre = self._qnow
+
+        if self._constrain and self._axes is not None:
+            self._axis = arcball_nearest_axis(self._vdown, self._axes)
+            self._vdown = arcball_constrain_to_axis(self._vdown, self._axis)
+        else:
+            self._axis = None
+
+    def drag(self, point):
+        """Update current cursor window coordinates."""
+        vnow = arcball_map_to_sphere(point, self._center, self._radius)
+
+        if self._axis is not None:
+            vnow = arcball_constrain_to_axis(vnow, self._axis)
+
+        self._qpre = self._qnow
+
+        t = numpy.cross(self._vdown, vnow)
+        if numpy.dot(t, t) < _EPS:
+            self._qnow = self._qdown
+        else:
+            q = [t[0], t[1], t[2], numpy.dot(self._vdown, vnow)]
+            self._qnow = quaternion_multiply(q, self._qdown)
+
+    def next(self, acceleration=0.0):
+        """Continue rotation in direction of last drag."""
+        q = quaternion_slerp(self._qpre, self._qnow, 2.0+acceleration, False)
+        self._qpre, self._qnow = self._qnow, q
+
+    def matrix(self):
+        """Return homogeneous rotation matrix."""
+        return quaternion_matrix(self._qnow)
+
+
+def arcball_map_to_sphere(point, center, radius):
+    """Return unit sphere coordinates from window coordinates."""
+    v = numpy.array(((point[0] - center[0]) / radius,
+                     (center[1] - point[1]) / radius,
+                     0.0), dtype=numpy.float64)
+    n = v[0]*v[0] + v[1]*v[1]
+    if n > 1.0:
+        v /= math.sqrt(n) # position outside of sphere
+    else:
+        v[2] = math.sqrt(1.0 - n)
+    return v
+
+
+def arcball_constrain_to_axis(point, axis):
+    """Return sphere point perpendicular to axis."""
+    v = numpy.array(point, dtype=numpy.float64, copy=True)
+    a = numpy.array(axis, dtype=numpy.float64, copy=True)
+    v -= a * numpy.dot(a, v) # on plane
+    n = vector_norm(v)
+    if n > _EPS:
+        if v[2] < 0.0:
+            v *= -1.0
+        v /= n
+        return v
+    if a[2] == 1.0:
+        return numpy.array([1, 0, 0], dtype=numpy.float64)
+    return unit_vector([-a[1], a[0], 0])
+
+
+def arcball_nearest_axis(point, axes):
+    """Return axis, which arc is nearest to point."""
+    point = numpy.array(point, dtype=numpy.float64, copy=False)
+    nearest = None
+    mx = -1.0
+    for axis in axes:
+        t = numpy.dot(arcball_constrain_to_axis(point, axis), point)
+        if t > mx:
+            nearest = axis
+            mx = t
+    return nearest
+
+
+# epsilon for testing whether a number is close to zero
+_EPS = numpy.finfo(float).eps * 4.0
+
+# axis sequences for Euler angles
+_NEXT_AXIS = [1, 2, 0, 1]
+
+# map axes strings to/from tuples of inner axis, parity, repetition, frame
+_AXES2TUPLE = {
+    'sxyz': (0, 0, 0, 0), 'sxyx': (0, 0, 1, 0), 'sxzy': (0, 1, 0, 0),
+    'sxzx': (0, 1, 1, 0), 'syzx': (1, 0, 0, 0), 'syzy': (1, 0, 1, 0),
+    'syxz': (1, 1, 0, 0), 'syxy': (1, 1, 1, 0), 'szxy': (2, 0, 0, 0),
+    'szxz': (2, 0, 1, 0), 'szyx': (2, 1, 0, 0), 'szyz': (2, 1, 1, 0),
+    'rzyx': (0, 0, 0, 1), 'rxyx': (0, 0, 1, 1), 'ryzx': (0, 1, 0, 1),
+    'rxzx': (0, 1, 1, 1), 'rxzy': (1, 0, 0, 1), 'ryzy': (1, 0, 1, 1),
+    'rzxy': (1, 1, 0, 1), 'ryxy': (1, 1, 1, 1), 'ryxz': (2, 0, 0, 1),
+    'rzxz': (2, 0, 1, 1), 'rxyz': (2, 1, 0, 1), 'rzyz': (2, 1, 1, 1)}
+
+_TUPLE2AXES = dict((v, k) for k, v in _AXES2TUPLE.items())
+
+# helper functions
+
+def 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
+
+    """
+    data = numpy.array(data, dtype=numpy.float64, copy=True)
+    if out is None:
+        if data.ndim == 1:
+            return math.sqrt(numpy.dot(data, data))
+        data *= data
+        out = numpy.atleast_1d(numpy.sum(data, axis=axis))
+        numpy.sqrt(out, out)
+        return out
+    else:
+        data *= data
+        numpy.sum(data, axis=axis, out=out)
+        numpy.sqrt(out, out)
+
+
+def 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]
+
+    """
+    if out is None:
+        data = numpy.array(data, dtype=numpy.float64, copy=True)
+        if data.ndim == 1:
+            data /= math.sqrt(numpy.dot(data, data))
+            return data
+    else:
+        if out is not data:
+            out[:] = numpy.array(data, copy=False)
+        data = out
+    length = numpy.atleast_1d(numpy.sum(data*data, axis))
+    numpy.sqrt(length, length)
+    if axis is not None:
+        length = numpy.expand_dims(length, axis)
+    data /= length
+    if out is None:
+        return data
+
+
+def 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
+
+    """
+    return numpy.random.random(size)
+
+
+def 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
+
+    """
+    return numpy.linalg.inv(matrix)
+
+
+def 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
+
+    """
+    M = numpy.identity(4)
+    for i in matrices:
+        M = numpy.dot(M, i)
+    return M
+
+
+def 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
+
+    """
+    matrix0 = numpy.array(matrix0, dtype=numpy.float64, copy=True)
+    matrix0 /= matrix0[3, 3]
+    matrix1 = numpy.array(matrix1, dtype=numpy.float64, copy=True)
+    matrix1 /= matrix1[3, 3]
+    return numpy.allclose(matrix0, matrix1)
+
+
+def _import_module(module_name, warn=True, prefix='_py_', ignore='_'):
+    """Try import all public attributes from module into global namespace.
+
+    Existing attributes with name clashes are renamed with prefix.
+    Attributes starting with underscore are ignored by default.
+
+    Return True on successful import.
+
+    """
+    try:
+        module = __import__(module_name)
+    except ImportError:
+        if warn:
+            warnings.warn("Failed to import module " + module_name)
+    else:
+        for attr in dir(module):
+            if ignore and attr.startswith(ignore):
+                continue
+            if prefix:
+                if attr in globals():
+                    globals()[prefix + attr] = globals()[attr]
+                elif warn:
+                    warnings.warn("No Python implementation of " + attr)
+            globals()[attr] = getattr(module, attr)
+        return True