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File: C:/Users/fred/anaconda3/Lib/site-packages/scipy/spatial/transform/_rotation.cp312-win_amd64.pyd
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�����������%�����������%������������%�����������%�����������%�����������%J�����������%B�����������%:�����������%2�����������%*�����������%"�����������%�����������%�����������%z�����������%r�����������%j�����������%j�����������%b�����������%¢������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������: ASCIIAll dimensions preceding dimension %d must be indexed and not slicedApply this rotation to a set of vectors.

        If the original frame rotates to the final frame by this rotation, then
        its application to a vector can be seen in two ways:

            - As a projection of vector components expressed in the final frame
              to the original frame.
            - As the physical rotation of a vector being glued to the original
              frame as it rotates. In this case the vector components are
              expressed in the original frame before and after the rotation.

        In terms of rotation matrices, this application is the same as
        ``self.as_matrix() @ vectors``.

        Parameters
        ----------
        vectors : array_like, shape (3,) or (N, 3)
            Each `vectors[i]` represents a vector in 3D space. A single vector
            can either be specified with shape `(3, )` or `(1, 3)`. The number
            of rotations and number of vectors given must follow standard numpy
            broadcasting rules: either one of them equals unity or they both
            equal each other.
        inverse : boolean, optional
            If True then the inverse of the rotation(s) is applied to the input
            vectors. Default is False.

        Returns
        -------
        rotated_vectors : ndarray, shape (3,) or (N, 3)
            Result of applying rotation on input vectors.
            Shape depends on the following cases:

                - If object contains a single rotation (as opposed to a stack
                  with a single rotation) and a single vector is specified with
                  shape ``(3,)``, then `rotated_vectors` has shape ``(3,)``.
                - In all other cases, `rotated_vectors` has shape ``(N, 3)``,
                  where ``N`` is either the number of rotations or vectors.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Single rotation applied on a single vector:

        >>> vector = np.array([1, 0, 0])
        >>> r = R.from_rotvec([0, 0, np.pi/2])
        >>> r.as_matrix()
        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.apply(vector)
        array([2.22044605e-16, 1.00000000e+00, 0.00000000e+00])
        >>> r.apply(vector).shape
        (3,)

        Single rotation applied on multiple vectors:

        >>> vectors = np.array([
        ... [1, 0, 0],
        ... [1, 2, 3]])
        >>> r = R.from_rotvec([0, 0, np.pi/4])
        >>> r.as_matrix()
        array([[ 0.70710678, -0.70710678,  0.        ],
               [ 0.70710678,  0.70710678,  0.        ],
               [ 0.        ,  0.        ,  1.        ]])
        >>> r.apply(vectors)
        array([[ 0.70710678,  0.70710678,  0.        ],
               [-0.70710678,  2.12132034,  3.        ]])
        >>> r.apply(vectors).shape
        (2, 3)

        Multiple rotations on a single vector:

        >>> r = R.from_rotvec([[0, 0, np.pi/4], [np.pi/2, 0, 0]])
        >>> vector = np.array([1,2,3])
        >>> r.as_matrix()
        array([[[ 7.07106781e-01, -7.07106781e-01,  0.00000000e+00],
                [ 7.07106781e-01,  7.07106781e-01,  0.00000000e+00],
                [ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]],
               [[ 1.00000000e+00,  0.00000000e+00,  0.00000000e+00],
                [ 0.00000000e+00,  2.22044605e-16, -1.00000000e+00],
                [ 0.00000000e+00,  1.00000000e+00,  2.22044605e-16]]])
        >>> r.apply(vector)
        array([[-0.70710678,  2.12132034,  3.        ],
               [ 1.        , -3.        ,  2.        ]])
        >>> r.apply(vector).shape
        (2, 3)

        Multiple rotations on multiple vectors. Each rotation is applied on the
        corresponding vector:

        >>> r = R.from_euler('zxy', [
        ... [0, 0, 90],
        ... [45, 30, 60]], degrees=True)
        >>> vectors = [
        ... [1, 2, 3],
        ... [1, 0, -1]]
        >>> r.apply(vectors)
        array([[ 3.        ,  2.        , -1.        ],
               [-0.09026039,  1.11237244, -0.86860844]])
        >>> r.apply(vectors).shape
        (2, 3)

        It is also possible to apply the inverse rotation:

        >>> r = R.from_euler('zxy', [
        ... [0, 0, 90],
        ... [45, 30, 60]], degrees=True)
        >>> vectors = [
        ... [1, 2, 3],
        ... [1, 0, -1]]
        >>> r.apply(vectors, inverse=True)
        array([[-3.        ,  2.        ,  1.        ],
               [ 1.09533535, -0.8365163 ,  0.3169873 ]])

        AssertionErrorAxes must be vectors of length 3.BBuffer view does not expose stridesCC:\b\abs_efv75hqhju\croot\scipy_1717521501389\work\scipy\spatial\transform\_rotation.pyxCan only create a buffer that is contiguous in memory.Cannot align zero length primary vectorsCannot assign to read-only memoryviewCannot create writable memory view from read-only memoryviewCannot index with type 'Cannot return sensitivity matrix with an infinite weight or one vector pairCannot transpose memoryview with indirect dimensionsCompose this rotation with itself `n` times.

        Composition of a rotation ``p`` with itself can be extended to
        non-integer ``n`` by considering the power ``n`` to be a scale factor
        applied to the angle of rotation about the rotation's fixed axis. The
        expression ``q = p ** n`` can also be expressed as
        ``q = Rotation.from_rotvec(n * p.as_rotvec())``.

        If ``n`` is negative, then the rotation is inverted before the power
        is applied. In other words, ``p ** -abs(n) == p.inv() ** abs(n)``.

        Parameters
        ----------
        n : float
            The number of times to compose the rotation with itself.
        modulus : None
            This overridden argument is not applicable to Rotations and must be
            ``None``.

        Returns
        -------
        power : `Rotation` instance
            If the input Rotation ``p`` contains ``N`` multiple rotations, then
            the output will contain ``N`` rotations where the ``i`` th rotation
            is equal to ``p[i] ** n``

        Notes
        -----
        For example, a power of 2 will double the angle of rotation, and a
        power of 0.5 will halve the angle. There are three notable cases: if
        ``n == 1`` then the original rotation is returned, if ``n == 0``
        then the identity rotation is returned, and if ``n == -1`` then
        ``p.inv()`` is returned.

        Note that fractional powers ``n`` which effectively take a root of
        rotation, do so using the shortest path smallest representation of that
        angle (the principal root). This means that powers of ``n`` and ``1/n``
        are not necessarily inverses of each other. For example, a 0.5 power of
        a +240 degree rotation will be calculated as the 0.5 power of a -120
        degree rotation, with the result being a rotation of -60 rather than
        +120 degrees.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Raising a rotation to a power:

        >>> p = R.from_rotvec([1, 0, 0])
        >>> q = p ** 2
        >>> q.as_rotvec()
        array([2., 0., 0.])
        >>> r = p ** 0.5
        >>> r.as_rotvec()
        array([0.5, 0., 0.])

        Inverse powers do not necessarily cancel out:

        >>> p = R.from_rotvec([0, 0, 120], degrees=True)
        >>> ((p ** 2) ** 0.5).as_rotvec(degrees=True)
        array([  -0.,   -0., -60.])

        Compose this rotation with the other.

        If `p` and `q` are two rotations, then the composition of 'q followed
        by p' is equivalent to `p * q`. In terms of rotation matrices,
        the composition can be expressed as
        ``p.as_matrix() @ q.as_matrix()``.

        Parameters
        ----------
        other : `Rotation` instance
            Object containing the rotations to be composed with this one. Note
            that rotation compositions are not commutative, so ``p * q`` is
            generally different from ``q * p``.

        Returns
        -------
        composition : `Rotation` instance
            This function supports composition of multiple rotations at a time.
            The following cases are possible:

            - Either ``p`` or ``q`` contains a single rotation. In this case
              `composition` contains the result of composing each rotation in
              the other object with the single rotation.
            - Both ``p`` and ``q`` contain ``N`` rotations. In this case each
              rotation ``p[i]`` is composed with the corresponding rotation
              ``q[i]`` and `output` contains ``N`` rotations.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Composition of two single rotations:

        >>> p = R.from_quat([0, 0, 1, 1])
        >>> q = R.from_quat([1, 0, 0, 1])
        >>> p.as_matrix()
        array([[ 0., -1.,  0.],
               [ 1.,  0.,  0.],
               [ 0.,  0.,  1.]])
        >>> q.as_matrix()
        array([[ 1.,  0.,  0.],
               [ 0.,  0., -1.],
               [ 0.,  1.,  0.]])
        >>> r = p * q
        >>> r.as_matrix()
        array([[0., 0., 1.],
               [1., 0., 0.],
               [0., 1., 0.]])

        Composition of two objects containing equal number of rotations:

        >>> p = R.from_quat([[0, 0, 1, 1], [1, 0, 0, 1]])
        >>> q = R.from_rotvec([[np.pi/4, 0, 0], [-np.pi/4, 0, np.pi/4]])
        >>> p.as_quat()
        array([[0.        , 0.        , 0.70710678, 0.70710678],
               [0.70710678, 0.        , 0.        , 0.70710678]])
        >>> q.as_quat()
        array([[ 0.38268343,  0.        ,  0.        ,  0.92387953],
               [-0.37282173,  0.        ,  0.37282173,  0.84971049]])
        >>> r = p * q
        >>> r.as_quat()
        array([[ 0.27059805,  0.27059805,  0.65328148,  0.65328148],
               [ 0.33721128, -0.26362477,  0.26362477,  0.86446082]])

        Concatenate a sequence of `Rotation` objects into a single object.

        This is useful if you want to, for example, take the mean of a set of
        rotations and need to pack them into a single object to do so.

        Parameters
        ----------
        rotations : sequence of `Rotation` objects
            The rotations to concatenate.

        Returns
        -------
        concatenated : `Rotation` instance
            The concatenated rotations.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> r1 = R.from_rotvec([0, 0, 1])
        >>> r2 = R.from_rotvec([0, 0, 2])
        >>> rc = R.concatenate([r1, r2])
        >>> rc.as_rotvec()
        array([[0., 0., 1.],
               [0., 0., 2.]])
        >>> rc.mean().as_rotvec()
        array([0., 0., 1.5])

        Note that it may be simpler to create the desired rotations by passing
        in a single list of the data during initialization, rather then by
        concatenating:

        >>> R.from_rotvec([[0, 0, 1], [0, 0, 2]]).as_rotvec()
        array([[0., 0., 1.],
               [0., 0., 2.]])

        Notes
        -----
        .. versionadded:: 1.8.0
        Consecutive axes must be orthogonal.Determine if another rotation is approximately equal to this one.

        Equality is measured by calculating the smallest angle between the
        rotations, and checking to see if it is smaller than `atol`.

        Parameters
        ----------
        other : `Rotation` instance
            Object containing the rotations to measure against this one.
        atol : float, optional
            The absolute angular tolerance, below which the rotations are
            considered equal. If not given, then set to 1e-8 radians by
            default.
        degrees : bool, optional
            If True and `atol` is given, then `atol` is measured in degrees. If
            False (default), then atol is measured in radians.

        Returns
        -------
        approx_equal : ndarray or bool
            Whether the rotations are approximately equal, bool if object
            contains a single rotation and ndarray if object contains multiple
            rotations.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np
        >>> p = R.from_quat([0, 0, 0, 1])
        >>> q = R.from_quat(np.eye(4))
        >>> p.approx_equal(q)
        array([False, False, False, True])

        Approximate equality for a single rotation:

        >>> p.approx_equal(q[0])
        False
        Dimension %d is not directEllipsisEmpty shape tuple for cython.arrayEstimate a rotation to optimally align two sets of vectors.

        Find a rotation between frames A and B which best aligns a set of
        vectors `a` and `b` observed in these frames. The following loss
        function is minimized to solve for the rotation matrix
        :math:`C`:

        .. math::

            L(C) = \frac{1}{2} \sum_{i = 1}^{n} w_i \lVert \mathbf{a}_i -
            C \mathbf{b}_i \rVert^2 ,

        where :math:`w_i`'s are the `weights` corresponding to each vector.

        The rotation is estimated with Kabsch algorithm [1]_, and solves what
        is known as the "pointing problem", or "Wahba's problem" [2]_.

        There are two special cases. The first is if a single vector is given
        for `a` and `b`, in which the shortest distance rotation that aligns
        `b` to `a` is returned.

        The second is when one of the weights is infinity. In this case, the
        shortest distance rotation between the primary infinite weight vectors
        is calculated as above. Then, the rotation about the aligned primary
        vectors is calculated such that the secondary vectors are optimally
        aligned per the above loss function. The result is the composition
        of these two rotations. The result via this process is the same as the
        Kabsch algorithm as the corresponding weight approaches infinity in
        the limit. For a single secondary vector this is known as the
        "align-constrain" algorithm [3]_.

        For both special cases (single vectors or an infinite weight), the
        sensitivity matrix does not have physical meaning and an error will be
        raised if it is requested. For an infinite weight, the primary vectors
        act as a constraint with perfect alignment, so their contribution to
        `rssd` will be forced to 0 even if they are of different lengths.

        Parameters
        ----------
        a : array_like, shape (3,) or (N, 3)
            Vector components observed in initial frame A. Each row of `a`
            denotes a vector.
        b : array_like, shape (3,) or (N, 3)
            Vector components observed in another frame B. Each row of `b`
            denotes a vector.
        weights : array_like shape (N,), optional
            Weights describing the relative importance of the vector
            observations. If None (default), then all values in `weights` are
            assumed to be 1. One and only one weight may be infinity, and
            weights must be positive.
        return_sensitivity : bool, optional
            Whether to return the sensitivity matrix. See Notes for details.
            Default is False.

        Returns
        -------
        rotation : `Rotation` instance
            Best estimate of the rotation that transforms `b` to `a`.
        rssd : float
            Stands for "root sum squared distance". Square root of the weighted
            sum of the squared distances between the given sets of vectors
            after alignment. It is equal to ``sqrt(2 * minimum_loss)``, where
            ``minimum_loss`` is the loss function evaluated for the found
            optimal rotation.
            Note that the result will also be weighted by the vectors'
            magnitudes, so perfectly aligned vector pairs will have nonzero
            `rssd` if they are not of the same length. So, depending on the
            use case it may be desirable to normalize the input vectors to unit
            length before calling this method.
        sensitivity_matrix : ndarray, shape (3, 3)
            Sensitivity matrix of the estimated rotation estimate as explained
            in Notes. Returned only when `return_sensitivity` is True. Not
            valid if aligning a single pair of vectors or if there is an
            infinite weight, in which cases an error will be raised.

        Notes
        -----
        The sensitivity matrix gives the sensitivity of the estimated rotation
        to small perturbations of the vector measurements. Specifically we
        consider the rotation estimate error as a small rotation vector of
        frame A. The sensitivity matrix is proportional to the covariance of
        this rotation vector assuming that the vectors in `a` was measured with
        errors significantly less than their lengths. To get the true
        covariance matrix, the returned sensitivity matrix must be multiplied
        by harmonic mean [4]_ of variance in each observation. Note that
        `weights` are supposed to be inversely proportional to the observation
        variances to get consistent results. For example, if all vectors are
        measured with the same accuracy of 0.01 (`weights` must be all equal),
        then you should multiple the sensitivity matrix by 0.01**2 to get the
        covariance.

        Refer to [5]_ for more rigorous discussion of the covariance
        estimation. See [6]_ for more discussion of the pointing problem and
        minimal proper pointing.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Kabsch_algorithm
        .. [2] https://en.wikipedia.org/wiki/Wahba%27s_problem
        .. [3] Magner, Robert,
                "Extending target tracking capabilities through trajectory and
                momentum setpoint optimization." Small Satellite Conference,
                2018.
        .. [4] https://en.wikipedia.org/wiki/Harmonic_mean
        .. [5] F. Landis Markley,
                "Attitude determination using vector observations: a fast
                optimal matrix algorithm", Journal of Astronautical Sciences,
                Vol. 41, No.2, 1993, pp. 261-280.
        .. [6] Bar-Itzhack, Itzhack Y., Daniel Hershkowitz, and Leiba Rodman,
                "Pointing in Real Euclidean Space", Journal of Guidance,
                Control, and Dynamics, Vol. 20, No. 5, 1997, pp. 916-922.

        Examples
        --------
        >>> import numpy as np
        >>> from scipy.spatial.transform import Rotation as R

        Here we run the baseline Kabsch algorithm to best align two sets of
        vectors, where there is noise on the last two vector measurements of
        the ``b`` set:

        >>> a = [[0, 1, 0], [0, 1, 1], [0, 1, 1]]
        >>> b = [[1, 0, 0], [1, 1.1, 0], [1, 0.9, 0]]
        >>> rot, rssd, sens = R.align_vectors(a, b, return_sensitivity=True)
        >>> rot.as_matrix()
        array([[0., 0., 1.],
               [1., 0., 0.],
               [0., 1., 0.]])

        When we apply the rotation to ``b``, we get vectors close to ``a``:

        >>> rot.apply(b)
        array([[0. , 1. , 0. ],
               [0. , 1. , 1.1],
               [0. , 1. , 0.9]])

        The error for the first vector is 0, and for the last two the error is
        magnitude 0.1. The `rssd` is the square root of the sum of the
        weighted squared errors, and the default weights are all 1, so in this
        case the `rssd` is calculated as
        ``sqrt(1 * 0**2 + 1 * 0.1**2 + 1 * (-0.1)**2) = 0.141421356237308``

        >>> a - rot.apply(b)
        array([[ 0., 0.,  0. ],
               [ 0., 0., -0.1],
               [ 0., 0.,  0.1]])
        >>> np.sqrt(np.sum(np.ones(3) @ (a - rot.apply(b))**2))
        0.141421356237308
        >>> rssd
        0.141421356237308

        The sensitivity matrix for this example is as follows:

        >>> sens
        array([[0.2, 0. , 0.],
               [0. , 1.5, 1.],
               [0. , 1. , 1.]])

        Special case 1: Find a minimum rotation between single vectors:

        >>> a = [1, 0, 0]
        >>> b = [0, 1, 0]
        >>> rot, _ = R.align_vectors(a, b)
        >>> rot.as_matrix()
        array([[0., 1., 0.],
               [-1., 0., 0.],
               [0., 0., 1.]])
        >>> rot.apply(b)
        array([1., 0., 0.])

        Special case 2: One infinite weight. Here we find a rotation between
        primary and secondary vectors that can align exactly:

        >>> a = [[0, 1, 0], [0, 1, 1]]
        >>> b = [[1, 0, 0], [1, 1, 0]]
        >>> rot, _ = R.align_vectors(a, b, weights=[np.inf, 1])
        >>> rot.as_matrix()
        array([[0., 0., 1.],
               [1., 0., 0.],
               [0., 1., 0.]])
        >>> rot.apply(b)
        array([[0., 1., 0.],
               [0., 1., 1.]])

        Here the secondary vectors must be best-fit:

        >>> a = [[0, 1, 0], [0, 1, 1]]
        >>> b = [[1, 0, 0], [1, 2, 0]]
        >>> rot, _ = R.align_vectors(a, b, weights=[np.inf, 1])
        >>> rot.as_matrix()
        array([[0., 0., 1.],
               [1., 0., 0.],
               [0., 1., 0.]])
        >>> rot.apply(b)
        array([[0., 1., 0.],
               [0., 1., 2.]])
        Expected 3 axes, got {}.Expected `angles` parameter to have shape (N, 1), got {}.Expected `angles` to be at most 2-dimensional with width equal to number of axes specified, got {} for shapeExpected angles to have shape (num_rotations, num_axes), got {}.Expected axes from `seq` to be from ['x', 'y', 'z'] or ['X', 'Y', 'Z'], got {}Expected axis specification to be a non-empty string of upto 3 characters, got {}Expected consecutive axes to be different, got {}Expected equal number of rotations in both or a single rotation in either object, got {} rotations in first and {} rotations in second object.Expected equal numbers of rotations and vectors , or a single rotation, or a single vector, got {} rotations and {} vectors.Expected float, 1D array, or 2D array for parameter `angles` corresponding to `seq`, got shape {}.Expected input `a` to have shape (3,) or (N, 3), got {}Expected input `b` to have shape (3,) or (N, 3), got {}Expected input of shape (3,) or (P, 3), got {}.Expected inputs `a` and `b` to have same shapes, got {} and {} respectively.Expected `matrix` to have shape (3, 3) or (N, 3, 3), got {}Expected `mrp` to have shape (3,) or (N, 3), got {}Expected number of rotations to be equal to number of timestamps given, got {} rotations and {} timestamps.Expected `quat` to have shape (4,) or (N, 4), got Expected `rot_vec` to have shape (3,) or (N, 3), got {}Expected times to be specified in a 1 dimensional array, got {} dimensions.Expected up to 3 axes, got {}Expected `weights` to be 1 dimensional, got shape {}.Expected `weights` to have number of values equal to number of rotations, got {} values and {} rotations.Expected `weights` to have number of values equal to number of input vectors, got {} values and {} vectors.Extract rotation(s) at given index(es) from object.

        Create a new `Rotation` instance containing a subset of rotations
        stored in this object.

        Parameters
        ----------
        indexer : index, slice, or index array
            Specifies which rotation(s) to extract. A single indexer must be
            specified, i.e. as if indexing a 1 dimensional array or list.

        Returns
        -------
        rotation : `Rotation` instance
            Contains
                - a single rotation, if `indexer` is a single index
                - a stack of rotation(s), if `indexer` is a slice, or and index
                  array.

        Raises
        ------
        TypeError if the instance was created as a single rotation.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> r = R.from_quat([
        ... [1, 1, 0, 0],
        ... [0, 1, 0, 1],
        ... [1, 1, -1, 0]])
        >>> r.as_quat()
        array([[ 0.70710678,  0.70710678,  0.        ,  0.        ],
               [ 0.        ,  0.70710678,  0.        ,  0.70710678],
               [ 0.57735027,  0.57735027, -0.57735027,  0.        ]])

        Indexing using a single index:

        >>> p = r[0]
        >>> p.as_quat()
        array([0.70710678, 0.70710678, 0.        , 0.        ])

        Array slicing:

        >>> q = r[1:3]
        >>> q.as_quat()
        array([[ 0.        ,  0.70710678,  0.        ,  0.70710678],
               [ 0.57735027,  0.57735027, -0.57735027,  0.        ]])

        Found zero norm quaternions in `quat`.Generate uniformly distributed rotations.

        Parameters
        ----------
        num : int or None, optional
            Number of random rotations to generate. If None (default), then a
            single rotation is generated.
        random_state : {None, int, `numpy.random.Generator`,
                        `numpy.random.RandomState`}, optional

            If `seed` is None (or `np.random`), the `numpy.random.RandomState`
            singleton is used.
            If `seed` is an int, a new ``RandomState`` instance is used,
            seeded with `seed`.
            If `seed` is already a ``Generator`` or ``RandomState`` instance
            then that instance is used.

        Returns
        -------
        random_rotation : `Rotation` instance
            Contains a single rotation if `num` is None. Otherwise contains a
            stack of `num` rotations.

        Notes
        -----
        This function is optimized for efficiently sampling random rotation
        matrices in three dimensions. For generating random rotation matrices
        in higher dimensions, see `scipy.stats.special_ortho_group`.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Sample a single rotation:

        >>> R.random().as_euler('zxy', degrees=True)
        array([-110.5976185 ,   55.32758512,   76.3289269 ])  # random

        Sample a stack of rotations:

        >>> R.random(5).as_euler('zxy', degrees=True)
        array([[-110.5976185 ,   55.32758512,   76.3289269 ],  # random
               [ -91.59132005,  -14.3629884 ,  -93.91933182],
               [  25.23835501,   45.02035145, -121.67867086],
               [ -51.51414184,  -15.29022692, -172.46870023],
               [ -81.63376847,  -27.39521579,    2.60408416]])

        See Also
        --------
        scipy.stats.special_ortho_group

       Get the magnitude(s) of the rotation(s).

        Returns
        -------
        magnitude : ndarray or float
            Angle(s) in radians, float if object contains a single rotation
            and ndarray if object contains multiple rotations. The magnitude
            will always be in the range [0, pi].

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np
        >>> r = R.from_quat(np.eye(4))
        >>> r.as_quat()
        array([[ 1., 0., 0., 0.],
               [ 0., 1., 0., 0.],
               [ 0., 0., 1., 0.],
               [ 0., 0., 0., 1.]])
        >>> r.magnitude()
        array([3.14159265, 3.14159265, 3.14159265, 0.        ])

        Magnitude of a single rotation:

        >>> r[0].magnitude()
        3.141592653589793
        Get the mean of the rotations.

        The mean used is the chordal L2 mean (also called the projected or
        induced arithmetic mean) [1]_. If ``A`` is a set of rotation matrices,
        then the mean ``M`` is the rotation matrix that minimizes the
        following loss function:

        .. math::

            L(M) = \sum_{i = 1}^{n} w_i \lVert \mathbf{A}_i -
            \mathbf{M} \rVert^2 ,

        where :math:`w_i`'s are the `weights` corresponding to each matrix.

        Parameters
        ----------
        weights : array_like shape (N,), optional
            Weights describing the relative importance of the rotations. If
            None (default), then all values in `weights` are assumed to be
            equal.

        Returns
        -------
        mean : `Rotation` instance
            Object containing the mean of the rotations in the current
            instance.

        References
        ----------
        .. [1] Hartley, Richard, et al.,
                "Rotation Averaging", International Journal of Computer Vision
                103, 2013, pp. 267-305.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> r = R.from_euler('zyx', [[0, 0, 0],
        ...                          [1, 0, 0],
        ...                          [0, 1, 0],
        ...                          [0, 0, 1]], degrees=True)
        >>> r.mean().as_euler('zyx', degrees=True)
        array([0.24945696, 0.25054542, 0.24945696])
        Gimbal lock detected. Setting third angle to zero since it is not possible to uniquely determine all angles.ImportErrorIncompatible checksums (0x%x vs (0x82a3537, 0x6ae9995, 0xb068931) = (name))Incompatible checksums (0x%x vs (0x0fbb6f7, 0x22c69a8, 0x14ed78b) = (_quat, _single))IndexErrorIndex out of bounds (axis %d)Indirect dimensions not supportedInitialize from Davenport angles.

        Rotations in 3-D can be represented by a sequence of 3
        rotations around a sequence of axes.

        The three rotations can either be in a global frame of reference
        (extrinsic) or in a body centred frame of reference (intrinsic), which
        is attached to, and moves with, the object under rotation [1]_.

        For both Euler angles and Davenport angles, consecutive axes must 
        be are orthogonal (``axis2`` is orthogonal to both ``axis1`` and 
        ``axis3``). For Euler angles, there is an additional relationship 
        between ``axis1`` or ``axis3``, with two possibilities:

            - ``axis1`` and ``axis3`` are also orthogonal (asymmetric sequence)
            - ``axis1 == axis3`` (symmetric sequence)

        For Davenport angles, this last relationship is relaxed [2]_, and only
        the consecutive orthogonal axes requirement is maintained.

        Parameters
        ----------
        axes : array_like, shape (3,) or ([1 or 2 or 3], 3)
            Axis of rotation, if one dimensional. If two dimensional, describes the 
            sequence of axes for rotations, where each axes[i, :] is the ith
            axis. If more than one axis is given, then the second axis must be
            orthogonal to both the first and third axes.
        order : string
            If it is equal to 'e' or 'extrinsic', the sequence will be 
            extrinsic. If it is equal to 'i' or 'intrinsic', sequence 
            will be treated as intrinsic.
        angles : float or array_like, shape (N,) or (N, [1 or 2 or 3])
            Euler angles specified in radians (`degrees` is False) or degrees
            (`degrees` is True).
            For a single axis, `angles` can be:

            - a single value
            - array_like with shape (N,), where each `angle[i]`
              corresponds to a single rotation
            - array_like with shape (N, 1), where each `angle[i, 0]`
              corresponds to a single rotation

            For 2 and 3 axes, `angles` can be:

            - array_like with shape (W,) where `W` is the number of rows of
              `axes`, which corresponds to a single rotation with `W` axes
            - array_like with shape (N, W) where each `angle[i]`
              corresponds to a sequence of Davenport angles describing a 
              single rotation

        degrees : bool, optional
            If True, then the given angles are assumed to be in degrees. 
            Default is False.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotation represented by the sequence of
            rotations around given axes with given angles.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations
        .. [2] Shuster, Malcolm & Markley, Landis. (2003). Generalization of
               the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Davenport angles are a generalization of Euler angles, when we use the
        canonical basis axes:

        >>> ex = [1, 0, 0]
        >>> ey = [0, 1, 0]
        >>> ez = [0, 0, 1]

        Initialize a single rotation with a given axis sequence:

        >>> axes = [ez, ey, ex]
        >>> r = R.from_davenport(axes, 'extrinsic', [90, 0, 0], degrees=True)
        >>> r.as_quat().shape
        (4,)

        It is equivalent to Euler angles in this case:

        >>> r.as_euler('zyx', degrees=True)
        array([90.,  0., -0.])

        Initialize multiple rotations in one object:

        >>> r = R.from_davenport(axes, 'extrinsic', [[90, 45, 30], [35, 45, 90]], degrees=True)
        >>> r.as_quat().shape
        (2, 4)

        Using only one or two axes is also possible:

        >>> r = R.from_davenport([ez, ex], 'extrinsic', [[90, 45], [35, 45]], degrees=True)
        >>> r.as_quat().shape
        (2, 4)

        Non-canonical axes are possible, and they do not need to be normalized,
        as long as consecutive axes are orthogonal:

        >>> e1 = [2, 0, 0]
        >>> e2 = [0, 1, 0]
        >>> e3 = [1, 0, 1]
        >>> axes = [e1, e2, e3]
        >>> r = R.from_davenport(axes, 'extrinsic', [90, 45, 30], degrees=True)
        >>> r.as_quat()
        [ 0.701057,  0.430459, -0.092296,  0.560986]
        Initialize from Euler angles.

        Rotations in 3-D can be represented by a sequence of 3
        rotations around a sequence of axes. In theory, any three axes spanning
        the 3-D Euclidean space are enough. In practice, the axes of rotation are
        chosen to be the basis vectors.

        The three rotations can either be in a global frame of reference
        (extrinsic) or in a body centred frame of reference (intrinsic), which
        is attached to, and moves with, the object under rotation [1]_.

        Parameters
        ----------
        seq : string
            Specifies sequence of axes for rotations. Up to 3 characters
            belonging to the set {'X', 'Y', 'Z'} for intrinsic rotations, or
            {'x', 'y', 'z'} for extrinsic rotations. Extrinsic and intrinsic
            rotations cannot be mixed in one function call.
        angles : float or array_like, shape (N,) or (N, [1 or 2 or 3])
            Euler angles specified in radians (`degrees` is False) or degrees
            (`degrees` is True).
            For a single character `seq`, `angles` can be:

            - a single value
            - array_like with shape (N,), where each `angle[i]`
              corresponds to a single rotation
            - array_like with shape (N, 1), where each `angle[i, 0]`
              corresponds to a single rotation

            For 2- and 3-character wide `seq`, `angles` can be:

            - array_like with shape (W,) where `W` is the width of
              `seq`, which corresponds to a single rotation with `W` axes
            - array_like with shape (N, W) where each `angle[i]`
              corresponds to a sequence of Euler angles describing a single
              rotation

        degrees : bool, optional
            If True, then the given angles are assumed to be in degrees.
            Default is False.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotation represented by the sequence of
            rotations around given axes with given angles.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Initialize a single rotation along a single axis:

        >>> r = R.from_euler('x', 90, degrees=True)
        >>> r.as_quat().shape
        (4,)

        Initialize a single rotation with a given axis sequence:

        >>> r = R.from_euler('zyx', [90, 45, 30], degrees=True)
        >>> r.as_quat().shape
        (4,)

        Initialize a stack with a single rotation around a single axis:

        >>> r = R.from_euler('x', [90], degrees=True)
        >>> r.as_quat().shape
        (1, 4)

        Initialize a stack with a single rotation with an axis sequence:

        >>> r = R.from_euler('zyx', [[90, 45, 30]], degrees=True)
        >>> r.as_quat().shape
        (1, 4)

        Initialize multiple elementary rotations in one object:

        >>> r = R.from_euler('x', [90, 45, 30], degrees=True)
        >>> r.as_quat().shape
        (3, 4)

        Initialize multiple rotations in one object:

        >>> r = R.from_euler('zyx', [[90, 45, 30], [35, 45, 90]], degrees=True)
        >>> r.as_quat().shape
        (2, 4)

        Initialize from Modified Rodrigues Parameters (MRPs).

        MRPs are a 3 dimensional vector co-directional to the axis of rotation and whose
        magnitude is equal to ``tan(theta / 4)``, where ``theta`` is the angle of rotation
        (in radians) [1]_.

        MRPs have a singularity at 360 degrees which can be avoided by ensuring the angle of
        rotation does not exceed 180 degrees, i.e. switching the direction of the rotation when
        it is past 180 degrees.

        Parameters
        ----------
        mrp : array_like, shape (N, 3) or (3,)
            A single vector or a stack of vectors, where `mrp[i]` gives
            the ith set of MRPs.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotations represented by input MRPs.

        References
        ----------
        .. [1] Shuster, M. D. "A Survey of Attitude Representations",
               The Journal of Astronautical Sciences, Vol. 41, No.4, 1993,
               pp. 475-476

        Notes
        -----

        .. versionadded:: 1.6.0

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Initialize a single rotation:

        >>> r = R.from_mrp([0, 0, 1])
        >>> r.as_euler('xyz', degrees=True)
        array([0.        , 0.        , 180.      ])
        >>> r.as_euler('xyz').shape
        (3,)

        Initialize multiple rotations in one object:

        >>> r = R.from_mrp([
        ... [0, 0, 1],
        ... [1, 0, 0]])
        >>> r.as_euler('xyz', degrees=True)
        array([[0.        , 0.        , 180.      ],
               [180.0     , 0.        , 0.        ]])
        >>> r.as_euler('xyz').shape
        (2, 3)

        It is also possible to have a stack of a single rotation:

        >>> r = R.from_mrp([[0, 0, np.pi/2]])
        >>> r.as_euler('xyz').shape
        (1, 3)

        Initialize from quaternions.

        3D rotations can be represented using unit-norm quaternions [1]_.

        Advanced users may be interested in the "double cover" of 3D space by
        the quaternion representation [2]_. As of version 1.11.0, the
        following subset (and only this subset) of operations on a `Rotation`
        ``r`` corresponding to a quaternion ``q`` are guaranteed to preserve
        the double cover property: ``r = Rotation.from_quat(q)``,
        ``r.as_quat(canonical=False)``, ``r.inv()``, and composition using the
        ``*`` operator such as ``r*r``.

        Parameters
        ----------
        quat : array_like, shape (N, 4) or (4,)
            Each row is a (possibly non-unit norm) quaternion representing an
            active rotation, in scalar-last (x, y, z, w) format. Each
            quaternion will be normalized to unit norm.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotations represented by input quaternions.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
        .. [2] Hanson, Andrew J. "Visualizing quaternions."
            Morgan Kaufmann Publishers Inc., San Francisco, CA. 2006.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Initialize a single rotation:

        >>> r = R.from_quat([1, 0, 0, 0])
        >>> r.as_quat()
        array([1., 0., 0., 0.])
        >>> r.as_quat().shape
        (4,)

        Initialize multiple rotations in a single object:

        >>> r = R.from_quat([
        ... [1, 0, 0, 0],
        ... [0, 0, 0, 1]
        ... ])
        >>> r.as_quat()
        array([[1., 0., 0., 0.],
               [0., 0., 0., 1.]])
        >>> r.as_quat().shape
        (2, 4)

        It is also possible to have a stack of a single rotation:

        >>> r = R.from_quat([[0, 0, 0, 1]])
        >>> r.as_quat()
        array([[0., 0., 0., 1.]])
        >>> r.as_quat().shape
        (1, 4)

        Quaternions are normalized before initialization.

        >>> r = R.from_quat([0, 0, 1, 1])
        >>> r.as_quat()
        array([0.        , 0.        , 0.70710678, 0.70710678])
        Initialize from rotation matrix.

        Rotations in 3 dimensions can be represented with 3 x 3 proper
        orthogonal matrices [1]_. If the input is not proper orthogonal,
        an approximation is created using the method described in [2]_.

        Parameters
        ----------
        matrix : array_like, shape (N, 3, 3) or (3, 3)
            A single matrix or a stack of matrices, where ``matrix[i]`` is
            the i-th matrix.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotations represented by the rotation
            matrices.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions
        .. [2] F. Landis Markley, "Unit Quaternion from Rotation Matrix",
               Journal of guidance, control, and dynamics vol. 31.2, pp.
               440-442, 2008.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Initialize a single rotation:

        >>> r = R.from_matrix([
        ... [0, -1, 0],
        ... [1, 0, 0],
        ... [0, 0, 1]])
        >>> r.as_matrix().shape
        (3, 3)

        Initialize multiple rotations in a single object:

        >>> r = R.from_matrix([
        ... [
        ...     [0, -1, 0],
        ...     [1, 0, 0],
        ...     [0, 0, 1],
        ... ],
        ... [
        ...     [1, 0, 0],
        ...     [0, 0, -1],
        ...     [0, 1, 0],
        ... ]])
        >>> r.as_matrix().shape
        (2, 3, 3)

        If input matrices are not special orthogonal (orthogonal with
        determinant equal to +1), then a special orthogonal estimate is stored:

        >>> a = np.array([
        ... [0, -0.5, 0],
        ... [0.5, 0, 0],
        ... [0, 0, 0.5]])
        >>> np.linalg.det(a)
        0.12500000000000003
        >>> r = R.from_matrix(a)
        >>> matrix = r.as_matrix()
        >>> matrix
        array([[-0.38461538, -0.92307692,  0.        ],
               [ 0.92307692, -0.38461538,  0.        ],
               [ 0.        ,  0.        ,  1.        ]])
        >>> np.linalg.det(matrix)
        1.0000000000000002

        It is also possible to have a stack containing a single rotation:

        >>> r = R.from_matrix([[
        ... [0, -1, 0],
        ... [1, 0, 0],
        ... [0, 0, 1]]])
        >>> r.as_matrix()
        array([[[ 0., -1.,  0.],
                [ 1.,  0.,  0.],
                [ 0.,  0.,  1.]]])
        >>> r.as_matrix().shape
        (1, 3, 3)

        Notes
        -----
        This function was called from_dcm before.

        .. versionadded:: 1.4.0
        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 [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.
        degrees : bool, optional
            If True, then the given magnitudes are assumed to be in degrees.
            Default is False.

            .. versionadded:: 1.7.0

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotations represented by input rotation
            vectors.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        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 a rotation in degrees, and view it in degrees:

        >>> r = R.from_rotvec(45 * np.array([0, 1, 0]), degrees=True)
        >>> r.as_rotvec(degrees=True)
        array([ 0., 45.,  0.])

        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)

        Interpolation times must be within the range [{}, {}], both inclusive.Invalid mode, expected 'c' or 'fortran', got Invalid shape in axis Invert this rotation.

        Composition of a rotation with its inverse results in an identity
        transformation.

        Returns
        -------
        inverse : `Rotation` instance
            Object containing inverse of the rotations in the current instance.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Inverting a single rotation:

        >>> p = R.from_euler('z', 45, degrees=True)
        >>> q = p.inv()
        >>> q.as_euler('zyx', degrees=True)
        array([-45.,   0.,   0.])

        Inverting multiple rotations:

        >>> p = R.from_rotvec([[0, 0, np.pi/3], [-np.pi/4, 0, 0]])
        >>> q = p.inv()
        >>> q.as_rotvec()
        array([[-0.        , -0.        , -1.04719755],
               [ 0.78539816, -0.        , -0.        ]])

        KMemoryError<MemoryView of %r at 0x%x><MemoryView of %r object>NNotImplementedErrorOOnly one infinite weight is allowedOptimal rotation is not uniquely or poorly defined for the given sets of vectors.Out of bounds on buffer access (axis PickleErrorRR_flipR_optR_priR_secRepresent as Davenport angles.

        Any orientation can be expressed as a composition of 3 elementary
        rotations.

        For both Euler angles and Davenport angles, consecutive axes must 
        be are orthogonal (``axis2`` is orthogonal to both ``axis1`` and 
        ``axis3``). For Euler angles, there is an additional relationship 
        between ``axis1`` or ``axis3``, with two possibilities:

            - ``axis1`` and ``axis3`` are also orthogonal (asymmetric sequence)
            - ``axis1 == axis3`` (symmetric sequence)

        For Davenport angles, this last relationship is relaxed [1]_, and only
        the consecutive orthogonal axes requirement is maintained.

        A slightly modified version of the algorithm from [2]_ has been used to
        calculate Davenport angles for the rotation about a given sequence of
        axes.

        Davenport angles, just like Euler angles, suffer from the problem of
        gimbal lock [3]_, where the representation loses a degree of freedom
        and it is not possible to determine the first and third angles
        uniquely. In this case, a warning is raised, and the third angle is set
        to zero. Note however that the returned angles still represent the
        correct rotation.

        Parameters
        ----------
        axes : array_like, shape (3,) or ([1 or 2 or 3], 3)
            Axis of rotation, if one dimensional. If two dimensional, describes the 
            sequence of axes for rotations, where each axes[i, :] is the ith
            axis. If more than one axis is given, then the second axis must be
            orthogonal to both the first and third axes.
        order : string
            If it belongs to the set {'e', 'extrinsic'}, the sequence will be 
            extrinsic. If if belongs to the set {'i', 'intrinsic'}, sequence 
            will be treated as intrinsic.
        degrees : boolean, optional
            Returned angles are in degrees if this flag is True, else they are
            in radians. Default is False.

        Returns
        -------
        angles : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used to initialize object.
            The returned angles are in the range:

            - First angle belongs to [-180, 180] degrees (both inclusive)
            - Third angle belongs to [-180, 180] degrees (both inclusive)
            - Second angle belongs to a set of size 180 degrees, 
              given by: ``[-abs(lambda), 180 - abs(lambda)]``, where ``lambda``
              is the angle between the first and third axes.

        References
        ----------
        .. [1] Shuster, Malcolm & Markley, Landis. (2003). Generalization of
               the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304.
        .. [2] Bernardes E, Viollet S (2022) Quaternion to Euler angles
               conversion: A direct, general and computationally efficient method.
               PLoS ONE 17(11): e0276302. 10.1371/journal.pone.0276302
        .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Davenport angles are a generalization of Euler angles, when we use the
        canonical basis axes:

        >>> ex = [1, 0, 0]
        >>> ey = [0, 1, 0]
        >>> ez = [0, 0, 1]

        Represent a single rotation:

        >>> r = R.from_rotvec([0, 0, np.pi/2])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
        array([90.,  0.,  0.])
        >>> r.as_euler('zxy', degrees=True)
        array([90.,  0.,  0.])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape
        (3,)

        Represent a stack of single rotation:

        >>> r = R.from_rotvec([[0, 0, np.pi/2]])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
        array([[90.,  0.,  0.]])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape
        (1, 3)

        Represent multiple rotations in a single object:

        >>> r = R.from_rotvec([
        ... [0, 0, 90],
        ... [45, 0, 0]], degrees=True)
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
        array([[90.,  0.,  0.],
               [ 0., 45.,  0.]])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape
        (2, 3)
        Represent as Euler angles.

        Any orientation can be expressed as a composition of 3 elementary
        rotations. Once the axis sequence has been chosen, Euler angles define
        the angle of rotation around each respective axis [1]_.

        The algorithm from [2]_ has been used to calculate Euler angles for the 
        rotation about a given sequence of axes.

        Euler angles suffer from the problem of gimbal lock [3]_, where the
        representation loses a degree of freedom and it is not possible to
        determine the first and third angles uniquely. In this case,
        a warning is raised, and the third angle is set to zero. Note however
        that the returned angles still represent the correct rotation.

        Parameters
        ----------
        seq : string, length 3
            3 characters belonging to the set {'X', 'Y', 'Z'} for intrinsic
            rotations, or {'x', 'y', 'z'} for extrinsic rotations [1]_.
            Adjacent axes cannot be the same.
            Extrinsic and intrinsic rotations cannot be mixed in one function
            call.
        degrees : boolean, optional
            Returned angles are in degrees if this flag is True, else they are
            in radians. Default is False.

        Returns
        -------
        angles : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used to initialize object.
            The returned angles are in the range:

            - First angle belongs to [-180, 180] degrees (both inclusive)
            - Third angle belongs to [-180, 180] degrees (both inclusive)
            - Second angle belongs to:

                - [-90, 90] degrees if all axes are different (like xyz)
                - [0, 180] degrees if first and third axes are the same
                  (like zxz)

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations
        .. [2] Bernardes E, Viollet S (2022) Quaternion to Euler angles 
               conversion: A direct, general and computationally efficient 
               method. PLoS ONE 17(11): e0276302. 
               https://doi.org/10.1371/journal.pone.0276302
        .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_rotvec([0, 0, np.pi/2])
        >>> r.as_euler('zxy', degrees=True)
        array([90.,  0.,  0.])
        >>> r.as_euler('zxy', degrees=True).shape
        (3,)

        Represent a stack of single rotation:

        >>> r = R.from_rotvec([[0, 0, np.pi/2]])
        >>> r.as_euler('zxy', degrees=True)
        array([[90.,  0.,  0.]])
        >>> r.as_euler('zxy', degrees=True).shape
        (1, 3)

        Represent multiple rotations in a single object:

        >>> r = R.from_rotvec([
        ... [0, 0, np.pi/2],
        ... [0, -np.pi/3, 0],
        ... [np.pi/4, 0, 0]])
        >>> r.as_euler('zxy', degrees=True)
        array([[ 90.,   0.,   0.],
               [  0.,   0., -60.],
               [  0.,  45.,   0.]])
        >>> r.as_euler('zxy', degrees=True).shape
        (3, 3)

        Represent as Modified Rodrigues Parameters (MRPs).

        MRPs are a 3 dimensional vector co-directional to the axis of rotation and whose
        magnitude is equal to ``tan(theta / 4)``, where ``theta`` is the angle of rotation
        (in radians) [1]_.

        MRPs have a singularity at 360 degrees which can be avoided by ensuring the angle of
        rotation does not exceed 180 degrees, i.e. switching the direction of the rotation when
        it is past 180 degrees. This function will always return MRPs corresponding to a rotation
        of less than or equal to 180 degrees.

        Returns
        -------
        mrps : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used for initialization.

        References
        ----------
        .. [1] Shuster, M. D. "A Survey of Attitude Representations",
               The Journal of Astronautical Sciences, Vol. 41, No.4, 1993,
               pp. 475-476

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_rotvec([0, 0, np.pi])
        >>> r.as_mrp()
        array([0.        , 0.        , 1.         ])
        >>> r.as_mrp().shape
        (3,)

        Represent a stack with a single rotation:

        >>> r = R.from_euler('xyz', [[180, 0, 0]], degrees=True)
        >>> r.as_mrp()
        array([[1.       , 0.        , 0.         ]])
        >>> r.as_mrp().shape
        (1, 3)

        Represent multiple rotations:

        >>> r = R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]])
        >>> r.as_mrp()
        array([[0.41421356, 0.        , 0.        ],
               [0.        , 0.        , 0.41421356]])
        >>> r.as_mrp().shape
        (2, 3)

        Notes
        -----

        .. versionadded:: 1.6.0
        Represent as quaternions.

        Active rotations in 3 dimensions can be represented using unit norm
        quaternions [1]_. The mapping from quaternions to rotations is
        two-to-one, i.e. quaternions ``q`` and ``-q``, where ``-q`` simply
        reverses the sign of each component, represent the same spatial
        rotation. The returned value is in scalar-last (x, y, z, w) format.

        Parameters
        ----------
        canonical : `bool`, default False
            Whether to map the redundant double cover of rotation space to a
            unique "canonical" single cover. If True, then the quaternion is
            chosen from {q, -q} such that the w term is positive. If the w term
            is 0, then the quaternion is chosen such that the first nonzero
            term of the x, y, and z terms is positive.

        Returns
        -------
        quat : `numpy.ndarray`, shape (4,) or (N, 4)
            Shape depends on shape of inputs used for initialization.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_matrix([[0, -1, 0],
        ...                    [1, 0, 0],
        ...                    [0, 0, 1]])
        >>> r.as_quat()
        array([0.        , 0.        , 0.70710678, 0.70710678])
        >>> r.as_quat().shape
        (4,)

        Represent a stack with a single rotation:

        >>> r = R.from_quat([[0, 0, 0, 1]])
        >>> r.as_quat().shape
        (1, 4)

        Represent multiple rotations in a single object:

        >>> r = R.from_rotvec([[np.pi, 0, 0], [0, 0, np.pi/2]])
        >>> r.as_quat().shape
        (2, 4)

        Quaternions can be mapped from a redundant double cover of the
        rotation space to a canonical representation with a positive w term.

        >>> r = R.from_quat([0, 0, 0, -1])
        >>> r.as_quat()
        array([0. , 0. , 0. , -1.])
        >>> r.as_quat(canonical=True)
        array([0. , 0. , 0. , 1.])
        Represent as rotation matrix.

        3D rotations can be represented using rotation matrices, which
        are 3 x 3 real orthogonal matrices with determinant equal to +1 [1]_.

        Returns
        -------
        matrix : ndarray, shape (3, 3) or (N, 3, 3)
            Shape depends on shape of inputs used for initialization.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_rotvec([0, 0, np.pi/2])
        >>> r.as_matrix()
        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_matrix().shape
        (3, 3)

        Represent a stack with a single rotation:

        >>> r = R.from_quat([[1, 1, 0, 0]])
        >>> r.as_matrix()
        array([[[ 0.,  1.,  0.],
                [ 1.,  0.,  0.],
                [ 0.,  0., -1.]]])
        >>> r.as_matrix().shape
        (1, 3, 3)

        Represent multiple rotations:

        >>> r = R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]])
        >>> r.as_matrix()
        array([[[ 1.00000000e+00,  0.00000000e+00,  0.00000000e+00],
                [ 0.00000000e+00,  2.22044605e-16, -1.00000000e+00],
                [ 0.00000000e+00,  1.00000000e+00,  2.22044605e-16]],
               [[ 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_matrix().shape
        (2, 3, 3)

        Notes
        -----
        This function was called as_dcm before.

        .. versionadded:: 1.4.0
        Represent as 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 [1]_.

        Parameters
        ----------
        degrees : boolean, optional
            Returned magnitudes are in degrees if this flag is True, else they are
            in radians. Default is False.

            .. versionadded:: 1.7.0

        Returns
        -------
        rotvec : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used for initialization.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_euler('z', 90, degrees=True)
        >>> r.as_rotvec()
        array([0.        , 0.        , 1.57079633])
        >>> r.as_rotvec().shape
        (3,)

        Represent a rotation in degrees:

        >>> r = R.from_euler('YX', (-90, -90), degrees=True)
        >>> s = r.as_rotvec(degrees=True)
        >>> s
        array([-69.2820323, -69.2820323, -69.2820323])
        >>> np.linalg.norm(s)
        120.00000000000001

        Represent a stack with a single rotation:

        >>> r = R.from_quat([[0, 0, 1, 1]])
        >>> r.as_rotvec()
        array([[0.        , 0.        , 1.57079633]])
        >>> r.as_rotvec().shape
        (1, 3)

        Represent multiple rotations in a single object:

        >>> r = R.from_quat([[0, 0, 1, 1], [1, 1, 0, 1]])
        >>> r.as_rotvec()
        array([[0.        , 0.        , 1.57079633],
               [1.35102172, 1.35102172, 0.        ]])
        >>> r.as_rotvec().shape
        (2, 3)

        RotationRotation.__getitem__ (line 2995)Rotation.__getstate__Rotation.__mul__ (line 2494)Rotation.__pow__ (line 2575)Rotation.__reduce_cython__Rotation.__setstate__Rotation.__setstate_cython__Rotation._as_euler_from_matrixRotation._compute_eulerRotation.align_vectorsRotation.align_vectors (line 3171)Rotation.applyRotation.apply (line 2342)Rotation.approx_equalRotation.approx_equal (line 2741)Rotation.as_davenportRotation.as_davenport (line 2069)Rotation.as_eulerRotation.as_euler (line 1980)Rotation.as_matrixRotation.as_matrix (line 1684)Rotation.as_mrpRotation.as_mrp (line 2215)Rotation.as_quatRotation.as_quat (line 1609)Rotation.as_rotvecRotation.as_rotvec (line 1789)Rotation.concatenateRotation.concatenate (line 2294)Rotation.create_groupRotation.from_davenportRotation.from_davenport (line 1351)Rotation.from_eulerRotation.from_euler (line 1230)Rotation.from_matrixRotation.from_matrix (line 967)Rotation.from_mrpRotation.from_mrp (line 1507)Rotation.from_quatRotation.from_quat (line 894)Rotation.from_rotvecRotation.from_rotvec (line 1120)Rotation.identityRotation.invRotation.inv (line 2656)Rotation.magnitudeRotation.magnitude (line 2697)Rotation.meanRotation.mean (line 2794)Rotation.randomRotation.random (line 3107)Rotation.reduceSequenceSingle rotation has no len().Single rotation is not subscriptable.SlerpSlerp.__call__Slerp.__init__Spherical Linear Interpolation of Rotations.

    The interpolation between consecutive rotations is performed as a rotation
    around a fixed axis with a constant angular velocity [1]_. This ensures
    that the interpolated rotations follow the shortest path between initial
    and final orientations.

    Parameters
    ----------
    times : array_like, shape (N,)
        Times of the known rotations. At least 2 times must be specified.
    rotations : `Rotation` instance
        Rotations to perform the interpolation between. Must contain N
        rotations.

    Methods
    -------
    __call__

    See Also
    --------
    Rotation

    Notes
    -----
    .. versionadded:: 1.2.0

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Slerp#Quaternion_Slerp

    Examples
    --------
    >>> from scipy.spatial.transform import Rotation as R
    >>> from scipy.spatial.transform import Slerp

    Setup the fixed keyframe rotations and times:

    >>> key_rots = R.random(5, random_state=2342345)
    >>> key_times = [0, 1, 2, 3, 4]

    Create the interpolator object:

    >>> slerp = Slerp(key_times, key_rots)

    Interpolate the rotations at the given times:

    >>> times = [0, 0.5, 0.25, 1, 1.5, 2, 2.75, 3, 3.25, 3.60, 4]
    >>> interp_rots = slerp(times)

    The keyframe rotations expressed as Euler angles:

    >>> key_rots.as_euler('xyz', degrees=True)
    array([[ 14.31443779, -27.50095894,  -3.7275787 ],
           [ -1.79924227, -24.69421529, 164.57701743],
           [146.15020772,  43.22849451, -31.34891088],
           [ 46.39959442,  11.62126073, -45.99719267],
           [-88.94647804, -49.64400082, -65.80546984]])

    The interpolated rotations expressed as Euler angles. These agree with the
    keyframe rotations at both endpoints of the range of keyframe times.

    >>> interp_rots.as_euler('xyz', degrees=True)
    array([[  14.31443779,  -27.50095894,   -3.7275787 ],
           [   4.74588574,  -32.44683966,   81.25139984],
           [  10.71094749,  -31.56690154,   38.06896408],
           [  -1.79924227,  -24.69421529,  164.57701743],
           [  11.72796022,   51.64207311, -171.7374683 ],
           [ 146.15020772,   43.22849451,  -31.34891088],
           [  68.10921869,   20.67625074,  -48.74886034],
           [  46.39959442,   11.62126073,  -45.99719267],
           [  12.35552615,    4.21525086,  -64.89288124],
           [ -30.08117143,  -19.90769513,  -78.98121326],
           [ -88.94647804,  -49.64400082,  -65.80546984]])

    Step may not be zero (axis %d)T{TTimes must be in strictly increasing order.TypeErrorUnable to convert item to objectValueErrorView.MemoryView^[XYZ]{1,3}$Z:_}(,?.*')^aa_esta_originala_pria_pri_norma_secabcabsalgorithmalign_vectorsallocate_bufferalpha and angleangle2anglesanyapplyapprox_equalargmaxargminargsarrayas_davenportas_euler_as_euler_from_matrixas_matrixas_mrpas_quatas_rotvecasarrayasyncio.coroutinesatleast_1datleast_2datolatol must be set to use the degrees flag, defaulting to 1e-8 radians.axesaxisbb_originalb_prib_pri_normb_secbasecc_sec__call__canonicalcheck_random_statechoice__class____class_getitem__cline_in_tracebackcloseclscmatrixcmrpcollectionscollections.abc_compute_euler_compute_euler.<locals>.genexprcompute_timesconcatenateconcatenate.<locals>.genexpr<contiguous and direct><contiguous and indirect>copycos_termcountcreate_groupcrosscross_normcrotvecddecisiondeg2raddegreesdenominatordet__dict___dictdiffdisabledivide__doc__dotdtypedtype_is_objecteeigheinsumenableencode__enter__enumerateerrorerrstate__exit__extrinsiceyeflagsformat_format_anglesfortranfrom_davenportfrom_eulerfrom_euler.<locals>.genexprfrom_matrixfrom_mrpfrom_quatfrom_rotvecgcgenexpr__getstate__ (got got differing extents in dimension groupii,j,ki,jx,kxididentityignoreij,ij->iijk,ik->ijikj,ik->ij__imatmul____import__indindex__init____init_subclass___initializinginput must contain Rotation objects onlyintrinsicinvinvalidinverse_is_coroutineis_singleisenabledisposinfitemsizeitemsize <= 0 for cython.arrayix,j,kxix,jx,kjji,jk->ikjoinkkappaleftleft_bestlinalglogical_orlowerlslvmagnitude__main__match__matmul__matrixmax_indmeanmemview__metaclass__mode__module__modulus not supportedmoveaxismrpmrp_squared_plus_1mrpsn1n2n3n_infn_rotationsn_vectorsname__name__ndim__new__newaxisno default __reduce__ due to non-trivial __cinit__normnormalnormalizenpnumnum_axesnum_rotationsnumpynumpy.core.multiarray failed to importnumpy.core.umath failed to importobjonesorderorder should be 'e'/'extrinsic' for extrinsic sequences or 'i'/'intrinsic' for intrinsic sequences, got {}otherppackphipipickle__prepare__pspv__pyx_PickleError__pyx_checksum__pyx_result__pyx_state__pyx_type__pyx_unpickle_Enum__pyx_unpickle_Rotation__pyx_vtable__qqiqs__qualname__quatquatsrrad2degrandomrandom_staterangere__reduce__reduce__reduce_cython____reduce_ex__reduce.<locals>.split_rotationreducedregisterrepeatreshaperesultretreturn_indicesreturn_sensitivityrightright_best__rmatmul___rotation_groupsrotations`rotations` must be a `Rotation` instance.`rotations` must be a sequence of at least 2 rotations.rotvecrotvecsrsrssdrvssamplescalescipy._lib._utilscipy.spatial.transform._rotationsearchsortedselfsendsensitivityseq__set_name____setstate____setstate_cython__shapesignsinsin_termsinglesingle_timesingle_vectorsize__spec__split_rotationsqrtstacklevelstartstatestepstop<strided and direct><strided and direct or indirect><strided and indirect><stringsource>structsumsupersvdsys__test__thetatheta2throwtimedeltatimes`times` must be at most 1-dimensional.toleranceuunable to allocate array data.unable to allocate shape and strides.unpackupdateuse_setstatevvalue must be a Rotation objectvectorsversion_infovhww2warnwarningsweight_is_infweightsweights_inf_zero`weights` may not contain negative values`weights` must be non-negative.weights_secxx2xwxy^[xyz]{1,3}$xyz,ix,jy,kzxzyy2ywyzzz2zeroszetazwNumber of rotations contained in this object.

        Multiple rotations can be stored in a single instance.

        Returns
        -------
        length : int
            Number of rotations stored in object.

        Raises
        ------
        TypeError if the instance was created as a single rotation.
        Compose this rotation with the other.

        If `p` and `q` are two rotations, then the composition of 'q followed
        by p' is equivalent to `p * q`. In terms of rotation matrices,
        the composition can be expressed as
        ``p.as_matrix() @ q.as_matrix()``.

        Parameters
        ----------
        other : `Rotation` instance
            Object containing the rotations to be composed with this one. Note
            that rotation compositions are not commutative, so ``p * q`` is
            generally different from ``q * p``.

        Returns
        -------
        composition : `Rotation` instance
            This function supports composition of multiple rotations at a time.
            The following cases are possible:

            - Either ``p`` or ``q`` contains a single rotation. In this case
              `composition` contains the result of composing each rotation in
              the other object with the single rotation.
            - Both ``p`` and ``q`` contain ``N`` rotations. In this case each
              rotation ``p[i]`` is composed with the corresponding rotation
              ``q[i]`` and `output` contains ``N`` rotations.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Composition of two single rotations:

        >>> p = R.from_quat([0, 0, 1, 1])
        >>> q = R.from_quat([1, 0, 0, 1])
        >>> p.as_matrix()
        array([[ 0., -1.,  0.],
               [ 1.,  0.,  0.],
               [ 0.,  0.,  1.]])
        >>> q.as_matrix()
        array([[ 1.,  0.,  0.],
               [ 0.,  0., -1.],
               [ 0.,  1.,  0.]])
        >>> r = p * q
        >>> r.as_matrix()
        array([[0., 0., 1.],
               [1., 0., 0.],
               [0., 1., 0.]])

        Composition of two objects containing equal number of rotations:

        >>> p = R.from_quat([[0, 0, 1, 1], [1, 0, 0, 1]])
        >>> q = R.from_rotvec([[np.pi/4, 0, 0], [-np.pi/4, 0, np.pi/4]])
        >>> p.as_quat()
        array([[0.        , 0.        , 0.70710678, 0.70710678],
               [0.70710678, 0.        , 0.        , 0.70710678]])
        >>> q.as_quat()
        array([[ 0.38268343,  0.        ,  0.        ,  0.92387953],
               [-0.37282173,  0.        ,  0.37282173,  0.84971049]])
        >>> r = p * q
        >>> r.as_quat()
        array([[ 0.27059805,  0.27059805,  0.65328148,  0.65328148],
               [ 0.33721128, -0.26362477,  0.26362477,  0.86446082]])

        Compose this rotation with itself `n` times.

        Composition of a rotation ``p`` with itself can be extended to
        non-integer ``n`` by considering the power ``n`` to be a scale factor
        applied to the angle of rotation about the rotation's fixed axis. The
        expression ``q = p ** n`` can also be expressed as
        ``q = Rotation.from_rotvec(n * p.as_rotvec())``.

        If ``n`` is negative, then the rotation is inverted before the power
        is applied. In other words, ``p ** -abs(n) == p.inv() ** abs(n)``.

        Parameters
        ----------
        n : float
            The number of times to compose the rotation with itself.
        modulus : None
            This overridden argument is not applicable to Rotations and must be
            ``None``.

        Returns
        -------
        power : `Rotation` instance
            If the input Rotation ``p`` contains ``N`` multiple rotations, then
            the output will contain ``N`` rotations where the ``i`` th rotation
            is equal to ``p[i] ** n``

        Notes
        -----
        For example, a power of 2 will double the angle of rotation, and a
        power of 0.5 will halve the angle. There are three notable cases: if
        ``n == 1`` then the original rotation is returned, if ``n == 0``
        then the identity rotation is returned, and if ``n == -1`` then
        ``p.inv()`` is returned.

        Note that fractional powers ``n`` which effectively take a root of
        rotation, do so using the shortest path smallest representation of that
        angle (the principal root). This means that powers of ``n`` and ``1/n``
        are not necessarily inverses of each other. For example, a 0.5 power of
        a +240 degree rotation will be calculated as the 0.5 power of a -120
        degree rotation, with the result being a rotation of -60 rather than
        +120 degrees.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Raising a rotation to a power:

        >>> p = R.from_rotvec([1, 0, 0])
        >>> q = p ** 2
        >>> q.as_rotvec()
        array([2., 0., 0.])
        >>> r = p ** 0.5
        >>> r.as_rotvec()
        array([0.5, 0., 0.])

        Inverse powers do not necessarily cancel out:

        >>> p = R.from_rotvec([0, 0, 120], degrees=True)
        >>> ((p ** 2) ** 0.5).as_rotvec(degrees=True)
        array([  -0.,   -0., -60.])

        Extract rotation(s) at given index(es) from object.

        Create a new `Rotation` instance containing a subset of rotations
        stored in this object.

        Parameters
        ----------
        indexer : index, slice, or index array
            Specifies which rotation(s) to extract. A single indexer must be
            specified, i.e. as if indexing a 1 dimensional array or list.

        Returns
        -------
        rotation : `Rotation` instance
            Contains
                - a single rotation, if `indexer` is a single index
                - a stack of rotation(s), if `indexer` is a slice, or and index
                  array.

        Raises
        ------
        TypeError if the instance was created as a single rotation.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> r = R.from_quat([
        ... [1, 1, 0, 0],
        ... [0, 1, 0, 1],
        ... [1, 1, -1, 0]])
        >>> r.as_quat()
        array([[ 0.70710678,  0.70710678,  0.        ,  0.        ],
               [ 0.        ,  0.70710678,  0.        ,  0.70710678],
               [ 0.57735027,  0.57735027, -0.57735027,  0.        ]])

        Indexing using a single index:

        >>> p = r[0]
        >>> p.as_quat()
        array([0.70710678, 0.70710678, 0.        , 0.        ])

        Array slicing:

        >>> q = r[1:3]
        >>> q.as_quat()
        array([[ 0.        ,  0.70710678,  0.        ,  0.70710678],
               [ 0.57735027,  0.57735027, -0.57735027,  0.        ]])

        Set rotation(s) at given index(es) from object.

        Parameters
        ----------
        indexer : index, slice, or index array
            Specifies which rotation(s) to replace. A single indexer must be
            specified, i.e. as if indexing a 1 dimensional array or list.

        value : `Rotation` instance
            The rotations to set.

        Raises
        ------
        TypeError if the instance was created as a single rotation.

        Notes
        -----

        .. versionadded:: 1.8.0
        Rotation.from_quat(cls, quat)
Initialize from quaternions.

        3D rotations can be represented using unit-norm quaternions [1]_.

        Advanced users may be interested in the "double cover" of 3D space by
        the quaternion representation [2]_. As of version 1.11.0, the
        following subset (and only this subset) of operations on a `Rotation`
        ``r`` corresponding to a quaternion ``q`` are guaranteed to preserve
        the double cover property: ``r = Rotation.from_quat(q)``,
        ``r.as_quat(canonical=False)``, ``r.inv()``, and composition using the
        ``*`` operator such as ``r*r``.

        Parameters
        ----------
        quat : array_like, shape (N, 4) or (4,)
            Each row is a (possibly non-unit norm) quaternion representing an
            active rotation, in scalar-last (x, y, z, w) format. Each
            quaternion will be normalized to unit norm.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotations represented by input quaternions.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
        .. [2] Hanson, Andrew J. "Visualizing quaternions."
            Morgan Kaufmann Publishers Inc., San Francisco, CA. 2006.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Initialize a single rotation:

        >>> r = R.from_quat([1, 0, 0, 0])
        >>> r.as_quat()
        array([1., 0., 0., 0.])
        >>> r.as_quat().shape
        (4,)

        Initialize multiple rotations in a single object:

        >>> r = R.from_quat([
        ... [1, 0, 0, 0],
        ... [0, 0, 0, 1]
        ... ])
        >>> r.as_quat()
        array([[1., 0., 0., 0.],
               [0., 0., 0., 1.]])
        >>> r.as_quat().shape
        (2, 4)

        It is also possible to have a stack of a single rotation:

        >>> r = R.from_quat([[0, 0, 0, 1]])
        >>> r.as_quat()
        array([[0., 0., 0., 1.]])
        >>> r.as_quat().shape
        (1, 4)

        Quaternions are normalized before initialization.

        >>> r = R.from_quat([0, 0, 1, 1])
        >>> r.as_quat()
        array([0.        , 0.        , 0.70710678, 0.70710678])
        Rotation.from_matrix(cls, matrix)
Initialize from rotation matrix.

        Rotations in 3 dimensions can be represented with 3 x 3 proper
        orthogonal matrices [1]_. If the input is not proper orthogonal,
        an approximation is created using the method described in [2]_.

        Parameters
        ----------
        matrix : array_like, shape (N, 3, 3) or (3, 3)
            A single matrix or a stack of matrices, where ``matrix[i]`` is
            the i-th matrix.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotations represented by the rotation
            matrices.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions
        .. [2] F. Landis Markley, "Unit Quaternion from Rotation Matrix",
               Journal of guidance, control, and dynamics vol. 31.2, pp.
               440-442, 2008.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Initialize a single rotation:

        >>> r = R.from_matrix([
        ... [0, -1, 0],
        ... [1, 0, 0],
        ... [0, 0, 1]])
        >>> r.as_matrix().shape
        (3, 3)

        Initialize multiple rotations in a single object:

        >>> r = R.from_matrix([
        ... [
        ...     [0, -1, 0],
        ...     [1, 0, 0],
        ...     [0, 0, 1],
        ... ],
        ... [
        ...     [1, 0, 0],
        ...     [0, 0, -1],
        ...     [0, 1, 0],
        ... ]])
        >>> r.as_matrix().shape
        (2, 3, 3)

        If input matrices are not special orthogonal (orthogonal with
        determinant equal to +1), then a special orthogonal estimate is stored:

        >>> a = np.array([
        ... [0, -0.5, 0],
        ... [0.5, 0, 0],
        ... [0, 0, 0.5]])
        >>> np.linalg.det(a)
        0.12500000000000003
        >>> r = R.from_matrix(a)
        >>> matrix = r.as_matrix()
        >>> matrix
        array([[-0.38461538, -0.92307692,  0.        ],
               [ 0.92307692, -0.38461538,  0.        ],
               [ 0.        ,  0.        ,  1.        ]])
        >>> np.linalg.det(matrix)
        1.0000000000000002

        It is also possible to have a stack containing a single rotation:

        >>> r = R.from_matrix([[
        ... [0, -1, 0],
        ... [1, 0, 0],
        ... [0, 0, 1]]])
        >>> r.as_matrix()
        array([[[ 0., -1.,  0.],
                [ 1.,  0.,  0.],
                [ 0.,  0.,  1.]]])
        >>> r.as_matrix().shape
        (1, 3, 3)

        Notes
        -----
        This function was called from_dcm before.

        .. versionadded:: 1.4.0
        Rotation.from_rotvec(cls, rotvec, degrees=False)
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 [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.
        degrees : bool, optional
            If True, then the given magnitudes are assumed to be in degrees.
            Default is False.

            .. versionadded:: 1.7.0

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotations represented by input rotation
            vectors.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        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 a rotation in degrees, and view it in degrees:

        >>> r = R.from_rotvec(45 * np.array([0, 1, 0]), degrees=True)
        >>> r.as_rotvec(degrees=True)
        array([ 0., 45.,  0.])

        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)

        Rotation.from_euler(cls, seq, angles, degrees=False)
Initialize from Euler angles.

        Rotations in 3-D can be represented by a sequence of 3
        rotations around a sequence of axes. In theory, any three axes spanning
        the 3-D Euclidean space are enough. In practice, the axes of rotation are
        chosen to be the basis vectors.

        The three rotations can either be in a global frame of reference
        (extrinsic) or in a body centred frame of reference (intrinsic), which
        is attached to, and moves with, the object under rotation [1]_.

        Parameters
        ----------
        seq : string
            Specifies sequence of axes for rotations. Up to 3 characters
            belonging to the set {'X', 'Y', 'Z'} for intrinsic rotations, or
            {'x', 'y', 'z'} for extrinsic rotations. Extrinsic and intrinsic
            rotations cannot be mixed in one function call.
        angles : float or array_like, shape (N,) or (N, [1 or 2 or 3])
            Euler angles specified in radians (`degrees` is False) or degrees
            (`degrees` is True).
            For a single character `seq`, `angles` can be:

            - a single value
            - array_like with shape (N,), where each `angle[i]`
              corresponds to a single rotation
            - array_like with shape (N, 1), where each `angle[i, 0]`
              corresponds to a single rotation

            For 2- and 3-character wide `seq`, `angles` can be:

            - array_like with shape (W,) where `W` is the width of
              `seq`, which corresponds to a single rotation with `W` axes
            - array_like with shape (N, W) where each `angle[i]`
              corresponds to a sequence of Euler angles describing a single
              rotation

        degrees : bool, optional
            If True, then the given angles are assumed to be in degrees.
            Default is False.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotation represented by the sequence of
            rotations around given axes with given angles.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Initialize a single rotation along a single axis:

        >>> r = R.from_euler('x', 90, degrees=True)
        >>> r.as_quat().shape
        (4,)

        Initialize a single rotation with a given axis sequence:

        >>> r = R.from_euler('zyx', [90, 45, 30], degrees=True)
        >>> r.as_quat().shape
        (4,)

        Initialize a stack with a single rotation around a single axis:

        >>> r = R.from_euler('x', [90], degrees=True)
        >>> r.as_quat().shape
        (1, 4)

        Initialize a stack with a single rotation with an axis sequence:

        >>> r = R.from_euler('zyx', [[90, 45, 30]], degrees=True)
        >>> r.as_quat().shape
        (1, 4)

        Initialize multiple elementary rotations in one object:

        >>> r = R.from_euler('x', [90, 45, 30], degrees=True)
        >>> r.as_quat().shape
        (3, 4)

        Initialize multiple rotations in one object:

        >>> r = R.from_euler('zyx', [[90, 45, 30], [35, 45, 90]], degrees=True)
        >>> r.as_quat().shape
        (2, 4)

        Rotation.from_davenport(cls, axes, order, angles, degrees=False)
Initialize from Davenport angles.

        Rotations in 3-D can be represented by a sequence of 3
        rotations around a sequence of axes.

        The three rotations can either be in a global frame of reference
        (extrinsic) or in a body centred frame of reference (intrinsic), which
        is attached to, and moves with, the object under rotation [1]_.

        For both Euler angles and Davenport angles, consecutive axes must 
        be are orthogonal (``axis2`` is orthogonal to both ``axis1`` and 
        ``axis3``). For Euler angles, there is an additional relationship 
        between ``axis1`` or ``axis3``, with two possibilities:

            - ``axis1`` and ``axis3`` are also orthogonal (asymmetric sequence)
            - ``axis1 == axis3`` (symmetric sequence)

        For Davenport angles, this last relationship is relaxed [2]_, and only
        the consecutive orthogonal axes requirement is maintained.

        Parameters
        ----------
        axes : array_like, shape (3,) or ([1 or 2 or 3], 3)
            Axis of rotation, if one dimensional. If two dimensional, describes the 
            sequence of axes for rotations, where each axes[i, :] is the ith
            axis. If more than one axis is given, then the second axis must be
            orthogonal to both the first and third axes.
        order : string
            If it is equal to 'e' or 'extrinsic', the sequence will be 
            extrinsic. If it is equal to 'i' or 'intrinsic', sequence 
            will be treated as intrinsic.
        angles : float or array_like, shape (N,) or (N, [1 or 2 or 3])
            Euler angles specified in radians (`degrees` is False) or degrees
            (`degrees` is True).
            For a single axis, `angles` can be:

            - a single value
            - array_like with shape (N,), where each `angle[i]`
              corresponds to a single rotation
            - array_like with shape (N, 1), where each `angle[i, 0]`
              corresponds to a single rotation

            For 2 and 3 axes, `angles` can be:

            - array_like with shape (W,) where `W` is the number of rows of
              `axes`, which corresponds to a single rotation with `W` axes
            - array_like with shape (N, W) where each `angle[i]`
              corresponds to a sequence of Davenport angles describing a 
              single rotation

        degrees : bool, optional
            If True, then the given angles are assumed to be in degrees. 
            Default is False.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotation represented by the sequence of
            rotations around given axes with given angles.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations
        .. [2] Shuster, Malcolm & Markley, Landis. (2003). Generalization of
               the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Davenport angles are a generalization of Euler angles, when we use the
        canonical basis axes:

        >>> ex = [1, 0, 0]
        >>> ey = [0, 1, 0]
        >>> ez = [0, 0, 1]

        Initialize a single rotation with a given axis sequence:

        >>> axes = [ez, ey, ex]
        >>> r = R.from_davenport(axes, 'extrinsic', [90, 0, 0], degrees=True)
        >>> r.as_quat().shape
        (4,)

        It is equivalent to Euler angles in this case:

        >>> r.as_euler('zyx', degrees=True)
        array([90.,  0., -0.])

        Initialize multiple rotations in one object:

        >>> r = R.from_davenport(axes, 'extrinsic', [[90, 45, 30], [35, 45, 90]], degrees=True)
        >>> r.as_quat().shape
        (2, 4)

        Using only one or two axes is also possible:

        >>> r = R.from_davenport([ez, ex], 'extrinsic', [[90, 45], [35, 45]], degrees=True)
        >>> r.as_quat().shape
        (2, 4)

        Non-canonical axes are possible, and they do not need to be normalized,
        as long as consecutive axes are orthogonal:

        >>> e1 = [2, 0, 0]
        >>> e2 = [0, 1, 0]
        >>> e3 = [1, 0, 1]
        >>> axes = [e1, e2, e3]
        >>> r = R.from_davenport(axes, 'extrinsic', [90, 45, 30], degrees=True)
        >>> r.as_quat()
        [ 0.701057,  0.430459, -0.092296,  0.560986]
        Rotation.from_mrp(cls, mrp)
Initialize from Modified Rodrigues Parameters (MRPs).

        MRPs are a 3 dimensional vector co-directional to the axis of rotation and whose
        magnitude is equal to ``tan(theta / 4)``, where ``theta`` is the angle of rotation
        (in radians) [1]_.

        MRPs have a singularity at 360 degrees which can be avoided by ensuring the angle of
        rotation does not exceed 180 degrees, i.e. switching the direction of the rotation when
        it is past 180 degrees.

        Parameters
        ----------
        mrp : array_like, shape (N, 3) or (3,)
            A single vector or a stack of vectors, where `mrp[i]` gives
            the ith set of MRPs.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the rotations represented by input MRPs.

        References
        ----------
        .. [1] Shuster, M. D. "A Survey of Attitude Representations",
               The Journal of Astronautical Sciences, Vol. 41, No.4, 1993,
               pp. 475-476

        Notes
        -----

        .. versionadded:: 1.6.0

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Initialize a single rotation:

        >>> r = R.from_mrp([0, 0, 1])
        >>> r.as_euler('xyz', degrees=True)
        array([0.        , 0.        , 180.      ])
        >>> r.as_euler('xyz').shape
        (3,)

        Initialize multiple rotations in one object:

        >>> r = R.from_mrp([
        ... [0, 0, 1],
        ... [1, 0, 0]])
        >>> r.as_euler('xyz', degrees=True)
        array([[0.        , 0.        , 180.      ],
               [180.0     , 0.        , 0.        ]])
        >>> r.as_euler('xyz').shape
        (2, 3)

        It is also possible to have a stack of a single rotation:

        >>> r = R.from_mrp([[0, 0, np.pi/2]])
        >>> r.as_euler('xyz').shape
        (1, 3)

        Rotation.as_quat(self, canonical=False)
Represent as quaternions.

        Active rotations in 3 dimensions can be represented using unit norm
        quaternions [1]_. The mapping from quaternions to rotations is
        two-to-one, i.e. quaternions ``q`` and ``-q``, where ``-q`` simply
        reverses the sign of each component, represent the same spatial
        rotation. The returned value is in scalar-last (x, y, z, w) format.

        Parameters
        ----------
        canonical : `bool`, default False
            Whether to map the redundant double cover of rotation space to a
            unique "canonical" single cover. If True, then the quaternion is
            chosen from {q, -q} such that the w term is positive. If the w term
            is 0, then the quaternion is chosen such that the first nonzero
            term of the x, y, and z terms is positive.

        Returns
        -------
        quat : `numpy.ndarray`, shape (4,) or (N, 4)
            Shape depends on shape of inputs used for initialization.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_matrix([[0, -1, 0],
        ...                    [1, 0, 0],
        ...                    [0, 0, 1]])
        >>> r.as_quat()
        array([0.        , 0.        , 0.70710678, 0.70710678])
        >>> r.as_quat().shape
        (4,)

        Represent a stack with a single rotation:

        >>> r = R.from_quat([[0, 0, 0, 1]])
        >>> r.as_quat().shape
        (1, 4)

        Represent multiple rotations in a single object:

        >>> r = R.from_rotvec([[np.pi, 0, 0], [0, 0, np.pi/2]])
        >>> r.as_quat().shape
        (2, 4)

        Quaternions can be mapped from a redundant double cover of the
        rotation space to a canonical representation with a positive w term.

        >>> r = R.from_quat([0, 0, 0, -1])
        >>> r.as_quat()
        array([0. , 0. , 0. , -1.])
        >>> r.as_quat(canonical=True)
        array([0. , 0. , 0. , 1.])
        Rotation.as_matrix(self)
Represent as rotation matrix.

        3D rotations can be represented using rotation matrices, which
        are 3 x 3 real orthogonal matrices with determinant equal to +1 [1]_.

        Returns
        -------
        matrix : ndarray, shape (3, 3) or (N, 3, 3)
            Shape depends on shape of inputs used for initialization.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_rotvec([0, 0, np.pi/2])
        >>> r.as_matrix()
        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_matrix().shape
        (3, 3)

        Represent a stack with a single rotation:

        >>> r = R.from_quat([[1, 1, 0, 0]])
        >>> r.as_matrix()
        array([[[ 0.,  1.,  0.],
                [ 1.,  0.,  0.],
                [ 0.,  0., -1.]]])
        >>> r.as_matrix().shape
        (1, 3, 3)

        Represent multiple rotations:

        >>> r = R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]])
        >>> r.as_matrix()
        array([[[ 1.00000000e+00,  0.00000000e+00,  0.00000000e+00],
                [ 0.00000000e+00,  2.22044605e-16, -1.00000000e+00],
                [ 0.00000000e+00,  1.00000000e+00,  2.22044605e-16]],
               [[ 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_matrix().shape
        (2, 3, 3)

        Notes
        -----
        This function was called as_dcm before.

        .. versionadded:: 1.4.0
        Rotation.as_rotvec(self, degrees=False)
Represent as 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 [1]_.

        Parameters
        ----------
        degrees : boolean, optional
            Returned magnitudes are in degrees if this flag is True, else they are
            in radians. Default is False.

            .. versionadded:: 1.7.0

        Returns
        -------
        rotvec : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used for initialization.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_euler('z', 90, degrees=True)
        >>> r.as_rotvec()
        array([0.        , 0.        , 1.57079633])
        >>> r.as_rotvec().shape
        (3,)

        Represent a rotation in degrees:

        >>> r = R.from_euler('YX', (-90, -90), degrees=True)
        >>> s = r.as_rotvec(degrees=True)
        >>> s
        array([-69.2820323, -69.2820323, -69.2820323])
        >>> np.linalg.norm(s)
        120.00000000000001

        Represent a stack with a single rotation:

        >>> r = R.from_quat([[0, 0, 1, 1]])
        >>> r.as_rotvec()
        array([[0.        , 0.        , 1.57079633]])
        >>> r.as_rotvec().shape
        (1, 3)

        Represent multiple rotations in a single object:

        >>> r = R.from_quat([[0, 0, 1, 1], [1, 1, 0, 1]])
        >>> r.as_rotvec()
        array([[0.        , 0.        , 1.57079633],
               [1.35102172, 1.35102172, 0.        ]])
        >>> r.as_rotvec().shape
        (2, 3)

        Rotation._compute_euler(self, seq, degrees, algorithm)Rotation._as_euler_from_matrix(self, seq, degrees=False)
Represent as Euler angles.

        Any orientation can be expressed as a composition of 3 elementary
        rotations. Once the axis sequence has been chosen, Euler angles define
        the angle of rotation around each respective axis [1]_.

        The algorithm from [2]_ has been used to calculate Euler angles for the
        rotation about a given sequence of axes.

        Euler angles suffer from the problem of gimbal lock [3]_, where the
        representation loses a degree of freedom and it is not possible to
        determine the first and third angles uniquely. In this case,
        a warning is raised, and the third angle is set to zero. Note however
        that the returned angles still represent the correct rotation.

        Parameters
        ----------
        seq : string, length 3
            3 characters belonging to the set {'X', 'Y', 'Z'} for intrinsic
            rotations, or {'x', 'y', 'z'} for extrinsic rotations [1]_.
            Adjacent axes cannot be the same.
            Extrinsic and intrinsic rotations cannot be mixed in one function
            call.
        degrees : boolean, optional
            Returned angles are in degrees if this flag is True, else they are
            in radians. Default is False.

        Returns
        -------
        angles : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used to initialize object.
            The returned angles are in the range:

            - First angle belongs to [-180, 180] degrees (both inclusive)
            - Third angle belongs to [-180, 180] degrees (both inclusive)
            - Second angle belongs to:

                - [-90, 90] degrees if all axes are different (like xyz)
                - [0, 180] degrees if first and third axes are the same
                  (like zxz)

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations
        .. [2] Malcolm D. Shuster, F. Landis Markley, "General formula for
               extraction the Euler angles", Journal of guidance, control, and
               dynamics, vol. 29.1, pp. 215-221. 2006
        .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics

        Rotation.as_euler(self, seq, degrees=False)
Represent as Euler angles.

        Any orientation can be expressed as a composition of 3 elementary
        rotations. Once the axis sequence has been chosen, Euler angles define
        the angle of rotation around each respective axis [1]_.

        The algorithm from [2]_ has been used to calculate Euler angles for the 
        rotation about a given sequence of axes.

        Euler angles suffer from the problem of gimbal lock [3]_, where the
        representation loses a degree of freedom and it is not possible to
        determine the first and third angles uniquely. In this case,
        a warning is raised, and the third angle is set to zero. Note however
        that the returned angles still represent the correct rotation.

        Parameters
        ----------
        seq : string, length 3
            3 characters belonging to the set {'X', 'Y', 'Z'} for intrinsic
            rotations, or {'x', 'y', 'z'} for extrinsic rotations [1]_.
            Adjacent axes cannot be the same.
            Extrinsic and intrinsic rotations cannot be mixed in one function
            call.
        degrees : boolean, optional
            Returned angles are in degrees if this flag is True, else they are
            in radians. Default is False.

        Returns
        -------
        angles : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used to initialize object.
            The returned angles are in the range:

            - First angle belongs to [-180, 180] degrees (both inclusive)
            - Third angle belongs to [-180, 180] degrees (both inclusive)
            - Second angle belongs to:

                - [-90, 90] degrees if all axes are different (like xyz)
                - [0, 180] degrees if first and third axes are the same
                  (like zxz)

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations
        .. [2] Bernardes E, Viollet S (2022) Quaternion to Euler angles 
               conversion: A direct, general and computationally efficient 
               method. PLoS ONE 17(11): e0276302. 
               https://doi.org/10.1371/journal.pone.0276302
        .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_rotvec([0, 0, np.pi/2])
        >>> r.as_euler('zxy', degrees=True)
        array([90.,  0.,  0.])
        >>> r.as_euler('zxy', degrees=True).shape
        (3,)

        Represent a stack of single rotation:

        >>> r = R.from_rotvec([[0, 0, np.pi/2]])
        >>> r.as_euler('zxy', degrees=True)
        array([[90.,  0.,  0.]])
        >>> r.as_euler('zxy', degrees=True).shape
        (1, 3)

        Represent multiple rotations in a single object:

        >>> r = R.from_rotvec([
        ... [0, 0, np.pi/2],
        ... [0, -np.pi/3, 0],
        ... [np.pi/4, 0, 0]])
        >>> r.as_euler('zxy', degrees=True)
        array([[ 90.,   0.,   0.],
               [  0.,   0., -60.],
               [  0.,  45.,   0.]])
        >>> r.as_euler('zxy', degrees=True).shape
        (3, 3)

        Rotation.as_davenport(self, axes, order, degrees=False)
Represent as Davenport angles.

        Any orientation can be expressed as a composition of 3 elementary
        rotations.

        For both Euler angles and Davenport angles, consecutive axes must 
        be are orthogonal (``axis2`` is orthogonal to both ``axis1`` and 
        ``axis3``). For Euler angles, there is an additional relationship 
        between ``axis1`` or ``axis3``, with two possibilities:

            - ``axis1`` and ``axis3`` are also orthogonal (asymmetric sequence)
            - ``axis1 == axis3`` (symmetric sequence)

        For Davenport angles, this last relationship is relaxed [1]_, and only
        the consecutive orthogonal axes requirement is maintained.

        A slightly modified version of the algorithm from [2]_ has been used to
        calculate Davenport angles for the rotation about a given sequence of
        axes.

        Davenport angles, just like Euler angles, suffer from the problem of
        gimbal lock [3]_, where the representation loses a degree of freedom
        and it is not possible to determine the first and third angles
        uniquely. In this case, a warning is raised, and the third angle is set
        to zero. Note however that the returned angles still represent the
        correct rotation.

        Parameters
        ----------
        axes : array_like, shape (3,) or ([1 or 2 or 3], 3)
            Axis of rotation, if one dimensional. If two dimensional, describes the 
            sequence of axes for rotations, where each axes[i, :] is the ith
            axis. If more than one axis is given, then the second axis must be
            orthogonal to both the first and third axes.
        order : string
            If it belongs to the set {'e', 'extrinsic'}, the sequence will be 
            extrinsic. If if belongs to the set {'i', 'intrinsic'}, sequence 
            will be treated as intrinsic.
        degrees : boolean, optional
            Returned angles are in degrees if this flag is True, else they are
            in radians. Default is False.

        Returns
        -------
        angles : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used to initialize object.
            The returned angles are in the range:

            - First angle belongs to [-180, 180] degrees (both inclusive)
            - Third angle belongs to [-180, 180] degrees (both inclusive)
            - Second angle belongs to a set of size 180 degrees, 
              given by: ``[-abs(lambda), 180 - abs(lambda)]``, where ``lambda``
              is the angle between the first and third axes.

        References
        ----------
        .. [1] Shuster, Malcolm & Markley, Landis. (2003). Generalization of
               the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304.
        .. [2] Bernardes E, Viollet S (2022) Quaternion to Euler angles
               conversion: A direct, general and computationally efficient method.
               PLoS ONE 17(11): e0276302. 10.1371/journal.pone.0276302
        .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Davenport angles are a generalization of Euler angles, when we use the
        canonical basis axes:

        >>> ex = [1, 0, 0]
        >>> ey = [0, 1, 0]
        >>> ez = [0, 0, 1]

        Represent a single rotation:

        >>> r = R.from_rotvec([0, 0, np.pi/2])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
        array([90.,  0.,  0.])
        >>> r.as_euler('zxy', degrees=True)
        array([90.,  0.,  0.])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape
        (3,)

        Represent a stack of single rotation:

        >>> r = R.from_rotvec([[0, 0, np.pi/2]])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
        array([[90.,  0.,  0.]])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape
        (1, 3)

        Represent multiple rotations in a single object:

        >>> r = R.from_rotvec([
        ... [0, 0, 90],
        ... [45, 0, 0]], degrees=True)
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True)
        array([[90.,  0.,  0.],
               [ 0., 45.,  0.]])
        >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape
        (2, 3)
        Rotation.as_mrp(self)
Represent as Modified Rodrigues Parameters (MRPs).

        MRPs are a 3 dimensional vector co-directional to the axis of rotation and whose
        magnitude is equal to ``tan(theta / 4)``, where ``theta`` is the angle of rotation
        (in radians) [1]_.

        MRPs have a singularity at 360 degrees which can be avoided by ensuring the angle of
        rotation does not exceed 180 degrees, i.e. switching the direction of the rotation when
        it is past 180 degrees. This function will always return MRPs corresponding to a rotation
        of less than or equal to 180 degrees.

        Returns
        -------
        mrps : ndarray, shape (3,) or (N, 3)
            Shape depends on shape of inputs used for initialization.

        References
        ----------
        .. [1] Shuster, M. D. "A Survey of Attitude Representations",
               The Journal of Astronautical Sciences, Vol. 41, No.4, 1993,
               pp. 475-476

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Represent a single rotation:

        >>> r = R.from_rotvec([0, 0, np.pi])
        >>> r.as_mrp()
        array([0.        , 0.        , 1.         ])
        >>> r.as_mrp().shape
        (3,)

        Represent a stack with a single rotation:

        >>> r = R.from_euler('xyz', [[180, 0, 0]], degrees=True)
        >>> r.as_mrp()
        array([[1.       , 0.        , 0.         ]])
        >>> r.as_mrp().shape
        (1, 3)

        Represent multiple rotations:

        >>> r = R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]])
        >>> r.as_mrp()
        array([[0.41421356, 0.        , 0.        ],
               [0.        , 0.        , 0.41421356]])
        >>> r.as_mrp().shape
        (2, 3)

        Notes
        -----

        .. versionadded:: 1.6.0
        Rotation.concatenate(cls, rotations)
Concatenate a sequence of `Rotation` objects into a single object.

        This is useful if you want to, for example, take the mean of a set of
        rotations and need to pack them into a single object to do so.

        Parameters
        ----------
        rotations : sequence of `Rotation` objects
            The rotations to concatenate.

        Returns
        -------
        concatenated : `Rotation` instance
            The concatenated rotations.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> r1 = R.from_rotvec([0, 0, 1])
        >>> r2 = R.from_rotvec([0, 0, 2])
        >>> rc = R.concatenate([r1, r2])
        >>> rc.as_rotvec()
        array([[0., 0., 1.],
               [0., 0., 2.]])
        >>> rc.mean().as_rotvec()
        array([0., 0., 1.5])

        Note that it may be simpler to create the desired rotations by passing
        in a single list of the data during initialization, rather then by
        concatenating:

        >>> R.from_rotvec([[0, 0, 1], [0, 0, 2]]).as_rotvec()
        array([[0., 0., 1.],
               [0., 0., 2.]])

        Notes
        -----
        .. versionadded:: 1.8.0
        Rotation.apply(self, vectors, inverse=False)
Apply this rotation to a set of vectors.

        If the original frame rotates to the final frame by this rotation, then
        its application to a vector can be seen in two ways:

            - As a projection of vector components expressed in the final frame
              to the original frame.
            - As the physical rotation of a vector being glued to the original
              frame as it rotates. In this case the vector components are
              expressed in the original frame before and after the rotation.

        In terms of rotation matrices, this application is the same as
        ``self.as_matrix() @ vectors``.

        Parameters
        ----------
        vectors : array_like, shape (3,) or (N, 3)
            Each `vectors[i]` represents a vector in 3D space. A single vector
            can either be specified with shape `(3, )` or `(1, 3)`. The number
            of rotations and number of vectors given must follow standard numpy
            broadcasting rules: either one of them equals unity or they both
            equal each other.
        inverse : boolean, optional
            If True then the inverse of the rotation(s) is applied to the input
            vectors. Default is False.

        Returns
        -------
        rotated_vectors : ndarray, shape (3,) or (N, 3)
            Result of applying rotation on input vectors.
            Shape depends on the following cases:

                - If object contains a single rotation (as opposed to a stack
                  with a single rotation) and a single vector is specified with
                  shape ``(3,)``, then `rotated_vectors` has shape ``(3,)``.
                - In all other cases, `rotated_vectors` has shape ``(N, 3)``,
                  where ``N`` is either the number of rotations or vectors.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Single rotation applied on a single vector:

        >>> vector = np.array([1, 0, 0])
        >>> r = R.from_rotvec([0, 0, np.pi/2])
        >>> r.as_matrix()
        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.apply(vector)
        array([2.22044605e-16, 1.00000000e+00, 0.00000000e+00])
        >>> r.apply(vector).shape
        (3,)

        Single rotation applied on multiple vectors:

        >>> vectors = np.array([
        ... [1, 0, 0],
        ... [1, 2, 3]])
        >>> r = R.from_rotvec([0, 0, np.pi/4])
        >>> r.as_matrix()
        array([[ 0.70710678, -0.70710678,  0.        ],
               [ 0.70710678,  0.70710678,  0.        ],
               [ 0.        ,  0.        ,  1.        ]])
        >>> r.apply(vectors)
        array([[ 0.70710678,  0.70710678,  0.        ],
               [-0.70710678,  2.12132034,  3.        ]])
        >>> r.apply(vectors).shape
        (2, 3)

        Multiple rotations on a single vector:

        >>> r = R.from_rotvec([[0, 0, np.pi/4], [np.pi/2, 0, 0]])
        >>> vector = np.array([1,2,3])
        >>> r.as_matrix()
        array([[[ 7.07106781e-01, -7.07106781e-01,  0.00000000e+00],
                [ 7.07106781e-01,  7.07106781e-01,  0.00000000e+00],
                [ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]],
               [[ 1.00000000e+00,  0.00000000e+00,  0.00000000e+00],
                [ 0.00000000e+00,  2.22044605e-16, -1.00000000e+00],
                [ 0.00000000e+00,  1.00000000e+00,  2.22044605e-16]]])
        >>> r.apply(vector)
        array([[-0.70710678,  2.12132034,  3.        ],
               [ 1.        , -3.        ,  2.        ]])
        >>> r.apply(vector).shape
        (2, 3)

        Multiple rotations on multiple vectors. Each rotation is applied on the
        corresponding vector:

        >>> r = R.from_euler('zxy', [
        ... [0, 0, 90],
        ... [45, 30, 60]], degrees=True)
        >>> vectors = [
        ... [1, 2, 3],
        ... [1, 0, -1]]
        >>> r.apply(vectors)
        array([[ 3.        ,  2.        , -1.        ],
               [-0.09026039,  1.11237244, -0.86860844]])
        >>> r.apply(vectors).shape
        (2, 3)

        It is also possible to apply the inverse rotation:

        >>> r = R.from_euler('zxy', [
        ... [0, 0, 90],
        ... [45, 30, 60]], degrees=True)
        >>> vectors = [
        ... [1, 2, 3],
        ... [1, 0, -1]]
        >>> r.apply(vectors, inverse=True)
        array([[-3.        ,  2.        ,  1.        ],
               [ 1.09533535, -0.8365163 ,  0.3169873 ]])

        Rotation.inv(self)
Invert this rotation.

        Composition of a rotation with its inverse results in an identity
        transformation.

        Returns
        -------
        inverse : `Rotation` instance
            Object containing inverse of the rotations in the current instance.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np

        Inverting a single rotation:

        >>> p = R.from_euler('z', 45, degrees=True)
        >>> q = p.inv()
        >>> q.as_euler('zyx', degrees=True)
        array([-45.,   0.,   0.])

        Inverting multiple rotations:

        >>> p = R.from_rotvec([[0, 0, np.pi/3], [-np.pi/4, 0, 0]])
        >>> q = p.inv()
        >>> q.as_rotvec()
        array([[-0.        , -0.        , -1.04719755],
               [ 0.78539816, -0.        , -0.        ]])

        Rotation.magnitude(self)
Get the magnitude(s) of the rotation(s).

        Returns
        -------
        magnitude : ndarray or float
            Angle(s) in radians, float if object contains a single rotation
            and ndarray if object contains multiple rotations. The magnitude
            will always be in the range [0, pi].

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np
        >>> r = R.from_quat(np.eye(4))
        >>> r.as_quat()
        array([[ 1., 0., 0., 0.],
               [ 0., 1., 0., 0.],
               [ 0., 0., 1., 0.],
               [ 0., 0., 0., 1.]])
        >>> r.magnitude()
        array([3.14159265, 3.14159265, 3.14159265, 0.        ])

        Magnitude of a single rotation:

        >>> r[0].magnitude()
        3.141592653589793
        Rotation.approx_equal(self, Rotation other, atol=None, degrees=False)
Determine if another rotation is approximately equal to this one.

        Equality is measured by calculating the smallest angle between the
        rotations, and checking to see if it is smaller than `atol`.

        Parameters
        ----------
        other : `Rotation` instance
            Object containing the rotations to measure against this one.
        atol : float, optional
            The absolute angular tolerance, below which the rotations are
            considered equal. If not given, then set to 1e-8 radians by
            default.
        degrees : bool, optional
            If True and `atol` is given, then `atol` is measured in degrees. If
            False (default), then atol is measured in radians.

        Returns
        -------
        approx_equal : ndarray or bool
            Whether the rotations are approximately equal, bool if object
            contains a single rotation and ndarray if object contains multiple
            rotations.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> import numpy as np
        >>> p = R.from_quat([0, 0, 0, 1])
        >>> q = R.from_quat(np.eye(4))
        >>> p.approx_equal(q)
        array([False, False, False, True])

        Approximate equality for a single rotation:

        >>> p.approx_equal(q[0])
        False
        Rotation.mean(self, weights=None)
Get the mean of the rotations.

        The mean used is the chordal L2 mean (also called the projected or
        induced arithmetic mean) [1]_. If ``A`` is a set of rotation matrices,
        then the mean ``M`` is the rotation matrix that minimizes the
        following loss function:

        .. math::

            L(M) = \sum_{i = 1}^{n} w_i \lVert \mathbf{A}_i -
            \mathbf{M} \rVert^2 ,

        where :math:`w_i`'s are the `weights` corresponding to each matrix.

        Parameters
        ----------
        weights : array_like shape (N,), optional
            Weights describing the relative importance of the rotations. If
            None (default), then all values in `weights` are assumed to be
            equal.

        Returns
        -------
        mean : `Rotation` instance
            Object containing the mean of the rotations in the current
            instance.

        References
        ----------
        .. [1] Hartley, Richard, et al.,
                "Rotation Averaging", International Journal of Computer Vision
                103, 2013, pp. 267-305.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R
        >>> r = R.from_euler('zyx', [[0, 0, 0],
        ...                          [1, 0, 0],
        ...                          [0, 1, 0],
        ...                          [0, 0, 1]], degrees=True)
        >>> r.mean().as_euler('zyx', degrees=True)
        array([0.24945696, 0.25054542, 0.24945696])
        Rotation.reduce(self, left=None, right=None, return_indices=False)
Reduce this rotation with the provided rotation groups.

        Reduction of a rotation ``p`` is a transformation of the form
        ``q = l * p * r``, where ``l`` and ``r`` are chosen from `left` and
        `right` respectively, such that rotation ``q`` has the smallest
        magnitude.

        If `left` and `right` are rotation groups representing symmetries of
        two objects rotated by ``p``, then ``q`` is the rotation of the
        smallest magnitude to align these objects considering their symmetries.

        Parameters
        ----------
        left : `Rotation` instance, optional
            Object containing the left rotation(s). Default value (None)
            corresponds to the identity rotation.
        right : `Rotation` instance, optional
            Object containing the right rotation(s). Default value (None)
            corresponds to the identity rotation.
        return_indices : bool, optional
            Whether to return the indices of the rotations from `left` and
            `right` used for reduction.

        Returns
        -------
        reduced : `Rotation` instance
            Object containing reduced rotations.
        left_best, right_best: integer ndarray
            Indices of elements from `left` and `right` used for reduction.
        Rotation.create_group(cls, group, axis=u'Z')
Create a 3D rotation group.

        Parameters
        ----------
        group : string
            The name of the group. Must be one of 'I', 'O', 'T', 'Dn', 'Cn',
            where `n` is a positive integer. The groups are:

                * I: Icosahedral group
                * O: Octahedral group
                * T: Tetrahedral group
                * D: Dicyclic group
                * C: Cyclic group

        axis : integer
            The cyclic rotation axis. Must be one of ['X', 'Y', 'Z'] (or
            lowercase). Default is 'Z'. Ignored for groups 'I', 'O', and 'T'.

        Returns
        -------
        rotation : `Rotation` instance
            Object containing the elements of the rotation group.

        Notes
        -----
        This method generates rotation groups only. The full 3-dimensional
        point groups [PointGroups]_ also contain reflections.

        References
        ----------
        .. [PointGroups] `Point groups
           <https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions>`_
           on Wikipedia.
        Rotation.identity(cls, num=None)
Get identity rotation(s).

        Composition with the identity rotation has no effect.

        Parameters
        ----------
        num : int or None, optional
            Number of identity rotations to generate. If None (default), then a
            single rotation is generated.

        Returns
        -------
        identity : Rotation object
            The identity rotation.
        Rotation.random(cls, num=None, random_state=None)
Generate uniformly distributed rotations.

        Parameters
        ----------
        num : int or None, optional
            Number of random rotations to generate. If None (default), then a
            single rotation is generated.
        random_state : {None, int, `numpy.random.Generator`,
                        `numpy.random.RandomState`}, optional

            If `seed` is None (or `np.random`), the `numpy.random.RandomState`
            singleton is used.
            If `seed` is an int, a new ``RandomState`` instance is used,
            seeded with `seed`.
            If `seed` is already a ``Generator`` or ``RandomState`` instance
            then that instance is used.

        Returns
        -------
        random_rotation : `Rotation` instance
            Contains a single rotation if `num` is None. Otherwise contains a
            stack of `num` rotations.

        Notes
        -----
        This function is optimized for efficiently sampling random rotation
        matrices in three dimensions. For generating random rotation matrices
        in higher dimensions, see `scipy.stats.special_ortho_group`.

        Examples
        --------
        >>> from scipy.spatial.transform import Rotation as R

        Sample a single rotation:

        >>> R.random().as_euler('zxy', degrees=True)
        array([-110.5976185 ,   55.32758512,   76.3289269 ])  # random

        Sample a stack of rotations:

        >>> R.random(5).as_euler('zxy', degrees=True)
        array([[-110.5976185 ,   55.32758512,   76.3289269 ],  # random
               [ -91.59132005,  -14.3629884 ,  -93.91933182],
               [  25.23835501,   45.02035145, -121.67867086],
               [ -51.51414184,  -15.29022692, -172.46870023],
               [ -81.63376847,  -27.39521579,    2.60408416]])

        See Also
        --------
        scipy.stats.special_ortho_group

       Rotation.align_vectors(cls, a, b, weights=None, return_sensitivity=False)
Estimate a rotation to optimally align two sets of vectors.

        Find a rotation between frames A and B which best aligns a set of
        vectors `a` and `b` observed in these frames. The following loss
        function is minimized to solve for the rotation matrix
        :math:`C`:

        .. math::

            L(C) = \frac{1}{2} \sum_{i = 1}^{n} w_i \lVert \mathbf{a}_i -
            C \mathbf{b}_i \rVert^2 ,

        where :math:`w_i`'s are the `weights` corresponding to each vector.

        The rotation is estimated with Kabsch algorithm [1]_, and solves what
        is known as the "pointing problem", or "Wahba's problem" [2]_.

        There are two special cases. The first is if a single vector is given
        for `a` and `b`, in which the shortest distance rotation that aligns
        `b` to `a` is returned.

        The second is when one of the weights is infinity. In this case, the
        shortest distance rotation between the primary infinite weight vectors
        is calculated as above. Then, the rotation about the aligned primary
        vectors is calculated such that the secondary vectors are optimally
        aligned per the above loss function. The result is the composition
        of these two rotations. The result via this process is the same as the
        Kabsch algorithm as the corresponding weight approaches infinity in
        the limit. For a single secondary vector this is known as the
        "align-constrain" algorithm [3]_.

        For both special cases (single vectors or an infinite weight), the
        sensitivity matrix does not have physical meaning and an error will be
        raised if it is requested. For an infinite weight, the primary vectors
        act as a constraint with perfect alignment, so their contribution to
        `rssd` will be forced to 0 even if they are of different lengths.

        Parameters
        ----------
        a : array_like, shape (3,) or (N, 3)
            Vector components observed in initial frame A. Each row of `a`
            denotes a vector.
        b : array_like, shape (3,) or (N, 3)
            Vector components observed in another frame B. Each row of `b`
            denotes a vector.
        weights : array_like shape (N,), optional
            Weights describing the relative importance of the vector
            observations. If None (default), then all values in `weights` are
            assumed to be 1. One and only one weight may be infinity, and
            weights must be positive.
        return_sensitivity : bool, optional
            Whether to return the sensitivity matrix. See Notes for details.
            Default is False.

        Returns
        -------
        rotation : `Rotation` instance
            Best estimate of the rotation that transforms `b` to `a`.
        rssd : float
            Stands for "root sum squared distance". Square root of the weighted
            sum of the squared distances between the given sets of vectors
            after alignment. It is equal to ``sqrt(2 * minimum_loss)``, where
            ``minimum_loss`` is the loss function evaluated for the found
            optimal rotation.
            Note that the result will also be weighted by the vectors'
            magnitudes, so perfectly aligned vector pairs will have nonzero
            `rssd` if they are not of the same length. So, depending on the
            use case it may be desirable to normalize the input vectors to unit
            length before calling this method.
        sensitivity_matrix : ndarray, shape (3, 3)
            Sensitivity matrix of the estimated rotation estimate as explained
            in Notes. Returned only when `return_sensitivity` is True. Not
            valid if aligning a single pair of vectors or if there is an
            infinite weight, in which cases an error will be raised.

        Notes
        -----
        The sensitivity matrix gives the sensitivity of the estimated rotation
        to small perturbations of the vector measurements. Specifically we
        consider the rotation estimate error as a small rotation vector of
        frame A. The sensitivity matrix is proportional to the covariance of
        this rotation vector assuming that the vectors in `a` was measured with
        errors significantly less than their lengths. To get the true
        covariance matrix, the returned sensitivity matrix must be multiplied
        by harmonic mean [4]_ of variance in each observation. Note that
        `weights` are supposed to be inversely proportional to the observation
        variances to get consistent results. For example, if all vectors are
        measured with the same accuracy of 0.01 (`weights` must be all equal),
        then you should multiple the sensitivity matrix by 0.01**2 to get the
        covariance.

        Refer to [5]_ for more rigorous discussion of the covariance
        estimation. See [6]_ for more discussion of the pointing problem and
        minimal proper pointing.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Kabsch_algorithm
        .. [2] https://en.wikipedia.org/wiki/Wahba%27s_problem
        .. [3] Magner, Robert,
                "Extending target tracking capabilities through trajectory and
                momentum setpoint optimization." Small Satellite Conference,
                2018.
        .. [4] https://en.wikipedia.org/wiki/Harmonic_mean
        .. [5] F. Landis Markley,
                "Attitude determination using vector observations: a fast
                optimal matrix algorithm", Journal of Astronautical Sciences,
                Vol. 41, No.2, 1993, pp. 261-280.
        .. [6] Bar-Itzhack, Itzhack Y., Daniel Hershkowitz, and Leiba Rodman,
                "Pointing in Real Euclidean Space", Journal of Guidance,
                Control, and Dynamics, Vol. 20, No. 5, 1997, pp. 916-922.

        Examples
        --------
        >>> import numpy as np
        >>> from scipy.spatial.transform import Rotation as R

        Here we run the baseline Kabsch algorithm to best align two sets of
        vectors, where there is noise on the last two vector measurements of
        the ``b`` set:

        >>> a = [[0, 1, 0], [0, 1, 1], [0, 1, 1]]
        >>> b = [[1, 0, 0], [1, 1.1, 0], [1, 0.9, 0]]
        >>> rot, rssd, sens = R.align_vectors(a, b, return_sensitivity=True)
        >>> rot.as_matrix()
        array([[0., 0., 1.],
               [1., 0., 0.],
               [0., 1., 0.]])

        When we apply the rotation to ``b``, we get vectors close to ``a``:

        >>> rot.apply(b)
        array([[0. , 1. , 0. ],
               [0. , 1. , 1.1],
               [0. , 1. , 0.9]])

        The error for the first vector is 0, and for the last two the error is
        magnitude 0.1. The `rssd` is the square root of the sum of the
        weighted squared errors, and the default weights are all 1, so in this
        case the `rssd` is calculated as
        ``sqrt(1 * 0**2 + 1 * 0.1**2 + 1 * (-0.1)**2) = 0.141421356237308``

        >>> a - rot.apply(b)
        array([[ 0., 0.,  0. ],
               [ 0., 0., -0.1],
               [ 0., 0.,  0.1]])
        >>> np.sqrt(np.sum(np.ones(3) @ (a - rot.apply(b))**2))
        0.141421356237308
        >>> rssd
        0.141421356237308

        The sensitivity matrix for this example is as follows:

        >>> sens
        array([[0.2, 0. , 0.],
               [0. , 1.5, 1.],
               [0. , 1. , 1.]])

        Special case 1: Find a minimum rotation between single vectors:

        >>> a = [1, 0, 0]
        >>> b = [0, 1, 0]
        >>> rot, _ = R.align_vectors(a, b)
        >>> rot.as_matrix()
        array([[0., 1., 0.],
               [-1., 0., 0.],
               [0., 0., 1.]])
        >>> rot.apply(b)
        array([1., 0., 0.])

        Special case 2: One infinite weight. Here we find a rotation between
        primary and secondary vectors that can align exactly:

        >>> a = [[0, 1, 0], [0, 1, 1]]
        >>> b = [[1, 0, 0], [1, 1, 0]]
        >>> rot, _ = R.align_vectors(a, b, weights=[np.inf, 1])
        >>> rot.as_matrix()
        array([[0., 0., 1.],
               [1., 0., 0.],
               [0., 1., 0.]])
        >>> rot.apply(b)
        array([[0., 1., 0.],
               [0., 1., 1.]])

        Here the secondary vectors must be best-fit:

        >>> a = [[0, 1, 0], [0, 1, 1]]
        >>> b = [[1, 0, 0], [1, 2, 0]]
        >>> rot, _ = R.align_vectors(a, b, weights=[np.inf, 1])
        >>> rot.as_matrix()
        array([[0., 0., 1.],
               [1., 0., 0.],
               [0., 1., 0.]])
        >>> rot.apply(b)
        array([[0., 1., 0.],
               [0., 1., 2.]])
        00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899Interpolate rotations.

        Compute the interpolated rotations at the given `times`.

        Parameters
        ----------
        times : array_like
            Times to compute the interpolations at. Can be a scalar or
            1-dimensional.

        Returns
        -------
        interpolated_rotation : `Rotation` instance
            Object containing the rotations computed at given `times`.

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import name %Scalling %R should have returned an instance of BaseException, not %R_ARRAY_API<cyfunction %U at %p><stringsource>scipy.spatial.transform._rotation._empty3scipy.spatial.transform._rotation._cross3scipy.spatial.transform._rotation._empty2scipy.spatial.transform._rotation._zeros2scipy.spatial.transform._rotation._empty1_cython_3_0_10.0Interpreter change detected - this module can only be loaded into one interpreter per process.close() -> raise GeneratorExit inside generator.Whether this instance represents a single rotation.send(arg) -> send 'arg' into generator,
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    This class provides an interface to initialize from and represent rotations
    with:

    - Quaternions
    - Rotation Matrices
    - Rotation Vectors
    - Modified Rodrigues Parameters
    - 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_...`` methods (see examples below).
    ``Rotation(...)`` is not supposed to be instantiated directly.

    Attributes
    ----------
    single

    Methods
    -------
    __len__
    from_quat
    from_matrix
    from_rotvec
    from_mrp
    from_euler
    from_davenport
    as_quat
    as_matrix
    as_rotvec
    as_mrp
    as_euler
    as_davenport
    concatenate
    apply
    __mul__
    __pow__
    inv
    magnitude
    approx_equal
    mean
    reduce
    create_group
    __getitem__
    identity
    random
    align_vectors

    See Also
    --------
    Slerp

    Notes
    -----
    .. versionadded:: 1.2.0

    Examples
    --------
    >>> from scipy.spatial.transform import Rotation as R
    >>> import numpy as np

    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_matrix()
    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 rotation matrix:

    >>> r = R.from_matrix([[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_matrix()
    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`` method 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_matrix()
    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`` method:

    >>> 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_matrix()
    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]])

    In fact it can be converted to numpy.array:

    >>> r_array = np.asarray(r)
    >>> r_array.shape
    (3,)
    >>> r_array[0].as_matrix()
    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]])

    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])

    A rotation can be composed with itself using the ``**`` operator:

    >>> p = R.from_rotvec([1, 0, 0])
    >>> q = p ** 2
    >>> q.as_rotvec()
    array([2., 0., 0.])

    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.]])

    The following function can be used to plot rotations with Matplotlib by
    showing how they transform the standard x, y, z coordinate axes:

    >>> import matplotlib.pyplot as plt

    >>> def plot_rotated_axes(ax, r, name=None, offset=(0, 0, 0), scale=1):
    ...     colors = ("#FF6666", "#005533", "#1199EE")  # Colorblind-safe RGB
    ...     loc = np.array([offset, offset])
    ...     for i, (axis, c) in enumerate(zip((ax.xaxis, ax.yaxis, ax.zaxis),
    ...                                       colors)):
    ...         axlabel = axis.axis_name
    ...         axis.set_label_text(axlabel)
    ...         axis.label.set_color(c)
    ...         axis.line.set_color(c)
    ...         axis.set_tick_params(colors=c)
    ...         line = np.zeros((2, 3))
    ...         line[1, i] = scale
    ...         line_rot = r.apply(line)
    ...         line_plot = line_rot + loc
    ...         ax.plot(line_plot[:, 0], line_plot[:, 1], line_plot[:, 2], c)
    ...         text_loc = line[1]*1.2
    ...         text_loc_rot = r.apply(text_loc)
    ...         text_plot = text_loc_rot + loc[0]
    ...         ax.text(*text_plot, axlabel.upper(), color=c,
    ...                 va="center", ha="center")
    ...     ax.text(*offset, name, color="k", va="center", ha="center",
    ...             bbox={"fc": "w", "alpha": 0.8, "boxstyle": "circle"})

    Create three rotations - the identity and two Euler rotations using
    intrinsic and extrinsic conventions:

    >>> r0 = R.identity()
    >>> r1 = R.from_euler("ZYX", [90, -30, 0], degrees=True)  # intrinsic
    >>> r2 = R.from_euler("zyx", [90, -30, 0], degrees=True)  # extrinsic

    Add all three rotations to a single plot:

    >>> ax = plt.figure().add_subplot(projection="3d", proj_type="ortho")
    >>> plot_rotated_axes(ax, r0, name="r0", offset=(0, 0, 0))
    >>> plot_rotated_axes(ax, r1, name="r1", offset=(3, 0, 0))
    >>> plot_rotated_axes(ax, r2, name="r2", offset=(6, 0, 0))
    >>> _ = ax.annotate(
    ...     "r0: Identity Rotation\n"
    ...     "r1: Intrinsic Euler Rotation (ZYX)\n"
    ...     "r2: Extrinsic Euler Rotation (zyx)",
    ...     xy=(0.6, 0.7), xycoords="axes fraction", ha="left"
    ... )
    >>> ax.set(xlim=(-1.25, 7.25), ylim=(-1.25, 1.25), zlim=(-1.25, 1.25))
    >>> ax.set(xticks=range(-1, 8), yticks=[-1, 0, 1], zticks=[-1, 0, 1])
    >>> ax.set_aspect("equal", adjustable="box")
    >>> ax.figure.set_size_inches(6, 5)
    >>> plt.tight_layout()

    Show the plot:

    >>> plt.show()

    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.

    if _cython_generator_type is not None:
    try: Generator = _module.Generator
    except AttributeError: pass
    else: Generator.register(_cython_generator_type)
if _cython_coroutine_type is not None:
    try: Coroutine = _module.Coroutine
    except AttributeError: pass
    else: Coroutine.register(_cython_coroutine_type)
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