text: # -*- coding: utf-8 -*-
  # transformations.py
  
  # Copyright (c) 2006-2018, Christoph Gohlke
  # Copyright (c) 2006-2018, The Regents of the University of California
  # Produced at the Laboratory for Fluorescence Dynamics
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  """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.
  
  :Author:
    `Christoph Gohlke <https://www.lfd.uci.edu/~gohlke/>`_
  
  :Organization:
    Laboratory for Fluorescence Dynamics, University of California, Irvine
  
  :Version: 2018.02.08
  
  Requirements
  ------------
  * `CPython 2.7 or 3.6 <http://www.python.org>`_
  * `Numpy 1.13 <http://www.numpy.org>`_
  * `Transformations.c 2018.02.08 <https://www.lfd.uci.edu/~gohlke/>`_
    (recommended for speedup of some functions)
  
  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.
  
  Other Python packages and modules for 3D transformations and quaternions:
  
  * `Transforms3d <https://pypi.python.org/pypi/transforms3d>`_
     includes most code of this module.
  * `Blender.mathutils <http://www.blender.org/api/blender_python_api>`_
  * `numpy-dtypes <https://github.com/numpy/numpy-dtypes>`_
  
  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
  
  """
  
  from __future__ import division, print_function
  
  import math
  
  import numpy
  
  __version__ = '2018.02.08'
  __docformat__ = 'restructuredtext en'
  __all__ = ()
  
  
  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))
      True
  
      """
      return numpy.identity(4)
  
  
  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.
      >>> 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
      w, V = numpy.linalg.eig(M[:3, :3])
      i = numpy.where(abs(numpy.real(w) + 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
      w, V = numpy.linalg.eig(M)
      i = numpy.where(abs(numpy.real(w) - 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.
  
      >>> R = rotation_matrix(math.pi/2, [0, 0, 1], [1, 0, 0])
      >>> numpy.allclose(numpy.dot(R, [0, 0, 0, 1]), [1, -1, 0, 1])
      True
      >>> angle = (random.random() - 0.5) * (2*math.pi)
      >>> direc = numpy.random.random(3) - 0.5
      >>> point = numpy.random.random(3) - 0.5
      >>> R0 = rotation_matrix(angle, direc, point)
      >>> R1 = rotation_matrix(angle-2*math.pi, direc, point)
      >>> is_same_transform(R0, R1)
      True
      >>> R0 = rotation_matrix(angle, direc, point)
      >>> R1 = rotation_matrix(-angle, -direc, point)
      >>> is_same_transform(R0, R1)
      True
      >>> I = numpy.identity(4, numpy.float64)
      >>> numpy.allclose(I, rotation_matrix(math.pi*2, direc))
      True
      >>> numpy.allclose(2, numpy.trace(rotation_matrix(math.pi/2,
      ...                                               direc, point)))
      True
  
      """
      sina = math.sin(angle)
      cosa = math.cos(angle)
      direction = unit_vector(direction[:3])
      # rotation matrix around unit vector
      R = numpy.diag([cosa, cosa, cosa])
      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]])
      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
      w, W = numpy.linalg.eig(R33.T)
      i = numpy.where(abs(numpy.real(w) - 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
      w, Q = numpy.linalg.eig(R)
      i = numpy.where(abs(numpy.real(w) - 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
      >>> 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)
  
      """
      if direction is None:
          # uniform scaling
          M = numpy.diag([factor, factor, factor, 1.0])
          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
          w, V = numpy.linalg.eig(M33)
          i = numpy.where(abs(numpy.real(w) - 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
      w, V = numpy.linalg.eig(M)
      i = numpy.where(abs(numpy.real(w) - 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
      >>> v0[3] = 1
      >>> v1 = numpy.dot(P, v0)
      >>> numpy.allclose(v1[1], v0[1])
      True
      >>> numpy.allclose(v1[0], 3-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]
      w, V = numpy.linalg.eig(M)
      i = numpy.where(abs(numpy.real(w) - 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
          w, V = numpy.linalg.eig(M33)
          i = numpy.where(abs(numpy.real(w)) < 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
          w, V = numpy.linalg.eig(M33.T)
          i = numpy.where(abs(numpy.real(w)) < 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(w)) > 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 frustum.
  
      The frustum 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 frustum.
  
      If perspective is True the frustum 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).
  
      >>> frustum = numpy.random.rand(6)
      >>> frustum[1] += frustum[0]
      >>> frustum[3] += frustum[2]
      >>> frustum[5] += frustum[4]
      >>> M = clip_matrix(perspective=False, *frustum)
      >>> numpy.dot(M, [frustum[0], frustum[2], frustum[4], 1])
      array([-1., -1., -1.,  1.])
      >>> numpy.dot(M, [frustum[1], frustum[3], frustum[5], 1])
      array([ 1.,  1.,  1.,  1.])
      >>> M = clip_matrix(perspective=True, *frustum)
      >>> v = numpy.dot(M, [frustum[0], frustum[2], frustum[4], 1])
      >>> v / v[3]
      array([-1., -1., -1.,  1.])
      >>> v = numpy.dot(M, [frustum[1], frustum[3], frustum[4], 1])
      >>> v / v[3]
      array([ 1.,  1., -1.,  1.])
  
      """
      if left >= right or bottom >= top or near >= far:
          raise ValueError('invalid frustum')
      if perspective:
          if near <= _EPS:
              raise ValueError('invalid frustum: near <= 0')
          t = 2.0 * near
          M = [[t/(left-right), 0.0, (right+left)/(right-left), 0.0],
               [0.0, t/(bottom-top), (top+bottom)/(top-bottom), 0.0],
               [0.0, 0.0, (far+near)/(near-far), 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)
  
  
  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, 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
      w, V = numpy.linalg.eig(M33)
      i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-4)[0]
      if len(i) < 2:
          raise ValueError('no two linear independent eigenvectors found %s' % w)
      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])
          w = vector_norm(n)
          if w > lenorm:
              lenorm = w
              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
      w, V = numpy.linalg.eig(M)
      i = numpy.where(abs(numpy.real(w) - 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, 0.0, 0.0, 1.0
      if not numpy.linalg.det(P):
          raise ValueError('matrix is singular')
  
      scale = numpy.zeros((3, ))
      shear = [0.0, 0.0, 0.0]
      angles = [0.0, 0.0, 0.0]
  
      if any(abs(M[:3, 3]) > _EPS):
          perspective = numpy.dot(M[:, 3], numpy.linalg.inv(P.T))
          M[:, 3] = 0.0, 0.0, 0.0, 1.0
      else:
          perspective = numpy.array([0.0, 0.0, 0.0, 1.0])
  
      translate = M[3, :3].copy()
      M[3, :3] = 0.0
  
      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:
          numpy.negative(scale, scale)
          numpy.negative(row, row)
  
      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]])
  
  
  def 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/Euclidean transformation matrix
      is returned.
  
      By default the algorithm by Hartley and Zissermann [15] is used.
      If usesvd is True, similarity and Euclidean 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()
  
      """
      v0 = numpy.array(v0, dtype=numpy.float64, copy=True)
      v1 = numpy.array(v1, dtype=numpy.float64, copy=True)
  
      ndims = v0.shape[0]
      if ndims < 2 or v0.shape[1] < ndims or v0.shape != v1.shape:
          raise ValueError('input arrays are of wrong shape or type')
  
      # move centroids to origin
      t0 = -numpy.mean(v0, axis=1)
      M0 = numpy.identity(ndims+1)
      M0[:ndims, ndims] = t0
      v0 += t0.reshape(ndims, 1)
      t1 = -numpy.mean(v1, axis=1)
      M1 = numpy.identity(ndims+1)
      M1[:ndims, ndims] = t1
      v1 += t1.reshape(ndims, 1)
  
      if shear:
          # Affine transformation
          A = numpy.concatenate((v0, v1), axis=0)
          u, s, vh = numpy.linalg.svd(A.T)
          vh = vh[:ndims].T
          B = vh[:ndims]
          C = vh[ndims:2*ndims]
          t = numpy.dot(C, numpy.linalg.pinv(B))
          t = numpy.concatenate((t, numpy.zeros((ndims, 1))), axis=1)
          M = numpy.vstack((t, ((0.0,)*ndims) + (1.0,)))
      elif usesvd or ndims != 3:
          # Rigid transformation via SVD 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[:, ndims-1], vh[ndims-1, :]*2.0)
              s[-1] *= -1.0
          # homogeneous transformation matrix
          M = numpy.identity(ndims+1)
          M[:ndims, :ndims] = R
      else:
          # Rigid transformation matrix via quaternion
          # 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, 0.0,      0.0,      0.0],
               [yz-zy,    xx-yy-zz, 0.0,      0.0],
               [zx-xz,    xy+yx,    yy-xx-zz, 0.0],
               [xy-yx,    zx+xz,    yz+zy,    zz-xx-yy]]
          # quaternion: eigenvector corresponding to most positive eigenvalue
          w, V = numpy.linalg.eigh(N)
          q = V[:, numpy.argmax(w)]
          q /= vector_norm(q)  # unit quaternion
          # homogeneous transformation matrix
          M = quaternion_matrix(q)
  
      if scale and not shear:
          # Affine transformation; scale is ratio of RMS deviations from centroid
          v0 *= v0
          v1 *= v1
          M[:ndims, :ndims] *= math.sqrt(numpy.sum(v1) / numpy.sum(v0))
  
      # move centroids back
      M = numpy.dot(numpy.linalg.inv(M1), numpy.dot(M, M0))
      M /= M[ndims, ndims]
      return M
  
  
  def 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 Euclidean 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
  
      """
      v0 = numpy.array(v0, dtype=numpy.float64, copy=False)[:3]
      v1 = numpy.array(v1, dtype=numpy.float64, copy=False)[:3]
      return affine_matrix_from_points(v0, v1, shear=False,
                                       scale=scale, usesvd=usesvd)
  
  
  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*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]  # validation
          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*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]  # validation
          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([0.99810947, 0.06146124, 0, 0])
      >>> 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, [0.435953, 0.310622, -0.718287, 0.444435])
      True
  
      """
      try:
          firstaxis, parity, repetition, frame = _AXES2TUPLE[axes.lower()]
      except (AttributeError, KeyError):
          _TUPLE2AXES[axes]  # validation
          firstaxis, parity, repetition, frame = axes
  
      i = firstaxis + 1
      j = _NEXT_AXIS[i+parity-1] + 1
      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
  
      q = numpy.empty((4, ))
      if repetition:
          q[0] = cj*(cc - ss)
          q[i] = cj*(cs + sc)
          q[j] = sj*(cc + ss)
          q[k] = sj*(cs - sc)
      else:
          q[0] = cj*cc + sj*ss
          q[i] = cj*sc - sj*cs
          q[j] = cj*ss + sj*cc
          q[k] = cj*cs - sj*sc
      if parity:
          q[j] *= -1.0
  
      return q
  
  
  def quaternion_about_axis(angle, axis):
      """Return quaternion for rotation about axis.
  
      >>> q = quaternion_about_axis(0.123, [1, 0, 0])
      >>> numpy.allclose(q, [0.99810947, 0.06146124, 0, 0])
      True
  
      """
      q = numpy.array([0.0, axis[0], axis[1], axis[2]])
      qlen = vector_norm(q)
      if qlen > _EPS:
          q *= math.sin(angle/2.0) / qlen
      q[0] = math.cos(angle/2.0)
      return q
  
  
  def 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
  
      """
      q = numpy.array(quaternion, dtype=numpy.float64, copy=True)
      n = numpy.dot(q, q)
      if n < _EPS:
          return numpy.identity(4)
      q *= math.sqrt(2.0 / n)
      q = numpy.outer(q, q)
      return numpy.array([
          [1.0-q[2, 2]-q[3, 3],     q[1, 2]-q[3, 0],     q[1, 3]+q[2, 0], 0.0],
          [    q[1, 2]+q[3, 0], 1.0-q[1, 1]-q[3, 3],     q[2, 3]-q[1, 0], 0.0],
          [    q[1, 3]-q[2, 0],     q[2, 3]+q[1, 0], 1.0-q[1, 1]-q[2, 2], 0.0],
          [                0.0,                 0.0,                 0.0, 1.0]])
  
  
  def 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
      >>> is_same_quaternion(quaternion_from_matrix(R, isprecise=False),
      ...                    quaternion_from_matrix(R, isprecise=True))
      True
      >>> R = euler_matrix(0.0, 0.0, numpy.pi/2.0)
      >>> is_same_quaternion(quaternion_from_matrix(R, isprecise=False),
      ...                    quaternion_from_matrix(R, isprecise=True))
      True
  
      """
      M = numpy.array(matrix, dtype=numpy.float64, copy=False)[:4, :4]
      if isprecise:
          q = numpy.empty((4, ))
          t = numpy.trace(M)
          if t > M[3, 3]:
              q[0] = t
              q[3] = M[1, 0] - M[0, 1]
              q[2] = M[0, 2] - M[2, 0]
              q[1] = 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 = q[[3, 0, 1, 2]]
          q *= 0.5 / math.sqrt(t * M[3, 3])
      else:
          m00 = M[0, 0]
          m01 = M[0, 1]
          m02 = M[0, 2]
          m10 = M[1, 0]
          m11 = M[1, 1]
          m12 = M[1, 2]
          m20 = M[2, 0]
          m21 = M[2, 1]
          m22 = M[2, 2]
          # symmetric matrix K
          K = numpy.array([[m00-m11-m22, 0.0,         0.0,         0.0],
                           [m01+m10,     m11-m00-m22, 0.0,         0.0],
                           [m02+m20,     m12+m21,     m22-m00-m11, 0.0],
                           [m21-m12,     m02-m20,     m10-m01,     m00+m11+m22]])
          K /= 3.0
          # quaternion is eigenvector of K that corresponds to largest eigenvalue
          w, V = numpy.linalg.eigh(K)
          q = V[[3, 0, 1, 2], numpy.argmax(w)]
      if q[0] < 0.0:
          numpy.negative(q, q)
      return q
  
  
  def 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
  
      """
      w0, x0, y0, z0 = quaternion0
      w1, x1, y1, z1 = quaternion1
      return numpy.array([
          -x1*x0 - y1*y0 - z1*z0 + w1*w0,
          x1*w0 + y1*z0 - z1*y0 + w1*x0,
          -x1*z0 + y1*w0 + z1*x0 + w1*y0,
          x1*y0 - y1*x0 + z1*w0 + w1*z0], dtype=numpy.float64)
  
  
  def 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
  
      """
      q = numpy.array(quaternion, dtype=numpy.float64, copy=True)
      numpy.negative(q[1:], q[1:])
      return q
  
  
  def 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
  
      """
      q = numpy.array(quaternion, dtype=numpy.float64, copy=True)
      numpy.negative(q[1:], q[1:])
      return q / numpy.dot(q, q)
  
  
  def quaternion_real(quaternion):
      """Return real part of quaternion.
  
      >>> quaternion_real([3, 0, 1, 2])
      3.0
  
      """
      return float(quaternion[0])
  
  
  def quaternion_imag(quaternion):
      """Return imaginary part of quaternion.
  
      >>> quaternion_imag([3, 0, 1, 2])
      array([ 0.,  1.,  2.])
  
      """
      return numpy.array(quaternion[1:4], dtype=numpy.float64, copy=True)
  
  
  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)
      >>> 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
  
      """
      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
          numpy.negative(q1, q1)
      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, vector_norm(q))
      True
      >>> q = random_quaternion(numpy.random.random(3))
      >>> len(q.shape), q.shape[0]==4
      (1, True)
  
      """
      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.cos(t2)*r2, numpy.sin(t1)*r1,
                          numpy.cos(t1)*r1, numpy.sin(t2)*r2])
  
  
  def 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
  
      """
      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=[1, 0, 0, 0])
      >>> ball.place([320, 320], 320)
      >>> ball.setaxes([1, 1, 0], [-1, 1, 0])
      >>> ball.constrain = 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, 0.0, 1.0])
          self._constrain = False
          if initial is None:
              self._qdown = numpy.array([1.0, 0.0, 0.0, 0.0])
          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]
  
      @property
      def constrain(self):
          """Return state of constrain to axis mode."""
          return self._constrain
  
      @constrain.setter
      def constrain(self, value):
          """Set state of constrain to axis mode."""
          self._constrain = bool(value)
  
      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 = [numpy.dot(self._vdown, vnow), t[0], t[1], t[2]]
              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."""
      v0 = (point[0] - center[0]) / radius
      v1 = (center[1] - point[1]) / radius
      n = v0*v0 + v1*v1
      if n > 1.0:
          # position outside of sphere
          n = math.sqrt(n)
          return numpy.array([v0/n, v1/n, 0.0])
      else:
          return numpy.array([v0, v1, math.sqrt(1.0 - n)])
  
  
  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:
              numpy.negative(v, v)
          v /= n
          return v
      if a[2] == 1.0:
          return numpy.array([1.0, 0.0, 0.0])
      return unit_vector([-a[1], a[0], 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())
  
  
  def vector_norm(data, axis=None, out=None):
      """Return length, i.e. Euclidean 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
  
      """
      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. Euclidean 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]
  
      """
      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) and numpy.all(v < 1)
      True
      >>> v0 = random_vector(10)
      >>> v1 = random_vector(10)
      >>> numpy.any(v0 == v1)
      False
  
      """
      return numpy.random.random(size)
  
  
  def 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
  
      """
      return numpy.cross(v0, v1, axis=axis)
  
  
  def 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
  
      """
      v0 = numpy.array(v0, dtype=numpy.float64, copy=False)
      v1 = numpy.array(v1, dtype=numpy.float64, copy=False)
      dot = numpy.sum(v0 * v1, axis=axis)
      dot /= vector_norm(v0, axis=axis) * vector_norm(v1, axis=axis)
      dot = numpy.clip(dot, -1.0, 1.0)
      return numpy.arccos(dot if directed else numpy.fabs(dot))
  
  
  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 is_same_quaternion(q0, q1):
      """Return True if two quaternions are equal."""
      q0 = numpy.array(q0)
      q1 = numpy.array(q1)
      return numpy.allclose(q0, q1) or numpy.allclose(q0, -q1)
  
  
  def _import_module(name, package=None, 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.
  
      """
      import warnings
      from importlib import import_module
      try:
          if not package:
              module = import_module(name)
          else:
              module = import_module('.' + name, package=package)
      except ImportError:
          if warn:
              warnings.warn('failed to import module %s' % 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
  
  
  _import_module('_transformations', warn=False)
  
  
  if __name__ == '__main__':
      import doctest
      import random  # noqa: used in doctests
      try:
          numpy.set_printoptions(suppress=True, precision=5, legacy='1.13')
      except TypeError:
          numpy.set_printoptions(suppress=True, precision=5)
      doctest.testmod()