class_docstr: Rotation in 3 dimensions.
  
  This class provides an interface to initialize from and represent rotations
  with:
  
      - Quaternions
      - Direction Cosine Matrices
      - Rotation Vectors
      - Euler angles
  
  The following operations on rotations are supported:
  
      - Application on vectors
      - Rotation Composition
      - Rotation Inversion
      - Rotation Indexing
  
  Indexing within a rotation is supported since multiple rotation transforms
  can be stored within a single `Rotation` instance.
  
  To create `Rotation` objects use `from_...` classmethods, `__init__` is not
  supposed to be used directly.
  
  Methods
  -------
  __len__
  from_quat
  from_dcm
  from_rotvec
  from_euler
  as_quat
  as_dcm
  as_rotvec
  as_euler
  apply
  __mul__
  inv
  __getitem__
  random
  
  Examples
  --------
  >>> from scipy.spatial.transform import Rotation as R
  
  A `Rotation` instance can be initialized in any of the above formats and
  converted to any of the others. The underlying object is independent of the
  representation used for initialization.
  
  Consider a counter-clockwise rotation of 90 degrees about the z-axis. This
  corresponds to the following quaternion (in scalar-last format):
  
  >>> r = R.from_quat([0, 0, np.sin(np.pi/4), np.cos(np.pi/4)])
  
  The rotation can be expressed in any of the other formats:
  
  >>> r.as_dcm()
  array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
  [ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
  [ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
  >>> r.as_rotvec()
  array([0.        , 0.        , 1.57079633])
  >>> r.as_euler('zyx', degrees=True)
  array([90.,  0.,  0.])
  
  The same rotation can be initialized using a direction cosine matrix:
  
  >>> r = R.from_dcm(np.array([
  ... [0, -1, 0],
  ... [1, 0, 0],
  ... [0, 0, 1]]))
  
  Representation in other formats:
  
  >>> r.as_quat()
  array([0.        , 0.        , 0.70710678, 0.70710678])
  >>> r.as_rotvec()
  array([0.        , 0.        , 1.57079633])
  >>> r.as_euler('zyx', degrees=True)
  array([90.,  0.,  0.])
  
  The rotation vector corresponding to this rotation is given by:
  
  >>> r = R.from_rotvec(np.pi/2 * np.array([0, 0, 1]))
  
  Representation in other formats:
  
  >>> r.as_quat()
  array([0.        , 0.        , 0.70710678, 0.70710678])
  >>> r.as_dcm()
  array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
         [ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
         [ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
  >>> r.as_euler('zyx', degrees=True)
  array([90.,  0.,  0.])
  
  The `from_euler` function is quite flexible in the range of input formats
  it supports. Here we initialize a single rotation about a single axis:
  
  >>> r = R.from_euler('z', 90, degrees=True)
  
  Again, the object is representation independent and can be converted to any
  other format:
  
  >>> r.as_quat()
  array([0.        , 0.        , 0.70710678, 0.70710678])
  >>> r.as_dcm()
  array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
         [ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
         [ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
  >>> r.as_rotvec()
  array([0.        , 0.        , 1.57079633])
  
  It is also possible to initialize multiple rotations in a single instance
  using any of the `from_...` functions. Here we initialize a stack of 3
  rotations using the `from_euler` function:
  
  >>> r = R.from_euler('zyx', [
  ... [90, 0, 0],
  ... [0, 45, 0],
  ... [45, 60, 30]], degrees=True)
  
  The other representations also now return a stack of 3 rotations. For
  example:
  
  >>> r.as_quat()
  array([[0.        , 0.        , 0.70710678, 0.70710678],
         [0.        , 0.38268343, 0.        , 0.92387953],
         [0.39190384, 0.36042341, 0.43967974, 0.72331741]])
  
  Applying the above rotations onto a vector:
  
  >>> v = [1, 2, 3]
  >>> r.apply(v)
  array([[-2.        ,  1.        ,  3.        ],
         [ 2.82842712,  2.        ,  1.41421356],
         [ 2.24452282,  0.78093109,  2.89002836]])
  
  A `Rotation` instance can be indexed and sliced as if it were a single
  1D array or list:
  
  >>> r.as_quat()
  array([[0.        , 0.        , 0.70710678, 0.70710678],
         [0.        , 0.38268343, 0.        , 0.92387953],
         [0.39190384, 0.36042341, 0.43967974, 0.72331741]])
  >>> p = r[0]
  >>> p.as_dcm()
  array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
         [ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
         [ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
  >>> q = r[1:3]
  >>> q.as_quat()
  array([[0.        , 0.38268343, 0.        , 0.92387953],
         [0.39190384, 0.36042341, 0.43967974, 0.72331741]])
  
  Multiple rotations can be composed using the `*` operator:
  
  >>> r1 = R.from_euler('z', 90, degrees=True)
  >>> r2 = R.from_rotvec([np.pi/4, 0, 0])
  >>> v = [1, 2, 3]
  >>> r2.apply(r1.apply(v))
  array([-2.        , -1.41421356,  2.82842712])
  >>> r3 = r2 * r1 # Note the order
  >>> r3.apply(v)
  array([-2.        , -1.41421356,  2.82842712])
  
  Finally, it is also possible to invert rotations:
  
  >>> r1 = R.from_euler('z', [90, 45], degrees=True)
  >>> r2 = r1.inv()
  >>> r2.as_euler('zyx', degrees=True)
  array([[-90.,   0.,   0.],
         [-45.,   0.,   0.]])
  
  These examples serve as an overview into the `Rotation` class and highlight
  major functionalities. For more thorough examples of the range of input and
  output formats supported, consult the individual method's examples.
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docstr: Initialize from rotation vectors.
  
  A rotation vector is a 3 dimensional vector which is co-directional to
  the axis of rotation and whose norm gives the angle of rotation (in
  radians) [1]_.
  
  Parameters
  ----------
  rotvec : array_like, shape (N, 3) or (3,)
      A single vector or a stack of vectors, where `rot_vec[i]` gives
      the ith rotation vector.
  
  Returns
  -------
  rotation : `Rotation` instance
      Object containing the rotations represented by input rotation
      vectors.
  
  References
  ----------
  .. [1] `Rotation Vectors
          <https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector>`_
  
  Examples
  --------
  >>> from scipy.spatial.transform import Rotation as R
  
  Initialize a single rotation:
  
  >>> r = R.from_rotvec(np.pi/2 * np.array([0, 0, 1]))
  >>> r.as_rotvec()
  array([0.        , 0.        , 1.57079633])
  >>> r.as_rotvec().shape
  (3,)
  
  Initialize multiple rotations in one object:
  
  >>> r = R.from_rotvec([
  ... [0, 0, np.pi/2],
  ... [np.pi/2, 0, 0]])
  >>> r.as_rotvec()
  array([[0.        , 0.        , 1.57079633],
         [1.57079633, 0.        , 0.        ]])
  >>> r.as_rotvec().shape
  (2, 3)
  
  It is also possible to have a stack of a single rotaton:
  
  >>> r = R.from_rotvec([[0, 0, np.pi/2]])
  >>> r.as_rotvec().shape
  (1, 3)
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code: def from_rotvec(cls, rotvec):
      """Initialize from rotation vectors.
  
      A rotation vector is a 3 dimensional vector which is co-directional to
      the axis of rotation and whose norm gives the angle of rotation (in
      radians) [1]_.
  
      Parameters
      ----------
      rotvec : array_like, shape (N, 3) or (3,)
          A single vector or a stack of vectors, where `rot_vec[i]` gives
          the ith rotation vector.
  
      Returns
      -------
      rotation : `Rotation` instance
          Object containing the rotations represented by input rotation
          vectors.
  
      References
      ----------
      .. [1] `Rotation Vectors
              <https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector>`_
  
      Examples
      --------
      >>> from scipy.spatial.transform import Rotation as R
  
      Initialize a single rotation:
  
      >>> r = R.from_rotvec(np.pi/2 * np.array([0, 0, 1]))
      >>> r.as_rotvec()
      array([0.        , 0.        , 1.57079633])
      >>> r.as_rotvec().shape
      (3,)
  
      Initialize multiple rotations in one object:
  
      >>> r = R.from_rotvec([
      ... [0, 0, np.pi/2],
      ... [np.pi/2, 0, 0]])
      >>> r.as_rotvec()
      array([[0.        , 0.        , 1.57079633],
             [1.57079633, 0.        , 0.        ]])
      >>> r.as_rotvec().shape
      (2, 3)
  
      It is also possible to have a stack of a single rotaton:
  
      >>> r = R.from_rotvec([[0, 0, np.pi/2]])
      >>> r.as_rotvec().shape
      (1, 3)
  
      """
      is_single = False
      rotvec = np.asarray(rotvec, dtype=float)
  
      if rotvec.ndim not in [1, 2] or rotvec.shape[-1] != 3:
          raise ValueError(
              "Expected `rot_vec` to have shape (3,) "
              "or (N, 3), got {}".format(rotvec.shape)
          )
  
      # If a single vector is given, convert it to a 2D 1 x 3 matrix but
      # set self._single to True so that we can return appropriate objects
      # in the `as_...` methods
      if rotvec.shape == (3,):
          rotvec = rotvec[None, :]
          is_single = True
  
      num_rotations = rotvec.shape[0]
  
      norms = np.linalg.norm(rotvec, axis=1)
      small_angle = norms <= 1e-3
      large_angle = ~small_angle
  
      scale = np.empty(num_rotations)
      scale[small_angle] = (
          0.5
          - norms[small_angle] ** 2 / 48
          + norms[small_angle] ** 4 / 3840
      )
      scale[large_angle] = (
          np.sin(norms[large_angle] / 2) / norms[large_angle]
      )
  
      quat = np.empty((num_rotations, 4))
      quat[:, :3] = scale[:, None] * rotvec
      quat[:, 3] = np.cos(norms / 2)
  
      if is_single:
          return cls(quat[0], normalized=True, copy=False)
      else:
          return cls(quat, normalized=True, copy=False)
  
imports: from __future__ import division, print_function, absolute_import
  import re
  import warnings
  import numpy
  import scipy.linalg
  from scipy._lib._util import check_random_state
label_desc: Use image node_id 5 for calling the Rotation obj's underlying member method code with example usage: obj.from_rotvec(cls, rotvec) and returns: cls, cls