DAMASK_EICMD/python/tests/test_Rotation.py

import pytest
import numpy as np
from scipy import stats

from damask import Rotation
from damask import Table
from damask import _rotation
from damask import grid_filters
from damask import tensor

n = 1000
atol=1.e-4

@pytest.fixture
def ref_path(ref_path_base):
    """Directory containing reference results."""
    return ref_path_base/'Rotation'

@pytest.fixture
def set_of_rotations(set_of_quaternions):
    return [Rotation.from_quaternion(s) for s in set_of_quaternions]

@pytest.fixture
def multidim_rotations(set_of_quaternions):
    L = len(set_of_quaternions)
    i = 0
    while L%(f:=np.random.randint(2,np.sqrt(L).astype(int))) > 0 and i<L:
        i += 1

    f = i if i == L else f
    return Rotation.from_quaternion(set_of_quaternions.reshape((L//f,f,-1)))


####################################################################################################
# Code below available according to the following conditions
####################################################################################################
# Copyright (c) 2017-2019, Martin Diehl/Max-Planck-Institut für Eisenforschung GmbH
# Copyright (c) 2013-2014, Marc De Graef/Carnegie Mellon University
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without modification, are
# permitted provided that the following conditions are met:
#
#     - Redistributions of source code must retain the above copyright notice, this list
#        of conditions and the following disclaimer.
#     - Redistributions in binary form must reproduce the above copyright notice, this
#        list of conditions and the following disclaimer in the documentation and/or
#        other materials provided with the distribution.
#     - Neither the names of Marc De Graef, Carnegie Mellon University nor the names
#        of its contributors may be used to endorse or promote products derived from
#        this software without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
# USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
####################################################################################################
_P = _rotation._P

# parameters for conversion from/to cubochoric
_sc   = _rotation._sc
_beta = _rotation._beta
_R1   = _rotation._R1

def iszero(a):
    return np.isclose(a,0.0,atol=1.0e-12,rtol=0.0)

#---------- Quaternion ----------
def qu2om(qu):
    """Quaternion to rotation matrix."""
    qq = qu[0]**2-(qu[1]**2 + qu[2]**2 + qu[3]**2)
    om = np.diag(qq + 2.0*np.array([qu[1],qu[2],qu[3]])**2)

    om[0,1] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3])
    om[1,0] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3])
    om[1,2] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1])
    om[2,1] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1])
    om[2,0] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2])
    om[0,2] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2])
    return om if _P < 0.0 else np.swapaxes(om,-1,-2)

def qu2eu(qu):
    """Quaternion to Bunge-Euler angles."""
    q03 = qu[0]**2+qu[3]**2
    q12 = qu[1]**2+qu[2]**2
    chi = np.sqrt(q03*q12)
    if   np.abs(q12) < 1.e-8:
        eu = np.array([np.arctan2(-_P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0,   0.0])
    elif np.abs(q03) < 1.e-8:
        eu = np.array([np.arctan2(   2.0*qu[1]*qu[2],qu[1]**2-qu[2]**2), np.pi, 0.0])
    else:
        eu = np.array([np.arctan2((-_P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]-qu[2]*qu[3])*chi ),
                       np.arctan2( 2.0*chi, q03-q12 ),
                       np.arctan2(( _P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]+qu[2]*qu[3])*chi )])
    # reduce Euler angles to definition range
    eu[np.abs(eu)<1.e-6] = 0.0
    eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
    return eu

def qu2ax(qu):
    """
    Quaternion to axis angle pair.

    Modified version of the original formulation, should be numerically more stable
    """
    if np.isclose(qu[0],1.,rtol=0.0):                                                               # set axis to [001] if the angle is 0/360
        ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
    elif qu[0] > 1.e-8:
        s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2)
        omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
        ax = ax = np.array([ qu[1]*s, qu[2]*s, qu[3]*s, omega ])
    else:
        ax = ax = np.array([ qu[1], qu[2], qu[3], np.pi])
    return ax

def qu2ro(qu):
    """Quaternion to Rodrigues-Frank vector."""
    if iszero(qu[0]):
        ro = np.array([qu[1], qu[2], qu[3], np.inf])
    else:
        s = np.linalg.norm(qu[1:4])
        ro = np.array([0.0,0.0,_P,0.0] if iszero(s) else \
                      [ qu[1]/s,  qu[2]/s,  qu[3]/s, np.tan(np.arccos(np.clip(qu[0],-1.0,1.0)))])
    return ro

def qu2ho(qu):
    """Quaternion to homochoric vector."""
    omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
    if np.abs(omega) < 1.0e-12:
        ho = np.zeros(3)
    else:
        ho = np.array([qu[1], qu[2], qu[3]])
        f  = 0.75 * ( omega - np.sin(omega) )
        ho = ho/np.linalg.norm(ho) * f**(1./3.)
    return ho


#---------- Rotation matrix ----------
def om2qu(om):
    trace = om.trace()
    if trace > 0:
        s = 0.5 / np.sqrt(trace+ 1.0)
        qu = np.array([0.25 / s,( om[2,1] - om[1,2] ) * s,( om[0,2] - om[2,0] ) * s,( om[1,0] - om[0,1] ) * s])
    else:
        if ( om[0,0] > om[1,1] and om[0,0] > om[2,2] ):
            s = 2.0 * np.sqrt( 1.0 + om[0,0] - om[1,1] - om[2,2])
            qu = np.array([ (om[2,1] - om[1,2]) / s,0.25 * s,(om[0,1] + om[1,0]) / s,(om[0,2] + om[2,0]) / s])
        elif (om[1,1] > om[2,2]):
            s = 2.0 * np.sqrt( 1.0 + om[1,1] - om[0,0] - om[2,2])
            qu = np.array([ (om[0,2] - om[2,0]) / s,(om[0,1] + om[1,0]) / s,0.25 * s,(om[1,2] + om[2,1]) / s])
        else:
            s = 2.0 * np.sqrt( 1.0 + om[2,2] - om[0,0] - om[1,1] )
            qu = np.array([ (om[1,0] - om[0,1]) / s,(om[0,2] + om[2,0]) / s,(om[1,2] + om[2,1]) / s,0.25 * s])
    if qu[0]<0: qu*=-1
    return qu*np.array([1.,_P,_P,_P])

def om2eu(om):
    """Rotation matrix to Bunge-Euler angles."""
    if not np.isclose(np.abs(om[2,2]),1.0,0.0):
        zeta = 1.0/np.sqrt(1.0-om[2,2]**2)
        eu = np.array([np.arctan2(om[2,0]*zeta,-om[2,1]*zeta),
                       np.arccos(om[2,2]),
                       np.arctan2(om[0,2]*zeta, om[1,2]*zeta)])
    else:
        eu = np.array([np.arctan2( om[0,1],om[0,0]), np.pi*0.5*(1-om[2,2]),0.0])                    # following the paper, not the reference implementation
    eu[np.abs(eu)<1.e-8] = 0.0
    eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
    return eu

def om2ax(om):
    """Rotation matrix to axis angle pair."""
    ax=np.empty(4)

    # first get the rotation angle
    t = 0.5*(om.trace() -1.0)
    ax[3] = np.arccos(np.clip(t,-1.0,1.0))
    if np.abs(ax[3])<1.e-8:
        ax = np.array([ 0.0, 0.0, 1.0, 0.0])
    else:
        w,vr = np.linalg.eig(om)
        # next, find the eigenvalue (1,0j)
        i = np.where(np.isclose(w,1.0+0.0j))[0][0]
        ax[0:3] = np.real(vr[0:3,i])
        diagDelta = -_P*np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]])
        ax[0:3] = np.where(np.abs(diagDelta)<1e-12, ax[0:3],np.abs(ax[0:3])*np.sign(diagDelta))
    return ax

#---------- Bunge-Euler angles ----------
def eu2qu(eu):
    """Bunge-Euler angles to quaternion."""
    ee = 0.5*eu
    cPhi = np.cos(ee[1])
    sPhi = np.sin(ee[1])
    qu = np.array([    cPhi*np.cos(ee[0]+ee[2]),
                   -_P*sPhi*np.cos(ee[0]-ee[2]),
                   -_P*sPhi*np.sin(ee[0]-ee[2]),
                   -_P*cPhi*np.sin(ee[0]+ee[2]) ])
    if qu[0] < 0.0: qu*=-1
    return qu

def eu2om(eu):
    """Bunge-Euler angles to rotation matrix."""
    c = np.cos(eu)
    s = np.sin(eu)

    om = np.array([[+c[0]*c[2]-s[0]*s[2]*c[1], +s[0]*c[2]+c[0]*s[2]*c[1], +s[2]*s[1]],
                   [-c[0]*s[2]-s[0]*c[2]*c[1], -s[0]*s[2]+c[0]*c[2]*c[1], +c[2]*s[1]],
                   [+s[0]*s[1],                -c[0]*s[1],                +c[1]     ]])
    om[np.abs(om)<1.e-12] = 0.0
    return om

def eu2ax(eu):
    """Bunge-Euler angles to axis angle pair."""
    t = np.tan(eu[1]*0.5)
    sigma = 0.5*(eu[0]+eu[2])
    delta = 0.5*(eu[0]-eu[2])
    tau   = np.linalg.norm([t,np.sin(sigma)])
    alpha = np.pi if iszero(np.cos(sigma)) else \
            2.0*np.arctan(tau/np.cos(sigma))

    if np.abs(alpha)<1.e-6:
        ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
    else:
        ax = -_P/tau * np.array([ t*np.cos(delta), t*np.sin(delta), np.sin(sigma) ])                # passive axis angle pair so a minus sign in front
        ax = np.append(ax,alpha)
        if alpha < 0.0: ax *= -1.0                                                                  # ensure alpha is positive
    return ax

def eu2ro(eu):
    """Bunge-Euler angles to Rodrigues-Frank vector."""
    ro = eu2ax(eu)                                                                                  # convert to axis angle pair representation
    if ro[3] >= np.pi:                                                                              # Differs from original implementation. check convention 5
        ro[3] = np.inf
    elif iszero(ro[3]):
        ro = np.array([ 0.0, 0.0, _P, 0.0 ])
    else:
        ro[3] = np.tan(ro[3]*0.5)
    return ro

#---------- Axis angle pair ----------
def ax2qu(ax):
    """Axis angle pair to quaternion."""
    if np.abs(ax[3])<1.e-6:
        qu = np.array([ 1.0, 0.0, 0.0, 0.0 ])
    else:
        c = np.cos(ax[3]*0.5)
        s = np.sin(ax[3]*0.5)
        qu = np.array([ c, ax[0]*s, ax[1]*s, ax[2]*s ])
    return qu

def ax2om(ax):
    """Axis angle pair to rotation matrix."""
    c = np.cos(ax[3])
    s = np.sin(ax[3])
    omc = 1.0-c
    om=np.diag(ax[0:3]**2*omc + c)

    for idx in [[0,1,2],[1,2,0],[2,0,1]]:
        q = omc*ax[idx[0]] * ax[idx[1]]
        om[idx[0],idx[1]] = q + s*ax[idx[2]]
        om[idx[1],idx[0]] = q - s*ax[idx[2]]
    return om if _P < 0.0 else om.T

def ax2ro(ax):
    """Axis angle pair to Rodrigues-Frank vector."""
    if np.abs(ax[3])<1.e-6:
        ro = [ 0.0, 0.0, _P, 0.0 ]
    else:
        ro = [ax[0], ax[1], ax[2]]
        # 180 degree case
        ro += [np.inf] if np.isclose(ax[3],np.pi,atol=1.0e-15,rtol=0.0) else \
              [np.tan(ax[3]*0.5)]
    ro = np.array(ro)
    return ro

def ax2ho(ax):
    """Axis angle pair to homochoric vector."""
    f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0)
    ho = ax[0:3] * f
    return ho


#---------- Rodrigues-Frank vector ----------
def ro2ax(ro):
    """Rodrigues-Frank vector to axis angle pair."""
    if np.abs(ro[3]) < 1.e-8:
        ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
    elif not np.isfinite(ro[3]):
        ax = np.array([ ro[0], ro[1], ro[2], np.pi ])
    else:
        angle = 2.0*np.arctan(ro[3])
        ta = np.linalg.norm(ro[0:3])
        ax = np.array([ ro[0]*ta, ro[1]*ta, ro[2]*ta, angle ])
    return ax

def ro2ho(ro):
    """Rodrigues-Frank vector to homochoric vector."""
    if np.sum(ro[0:3]**2.0) < 1.e-8:
        ho = np.zeros(3)
    else:
        f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
        ho = ro[0:3] * (0.75*f)**(1.0/3.0)
    return ho

#---------- Homochoric vector----------
def ho2ax(ho):
    """Homochoric vector to axis angle pair."""
    tfit = np.array([+0.9999999999999968,     -0.49999999999986866,     -0.025000000000632055,
                     -0.003928571496460683,   -0.0008164666077062752,   -0.00019411896443261646,
                     -0.00004985822229871769, -0.000014164962366386031, -1.9000248160936107e-6,
                     -5.72184549898506e-6,    +7.772149920658778e-6,    -0.00001053483452909705,
                     +9.528014229335313e-6,   -5.660288876265125e-6,    +1.2844901692764126e-6,
                     +1.1255185726258763e-6,  -1.3834391419956455e-6,   +7.513691751164847e-7,
                     -2.401996891720091e-7,   +4.386887017466388e-8,    -3.5917775353564864e-9])
    # normalize h and store the magnitude
    hmag_squared = np.sum(ho**2.)
    if iszero(hmag_squared):
        ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
    else:
        hm = np.ones_like(hmag_squared)

        # convert the magnitude to the rotation angle
        s = 0.0
        for t in tfit:
            s += t * hm
            hm *= hmag_squared
        ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0)))
    return ax

def ho2cu(ho):
    """
    Homochoric vector to cubochoric vector.

    References
    ----------
    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
    https://doi.org/10.1088/0965-0393/22/7/075013

    """
    rs = np.linalg.norm(ho)

    if np.allclose(ho,0.0,rtol=0.0,atol=1.0e-16):
        cu = np.zeros(3)
    else:
        xyz3 = ho[_get_pyramid_order(ho,'forward')]

        # inverse M_3
        xyz2 = xyz3[0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[2])) )

        # inverse M_2
        qxy = np.sum(xyz2**2)

        if np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-16):
            Tinv = np.zeros(2)
        else:
            q2 = qxy + np.max(np.abs(xyz2))**2
            sq2 = np.sqrt(q2)
            q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2))*sq2))
            tt = np.clip((np.min(np.abs(xyz2))**2+np.max(np.abs(xyz2))*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
            Tinv = np.array([1.0,np.arccos(tt)/np.pi*12.0]) if np.abs(xyz2[1]) <= np.abs(xyz2[0]) else \
                   np.array([np.arccos(tt)/np.pi*12.0,1.0])
            Tinv = q * np.where(xyz2<0.0,-Tinv,Tinv)

        # inverse M_1
        cu = np.array([ Tinv[0], Tinv[1],  (-1.0 if xyz3[2] < 0.0 else 1.0) * rs / np.sqrt(6.0/np.pi) ]) /_sc
        cu = cu[_get_pyramid_order(ho,'backward')]
    return cu

#---------- Cubochoric ----------
def cu2ho(cu):
    """
    Cubochoric vector to homochoric vector.

    References
    ----------
    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
    https://doi.org/10.1088/0965-0393/22/7/075013

    """
    # transform to the sphere grid via the curved square, and intercept the zero point
    if np.allclose(cu,0.0,rtol=0.0,atol=1.0e-16):
        ho = np.zeros(3)
    else:
        # get pyramide and scale by grid parameter ratio
        XYZ = cu[_get_pyramid_order(cu,'forward')] * _sc

        # intercept all the points along the z-axis
        if np.allclose(XYZ[0:2],0.0,rtol=0.0,atol=1.0e-16):
            ho = np.array([0.0, 0.0, np.sqrt(6.0/np.pi) * XYZ[2]])
        else:
            order = [1,0] if np.abs(XYZ[1]) <= np.abs(XYZ[0]) else [0,1]
            q = np.pi/12.0 * XYZ[order[0]]/XYZ[order[1]]
            c = np.cos(q)
            s = np.sin(q)
            q = _R1*2.0**0.25/_beta * XYZ[order[1]] / np.sqrt(np.sqrt(2.0)-c)
            T = np.array([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q

            # transform to sphere grid (inverse Lambert)
            # note that there is no need to worry about dividing by zero, since XYZ[2] can not become zero
            c = np.sum(T**2)
            s = c *         np.pi/24.0 /XYZ[2]**2
            c = c * np.sqrt(np.pi/24.0)/XYZ[2]

            q = np.sqrt( 1.0 - s )
            ho = np.array([ T[order[1]] * q, T[order[0]] * q, np.sqrt(6.0/np.pi) * XYZ[2] - c ])

        ho = ho[_get_pyramid_order(cu,'backward')]
    return ho

def _get_pyramid_order(xyz,direction=None):
    """
    Get order of the coordinates.

    Depending on the pyramid in which the point is located, the order need to be adjusted.

    Parameters
    ----------
    xyz : numpy.ndarray
       coordinates of a point on a uniform refinable grid on a ball or
       in a uniform refinable cubical grid.

    References
    ----------
    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
    https://doi.org/10.1088/0965-0393/22/7/075013

    """
    order = {'forward':np.array([[0,1,2],[1,2,0],[2,0,1]]),
             'backward':np.array([[0,1,2],[2,0,1],[1,2,0]])}
    if   np.maximum(abs(xyz[0]),abs(xyz[1])) <= xyz[2] or \
         np.maximum(abs(xyz[0]),abs(xyz[1])) <=-xyz[2]:
        p = 0
    elif np.maximum(abs(xyz[1]),abs(xyz[2])) <= xyz[0] or \
         np.maximum(abs(xyz[1]),abs(xyz[2])) <=-xyz[0]:
        p = 1
    elif np.maximum(abs(xyz[2]),abs(xyz[0])) <= xyz[1] or \
         np.maximum(abs(xyz[2]),abs(xyz[0])) <=-xyz[1]:
        p = 2
    return order[direction][p]

####################################################################################################
####################################################################################################

def mul(me, other):
    """
    Multiplication.

    Parameters
    ----------
    other : numpy.ndarray or Rotation
        Vector, second or fourth order tensor, or rotation object that is rotated.

    """
    if me.quaternion.shape != (4,):
        raise NotImplementedError('Support for multiple rotations missing')
    if isinstance(other, Rotation):
        me_q  = me.quaternion[0]
        me_p  = me.quaternion[1:]
        other_q = other.quaternion[0]
        other_p = other.quaternion[1:]
        R = me.__class__(np.append(me_q*other_q - np.dot(me_p,other_p),
                                   me_q*other_p + other_q*me_p + _P * np.cross(me_p,other_p)))
        return R._standardize()
    elif isinstance(other, np.ndarray):
        if other.shape == (3,):
            A = me.quaternion[0]**2.0 - np.dot(me.quaternion[1:],me.quaternion[1:])
            B = 2.0 * np.dot(me.quaternion[1:],other)
            C = 2.0 * _P *   me.quaternion[0]

            return A*other + B*me.quaternion[1:] + C * np.cross(me.quaternion[1:],other)

        elif other.shape == (3,3,):
            R = me.as_matrix()
            return np.dot(R,np.dot(other,R.T))
        elif other.shape == (3,3,3,3,):
            R = me.as_matrix()
            return np.einsum('ia,jb,kc,ld,abcd->ijkl',R,R,R,R,other)
            RR = np.outer(R, R)
            RRRR = np.outer(RR, RR).reshape(4 * (3,3))
            axes = ((0, 2, 4, 6), (0, 1, 2, 3))
            return np.tensordot(RRRR, other, axes)
        else:
            raise ValueError('Can only rotate vectors, 2nd order tensors, and 4th order tensors')
    else:
        raise TypeError(f'Cannot rotate {type(other)}')


class TestRotation:

    @pytest.mark.parametrize('forward,backward',[(Rotation._qu2om,Rotation._om2qu),
                                                 (Rotation._qu2eu,Rotation._eu2qu),
                                                 (Rotation._qu2ax,Rotation._ax2qu),
                                                 (Rotation._qu2ro,Rotation._ro2qu),
                                                 (Rotation._qu2ho,Rotation._ho2qu),
                                                 (Rotation._qu2cu,Rotation._cu2qu)])
    def test_quaternion_internal(self,set_of_rotations,forward,backward):
        """Ensure invariance of conversion from quaternion and back."""
        for rot in set_of_rotations:
            m = rot.as_quaternion()
            o = backward(forward(m))
            ok = np.allclose(m,o,atol=atol)
            if np.isclose(rot.as_quaternion()[0],0.0,atol=atol):
                ok |= np.allclose(m*-1.,o,atol=atol)
            assert ok and np.isclose(np.linalg.norm(o),1.0), f'{m},{o},{rot.as_quaternion()}'

    @pytest.mark.parametrize('forward,backward',[(Rotation._om2qu,Rotation._qu2om),
                                                 (Rotation._om2eu,Rotation._eu2om),
                                                 (Rotation._om2ax,Rotation._ax2om),
                                                 (Rotation._om2ro,Rotation._ro2om),
                                                 (Rotation._om2ho,Rotation._ho2om),
                                                 (Rotation._om2cu,Rotation._cu2om)])
    def test_matrix_internal(self,set_of_rotations,forward,backward):
        """Ensure invariance of conversion from rotation matrix and back."""
        for rot in set_of_rotations:
            m = rot.as_matrix()
            o = backward(forward(m))
            ok = np.allclose(m,o,atol=atol)
            assert ok and np.isclose(np.linalg.det(o),1.0), f'{m},{o},{rot.as_quaternion()}'

    @pytest.mark.parametrize('forward,backward',[(Rotation._eu2qu,Rotation._qu2eu),
                                                 (Rotation._eu2om,Rotation._om2eu),
                                                 (Rotation._eu2ax,Rotation._ax2eu),
                                                 (Rotation._eu2ro,Rotation._ro2eu),
                                                 (Rotation._eu2ho,Rotation._ho2eu),
                                                 (Rotation._eu2cu,Rotation._cu2eu)])
    def test_Eulers_internal(self,set_of_rotations,forward,backward):
        """Ensure invariance of conversion from Euler angles and back."""
        for rot in set_of_rotations:
            m = rot.as_Euler_angles()
            o = backward(forward(m))
            u = np.array([np.pi*2,np.pi,np.pi*2])
            ok = np.allclose(m,o,atol=atol)
            ok |= np.allclose(np.where(np.isclose(m,u),m-u,m),np.where(np.isclose(o,u),o-u,o),atol=atol)
            if np.isclose(m[1],0.0,atol=atol) or np.isclose(m[1],np.pi,atol=atol):
                sum_phi = np.unwrap([m[0]+m[2],o[0]+o[2]])
                ok |= np.isclose(sum_phi[0],sum_phi[1],atol=atol)
            assert ok and (np.zeros(3)-1.e-9 <= o).all() \
                      and (o <= np.array([np.pi*2.,np.pi,np.pi*2.])+1.e-9).all(), f'{m},{o},{rot.as_quaternion()}'

    @pytest.mark.parametrize('forward,backward',[(Rotation._ax2qu,Rotation._qu2ax),
                                                 (Rotation._ax2om,Rotation._om2ax),
                                                 (Rotation._ax2eu,Rotation._eu2ax),
                                                 (Rotation._ax2ro,Rotation._ro2ax),
                                                 (Rotation._ax2ho,Rotation._ho2ax),
                                                 (Rotation._ax2cu,Rotation._cu2ax)])
    def test_axis_angle_internal(self,set_of_rotations,forward,backward):
        """Ensure invariance of conversion from axis angle angles pair and back."""
        for rot in set_of_rotations:
            m = rot.as_axis_angle()
            o = backward(forward(m))
            ok = np.allclose(m,o,atol=atol)
            if np.isclose(m[3],np.pi,atol=atol):
                ok |= np.allclose(m*np.array([-1.,-1.,-1.,1.]),o,atol=atol)
            assert ok and np.isclose(np.linalg.norm(o[:3]),1.0) and o[3]<=np.pi+1.e-9, f'{m},{o},{rot.as_quaternion()}'

    @pytest.mark.parametrize('forward,backward',[(Rotation._ro2qu,Rotation._qu2ro),
                                                 (Rotation._ro2om,Rotation._om2ro),
                                                 (Rotation._ro2eu,Rotation._eu2ro),
                                                 (Rotation._ro2ax,Rotation._ax2ro),
                                                 (Rotation._ro2ho,Rotation._ho2ro),
                                                 (Rotation._ro2cu,Rotation._cu2ro)])
    def test_Rodrigues_internal(self,set_of_rotations,forward,backward):
        """Ensure invariance of conversion from Rodrigues-Frank vector and back."""
        cutoff = np.tan(np.pi*.5*(1.-1e-5))
        for rot in set_of_rotations:
            m = rot.as_Rodrigues_vector()
            o = backward(forward(m))
            ok = np.allclose(np.clip(m,None,cutoff),np.clip(o,None,cutoff),atol=atol)
            ok |= np.isclose(m[3],0.0,atol=atol)
            if m[3] > cutoff:
                ok |= np.allclose(m[:3],-1*o[:3])

            assert ok and np.isclose(np.linalg.norm(o[:3]),1.0), f'{m},{o},{rot.as_quaternion()}'

    @pytest.mark.parametrize('forward,backward',[(Rotation._ho2qu,Rotation._qu2ho),
                                                 (Rotation._ho2om,Rotation._om2ho),
                                                 #(Rotation._ho2eu,Rotation._eu2ho),
                                                 (Rotation._ho2ax,Rotation._ax2ho),
                                                 (Rotation._ho2ro,Rotation._ro2ho),
                                                 (Rotation._ho2cu,Rotation._cu2ho)])
    def test_homochoric_internal(self,set_of_rotations,forward,backward):
        """Ensure invariance of conversion from homochoric vector and back."""
        for rot in set_of_rotations:
            m = rot.as_homochoric()
            o = backward(forward(m))
            ok = np.allclose(m,o,atol=atol)
            assert ok and np.linalg.norm(o) < _R1 + 1.e-9, f'{m},{o},{rot.as_quaternion()}'

    @pytest.mark.parametrize('forward,backward',[(Rotation._cu2qu,Rotation._qu2cu),
                                                 (Rotation._cu2om,Rotation._om2cu),
                                                 (Rotation._cu2eu,Rotation._eu2cu),
                                                 (Rotation._cu2ax,Rotation._ax2cu),
                                                 (Rotation._cu2ro,Rotation._ro2cu),
                                                 (Rotation._cu2ho,Rotation._ho2cu)])
    def test_cubochoric_internal(self,set_of_rotations,forward,backward):
        """Ensure invariance of conversion from cubochoric vector and back."""
        for rot in set_of_rotations:
            m = rot.as_cubochoric()
            o = backward(forward(m))
            ok = np.allclose(m,o,atol=atol)
            if np.count_nonzero(np.isclose(np.abs(o),np.pi**(2./3.)*.5)):
                ok |= np.allclose(m*-1.,o,atol=atol)
            assert ok and np.max(np.abs(o)) < np.pi**(2./3.) * 0.5 + 1.e-9, f'{m},{o},{rot.as_quaternion()}'

    @pytest.mark.parametrize('vectorized, single',[(Rotation._qu2om,qu2om),
                                                   (Rotation._qu2eu,qu2eu),
                                                   (Rotation._qu2ax,qu2ax),
                                                   (Rotation._qu2ro,qu2ro),
                                                   (Rotation._qu2ho,qu2ho)])
    def test_quaternion_vectorization(self,set_of_quaternions,vectorized,single):
        """Check vectorized implementation for quaternion against single point calculation."""
        qu = np.array(set_of_quaternions)
        vectorized(qu.reshape(qu.shape[0]//2,-1,4))
        co = vectorized(qu)
        for q,c in zip(qu,co):
            assert np.allclose(single(q),c) and np.allclose(single(q),vectorized(q)), f'{q},{c}'


    @pytest.mark.parametrize('vectorized, single',[(Rotation._om2qu,om2qu),
                                                   (Rotation._om2eu,om2eu),
                                                   (Rotation._om2ax,om2ax)])
    def test_matrix_vectorization(self,set_of_rotations,vectorized,single):
        """Check vectorized implementation for rotation matrix against single point calculation."""
        om = np.array([rot.as_matrix() for rot in set_of_rotations])
        vectorized(om.reshape(om.shape[0]//2,-1,3,3))
        co = vectorized(om)
        for o,c in zip(om,co):
            assert np.allclose(single(o),c) and np.allclose(single(o),vectorized(o)), f'{o},{c}'

    @pytest.mark.parametrize('vectorized, single',[(Rotation._eu2qu,eu2qu),
                                                   (Rotation._eu2om,eu2om),
                                                   (Rotation._eu2ax,eu2ax),
                                                   (Rotation._eu2ro,eu2ro)])
    def test_Eulers_vectorization(self,set_of_rotations,vectorized,single):
        """Check vectorized implementation for Euler angles against single point calculation."""
        eu = np.array([rot.as_Euler_angles() for rot in set_of_rotations])
        vectorized(eu.reshape(eu.shape[0]//2,-1,3))
        co = vectorized(eu)
        for e,c in zip(eu,co):
            assert np.allclose(single(e),c) and np.allclose(single(e),vectorized(e)), f'{e},{c}'

    @pytest.mark.parametrize('vectorized, single',[(Rotation._ax2qu,ax2qu),
                                                   (Rotation._ax2om,ax2om),
                                                   (Rotation._ax2ro,ax2ro),
                                                   (Rotation._ax2ho,ax2ho)])
    def test_axis_angle_vectorization(self,set_of_rotations,vectorized,single):
        """Check vectorized implementation for axis angle pair against single point calculation."""
        ax = np.array([rot.as_axis_angle() for rot in set_of_rotations])
        vectorized(ax.reshape(ax.shape[0]//2,-1,4))
        co = vectorized(ax)
        for a,c in zip(ax,co):
            assert np.allclose(single(a),c) and np.allclose(single(a),vectorized(a)), f'{a},{c}'


    @pytest.mark.parametrize('vectorized, single',[(Rotation._ro2ax,ro2ax),
                                                   (Rotation._ro2ho,ro2ho)])
    def test_Rodrigues_vectorization(self,set_of_rotations,vectorized,single):
        """Check vectorized implementation for Rodrigues-Frank vector against single point calculation."""
        ro = np.array([rot.as_Rodrigues_vector() for rot in set_of_rotations])
        vectorized(ro.reshape(ro.shape[0]//2,-1,4))
        co = vectorized(ro)
        for r,c in zip(ro,co):
            assert np.allclose(single(r),c) and np.allclose(single(r),vectorized(r)), f'{r},{c}'

    @pytest.mark.parametrize('vectorized, single',[(Rotation._ho2ax,ho2ax),
                                                   (Rotation._ho2cu,ho2cu)])
    def test_homochoric_vectorization(self,set_of_rotations,vectorized,single):
        """Check vectorized implementation for homochoric vector against single point calculation."""
        ho = np.array([rot.as_homochoric() for rot in set_of_rotations])
        vectorized(ho.reshape(ho.shape[0]//2,-1,3))
        co = vectorized(ho)
        for h,c in zip(ho,co):
            assert np.allclose(single(h),c) and np.allclose(single(h),vectorized(h)), f'{h},{c}'

    @pytest.mark.parametrize('vectorized, single',[(Rotation._cu2ho,cu2ho)])
    def test_cubochoric_vectorization(self,set_of_rotations,vectorized,single):
        """Check vectorized implementation for cubochoric vector against single point calculation."""
        cu = np.array([rot.as_cubochoric() for rot in set_of_rotations])
        vectorized(cu.reshape(cu.shape[0]//2,-1,3))
        co = vectorized(cu)
        for u,c in zip(cu,co):
            assert np.allclose(single(u),c) and np.allclose(single(u),vectorized(u)), f'{u},{c}'

    @pytest.mark.parametrize('func',[Rotation.from_axis_angle])
    def test_normalization_vectorization(self,func):
        """Check vectorized implementation normalization."""
        vec = np.random.rand(5,4)
        ori = func(vec,normalize=True)
        for v,o in zip(vec,ori):
            assert np.allclose(func(v,normalize=True).as_quaternion(),o.as_quaternion())

    def test_invalid_init(self):
        with pytest.raises(TypeError):
            Rotation(np.ones(3))

    def test_to_numpy(self):
        r = Rotation.from_random(np.random.randint(0,10,4))
        assert (r.as_quaternion() == np.array(r)).all()

    def test_bounds(self,multidim_rotations):
        m = multidim_rotations

        q = m.as_quaternion()
        assert np.allclose(1.,np.linalg.norm(q,axis=-1))

        v = m.as_Rodrigues_vector(compact=False)
        assert np.allclose(1.,np.linalg.norm(v[...,:3],axis=-1))

        v = m.as_axis_angle(degrees=False)
        assert np.allclose(1.,np.linalg.norm(v[...,:3],axis=-1))
        assert (v[...,3] >= 0.).all and (v < np.pi+1.e-9).all()

        r = m.as_matrix()
        assert np.allclose(1.,np.linalg.det(r))

        e = m.as_Euler_angles(degrees=False)
        assert (e >= 0.).all and (e < np.pi*np.array([2.,1.,2.])+1.e-9).all()

        c = m.as_cubochoric()
        assert (np.linalg.norm(c,ord=np.inf,axis=-1) < np.pi**(2./3.)*0.5+1.e-9).all()

        h = m.as_homochoric()
        assert (np.linalg.norm(h,axis=-1) < (3.*np.pi/4.)**(1./3.) + 1.e-9).all()


    @pytest.mark.parametrize('accept_homomorph',[True,False])
    @pytest.mark.parametrize('normalize',[True,False])
    @pytest.mark.parametrize('P',[1,-1])
    def test_quaternion(self,multidim_rotations,accept_homomorph,normalize,P):
        c = np.array([1,-P,-P,-P]) * (-1 if accept_homomorph else 1) * (0.9 if normalize else 1.0)
        m = multidim_rotations
        o = Rotation.from_quaternion(m.as_quaternion()*c,
                                     accept_homomorph=accept_homomorph,
                                     normalize=normalize,
                                     P=P)
        f = Rotation(np.where(np.isclose(m.as_quaternion()[...,0],0.0,atol=atol)[...,np.newaxis],~o,o))
        assert np.logical_or(m.isclose(o,atol=atol),
                             m.isclose(f,atol=atol)
                            ).all()


    @pytest.mark.parametrize('degrees',[True,False])
    def test_Eulers(self,multidim_rotations,degrees):
        m = multidim_rotations
        o = Rotation.from_Euler_angles(m.as_Euler_angles(degrees),
                                       degrees=degrees)
        f = Rotation(np.where(np.isclose(m.as_quaternion()[...,0],0.0,atol=atol)[...,np.newaxis],~o,o))
        assert np.logical_or(m.isclose(o,atol=atol),
                             m.isclose(f,atol=atol)
                            ).all()


    @pytest.mark.parametrize('degrees',[True,False])
    @pytest.mark.parametrize('normalize',[True,False])
    @pytest.mark.parametrize('P',[1,-1])
    def test_axis_angle(self,multidim_rotations,degrees,normalize,P):
        c = np.array([-P,-P,-P,1.])
        c[:3] *= 0.9 if normalize else 1.0

        m = multidim_rotations
        o = Rotation.from_axis_angle(m.as_axis_angle(degrees)*c,
                                     degrees=degrees,
                                     normalize=normalize,
                                     P=P)
        f = Rotation(np.where(np.isclose(m.as_quaternion()[...,0],0.0,atol=atol)[...,np.newaxis],~o,o))
        assert np.logical_or(m.isclose(o,atol=atol),
                             m.isclose(f,atol=atol)
                            ).all()


    @pytest.mark.parametrize('normalize',[True,False])
    def test_matrix(self,multidim_rotations,normalize):
        m = multidim_rotations
        o = Rotation.from_matrix(m.as_matrix()*(0.9 if normalize else 1.0),
                                 normalize=normalize)
        f = Rotation(np.where(np.isclose(m.as_quaternion()[...,0],0.0,atol=atol)[...,np.newaxis],~o,o))
        assert np.logical_or(m.isclose(o,atol=atol),
                             m.isclose(f,atol=atol)
                            ).all()


    def test_parallel(self,multidim_rotations):
        m = multidim_rotations
        a = np.broadcast_to(np.array([[1.0,0.0,0.0],
                                      [0.0,1.0,0.0]]),m.shape+(2,3))
        assert m.allclose(Rotation.from_parallel(a,m.broadcast_to(m.shape+(2,))@a))


    @pytest.mark.parametrize('normalize',[True,False])
    @pytest.mark.parametrize('P',[1,-1])
    def test_Rodrigues(self,multidim_rotations,normalize,P):
        c = np.array([-P,-P,-P,1.])
        c[:3] *= 0.9 if normalize else 1.0
        m = multidim_rotations
        o = Rotation.from_Rodrigues_vector(m.as_Rodrigues_vector()*c,
                                           normalize=normalize,
                                           P=P)
        f = Rotation(np.where(np.isclose(m.as_quaternion()[...,0],0.0,atol=atol)[...,np.newaxis],~o,o))
        assert np.logical_or(m.isclose(o,atol=atol),
                             m.isclose(f,atol=atol)
                            ).all()


    def test_Rodrigues_compact(self,multidim_rotations):
        m = multidim_rotations
        c = m.as_Rodrigues_vector(compact=True)
        r = m.as_Rodrigues_vector(compact=False)
        assert np.allclose(r[...,:3]*r[...,3:], c, equal_nan=True)


    @pytest.mark.parametrize('P',[1,-1])
    def test_homochoric(self,multidim_rotations,P):
        m = multidim_rotations
        o = Rotation.from_homochoric(m.as_homochoric()*-P,
                                     P=P)
        f = Rotation(np.where(np.isclose(m.as_quaternion()[...,0],0.0,atol=atol)[...,np.newaxis],~o,o))
        assert np.logical_or(m.isclose(o,atol=atol),
                             m.isclose(f,atol=atol)
                            ).all()


    @pytest.mark.parametrize('P',[1,-1])
    def test_cubochoric(self,multidim_rotations,P):
        m = multidim_rotations
        o = Rotation.from_cubochoric(m.as_cubochoric()*-P,
                                     P=P)
        f = Rotation(np.where(np.isclose(m.as_quaternion()[...,0],0.0,atol=atol)[...,np.newaxis],~o,o))
        assert np.logical_or(m.isclose(o,atol=atol),
                             m.isclose(f,atol=atol)
                            ).all()


    @pytest.mark.parametrize('reciprocal',[True,False])
    def test_basis(self,multidim_rotations,reciprocal):
        m = multidim_rotations
        r = m.as_matrix()
        r = np.linalg.inv(tensor.transpose(r)/np.pi) if reciprocal else r
        o = Rotation.from_basis(r,
                                reciprocal=reciprocal)
        f = Rotation(np.where(np.isclose(m.as_quaternion()[...,0],0.0,atol=atol)[...,np.newaxis],~o,o))
        assert np.logical_or(m.isclose(o,atol=atol),
                             m.isclose(f,atol=atol)
                            ).all()


    @pytest.mark.parametrize('shape',[None,1,(4,4)])
    def test_random(self,shape):
        r = Rotation.from_random(shape)
        assert r.shape == () if shape is None else (1,) if shape == 1 else shape

    @pytest.mark.parametrize('shape',[None,5,(4,6)])
    def test_equal(self,shape):
        R = Rotation.from_random(shape,rng_seed=1)
        assert R == R if shape is None else (R == R).all()

    @pytest.mark.parametrize('shape',[None,5,(4,6)])
    def test_allclose(self,shape):
        R = Rotation.from_random(shape,rng_seed=1)
        assert R.allclose(R)

    @pytest.mark.parametrize('shape',[None,5,(4,6)])
    def test_unequal(self,shape):
        R = Rotation.from_random(shape,rng_seed=1)
        assert not (R != R if shape is None else (R != R).any())


    def test_equal_ambiguous(self):
        qu = np.random.rand(10,4)
        qu[:,0] = 0.
        qu /= np.linalg.norm(qu,axis=1,keepdims=True)
        assert (Rotation(qu) == Rotation(-qu)).all()

    def test_inversion(self):
        r = Rotation.from_random()
        assert r == ~~r

    @pytest.mark.parametrize('shape',[1,(1,),(4,2),(1,1,1),tuple(np.random.randint(0,10,4))])
    def test_size(self,shape):
        assert Rotation.from_random(shape).size == np.prod(shape)

    @pytest.mark.parametrize('shape',[None,1,(1,),(4,2),(1,1,1),tuple(np.random.randint(0,10,4))])
    def test_shape(self,shape):
        r = Rotation.from_random(shape=shape)
        assert r.shape == (shape if isinstance(shape,tuple) else (shape,) if shape else ())

    @pytest.mark.parametrize('shape',[None,1,(1,),(4,2),(3,3,2)])
    def test_append(self,shape):
        r = Rotation.from_random(shape=shape)
        p = Rotation.from_random(shape=shape)
        s = r.append(p)
        print(f'append 2x {shape} --> {s.shape}')
        assert np.logical_and(s[0,...] == r[0,...], s[-1,...] == p[-1,...]).all()

    @pytest.mark.parametrize('shape',[None,1,(1,),(4,2),(3,3,2)])
    def test_append_list(self,shape):
        r = Rotation.from_random(shape=shape)
        p = Rotation.from_random(shape=shape)
        s = r.append([r,p])
        print(f'append 3x {shape} --> {s.shape}')
        assert np.logical_and(s[0,...] == r[0,...], s[-1,...] == p[-1,...]).all()

    @pytest.mark.parametrize('quat,standardized',[
                                                  ([-1,0,0,0],[1,0,0,0]),
                                                  ([-0.5,-0.5,-0.5,-0.5],[0.5,0.5,0.5,0.5]),
                                                 ])
    def test_standardization(self,quat,standardized):
        assert Rotation(quat)._standardize() == Rotation(standardized)

    @pytest.mark.parametrize('shape,length',[
                                          ((2,3,4),2),
                                          (4,4),
                                          ((),0)
                                         ])
    def test_len(self,shape,length):
        r = Rotation.from_random(shape=shape)
        assert len(r) == length

    @pytest.mark.parametrize('shape',[(4,6),(2,3,4),(3,3,3)])
    @pytest.mark.parametrize('order',['C','F'])
    def test_flatten_reshape(self,shape,order):
        r = Rotation.from_random(shape=shape)
        assert (r == r.flatten(order).reshape(shape,order)).all()

    @pytest.mark.parametrize('function',[Rotation.from_quaternion,
                                         Rotation.from_Euler_angles,
                                         Rotation.from_axis_angle,
                                         Rotation.from_matrix,
                                         Rotation.from_Rodrigues_vector,
                                         Rotation.from_homochoric,
                                         Rotation.from_cubochoric])
    def test_invalid_shape(self,function):
        invalid_shape = np.random.random(np.random.randint(8,32,(3)))
        with pytest.raises(ValueError):
            function(invalid_shape)

    def test_invalid_shape_parallel(self):
        invalid_a = np.random.random(np.random.randint(8,32,(3)))
        invalid_b = np.random.random(np.random.randint(8,32,(3)))
        with pytest.raises(ValueError):
            Rotation.from_parallel(invalid_a,invalid_b)


    @pytest.mark.parametrize('fr,to',[(Rotation.from_quaternion,'as_quaternion'),
                                      (Rotation.from_axis_angle,'as_axis_angle'),
                                      (Rotation.from_Rodrigues_vector, 'as_Rodrigues_vector'),
                                      (Rotation.from_homochoric,'as_homochoric'),
                                      (Rotation.from_cubochoric,'as_cubochoric')])
    def test_invalid_P(self,fr,to):
        R = Rotation.from_random(np.random.randint(8,32,(3)))                                       # noqa
        with pytest.raises(ValueError):
            fr(eval(f'R.{to}()'),P=-30)


    def test_invalid_multiplication(self):
        rot = Rotation.from_random()
        with pytest.raises(TypeError):
            rot@Rotation.from_random()
        with pytest.raises(TypeError):
            rot@[1,2,3,4]


    @pytest.mark.parametrize('shape',[None,(3,),(4,2)])
    def test_broadcast(self,shape):
        rot = Rotation.from_random(shape)
        new_shape = tuple(np.random.randint(8,32,(3))) if shape is None else \
                    rot.shape + (np.random.randint(8,32),)
        rot_broadcast = rot.broadcast_to(tuple(new_shape))
        for i in range(rot_broadcast.shape[-1]):
            assert np.allclose(rot_broadcast.quaternion[...,i,:], rot.quaternion)


    @pytest.mark.parametrize('function,invalid',[(Rotation.from_quaternion,        np.array([-1,0,0,0])),
                                                 (Rotation.from_quaternion,        np.array([1,1,1,0])),
                                                 (Rotation.from_Euler_angles,      np.array([1,4,0])),
                                                 (Rotation.from_Euler_angles,      np.array([-1,0,0])),
                                                 (Rotation.from_axis_angle,        np.array([1,0,0,4])),
                                                 (Rotation.from_axis_angle,        np.array([1,1,0,1])),
                                                 (Rotation.from_matrix,            np.random.rand(3,3)),
                                                 (Rotation.from_matrix,            np.array([[1,1,0],[1,2,0],[0,0,1]])),
                                                 (Rotation.from_Rodrigues_vector,  np.array([1,0,0,-1])),
                                                 (Rotation.from_Rodrigues_vector,  np.array([1,1,0,1])),
                                                 (Rotation.from_homochoric,        np.array([2,2,2])),
                                                 (Rotation.from_cubochoric,        np.array([1.1,0,0])),
                                                ])
    def test_invalid_value(self,function,invalid):
        with pytest.raises(ValueError):
            function(invalid)

    @pytest.mark.parametrize('direction',['forward',
                                          'backward'])
    def test_pyramid_vectorization(self,direction):
        p = np.random.rand(n,3)
        o = Rotation._get_pyramid_order(p,direction)
        for i,o_i in enumerate(o):
            assert (o_i==Rotation._get_pyramid_order(p[i],direction)).all()

    def test_pyramid_invariant(self):
        a = np.random.rand(n,3)
        f = Rotation._get_pyramid_order(a,'forward')
        b = Rotation._get_pyramid_order(a,'backward')
        assert (np.take_along_axis(np.take_along_axis(a,f,-1),b,-1) == a).all()


    @pytest.mark.parametrize('data',[np.random.rand(5,3),
                                     np.random.rand(5,3,3),
                                     np.random.rand(5,3,3,3,3)])
    def test_rotate_vectorization(self,set_of_rotations,data):
        for rot in set_of_rotations:
            v = rot.broadcast_to((5,)) @ data
            for i in range(data.shape[0]):
                assert np.allclose(mul(rot,data[i]),v[i]), f'{i-data[i]}'


    @pytest.mark.parametrize('data',[np.random.rand(3),
                                     np.random.rand(3,3),
                                     np.random.rand(3,3,3,3)])
    def test_rotate_identity(self,data):
        R = Rotation()
        print(R,data)
        assert np.allclose(data,R@data)

    @pytest.mark.parametrize('data',[np.random.rand(3),
                                     np.random.rand(3,3),
                                     np.random.rand(3,3,3,3)])
    def test_rotate_360deg(self,data):
        phi_1 = np.random.random() * np.pi
        phi_2 = 2*np.pi - phi_1
        R_1 = Rotation.from_Euler_angles(np.array([phi_1,0.,0.]))
        R_2 = Rotation.from_Euler_angles(np.array([0.,0.,phi_2]))
        assert np.allclose(data,R_2@(R_1@data))

    @pytest.mark.parametrize('pwr',[-10,0,1,2.5,np.pi,np.random.random()])
    def test_rotate_power(self,pwr):
        R = Rotation.from_random()
        axis_angle = R.as_axis_angle()
        axis_angle[ 3] = (pwr*axis_angle[-1])%(2.*np.pi)
        if axis_angle[3] > np.pi:
            axis_angle[3] -= 2.*np.pi
            axis_angle    *= -1
        assert (R**pwr).isclose(Rotation.from_axis_angle(axis_angle))

    def test_rotate_inverse(self):
        R = Rotation.from_random()
        assert np.allclose(np.eye(3),(~R*R).as_matrix())

    @pytest.mark.parametrize('data',[np.random.rand(3),
                                     np.random.rand(3,3),
                                     np.random.rand(3,3,3,3)])
    def test_rotate_inverse_array(self,data):
        R = Rotation.from_random()
        assert np.allclose(data,~R@(R@data))

    @pytest.mark.parametrize('data',[np.random.rand(4),
                                     np.random.rand(3,2),
                                     np.random.rand(3,2,3,3)])
    def test_rotate_invalid_shape(self,data):
        R = Rotation.from_random()
        with pytest.raises(ValueError):
            R@data

    @pytest.mark.parametrize('data',['does_not_work',
                                     (1,2),
                                     5])
    def test_rotate_invalid_type(self,data):
        R = Rotation.from_random()
        with pytest.raises(TypeError):
            R@data

    def test_misorientation_invariant(self):
        R = Rotation.from_random()
        assert np.allclose(R.misorientation(R).as_matrix(),np.eye(3))

    def test_misorientation_average(self):
        """2 times the average is the misorientation."""
        r = Rotation.from_random(2)
        a = r[0].misorientation(r[1]).as_axis_angle()
        b = r.average().misorientation(r[1]).as_axis_angle()
        b[3] = (b[3]*2)%np.pi
        assert np.allclose(a,b)

    def test_misorientation_360deg(self):
        R_1 = Rotation()
        R_2 = Rotation.from_Euler_angles([360,0,0],degrees=True)
        assert np.allclose(R_1.misorientation(R_2).as_matrix(),np.eye(3))

    def test_composition(self):
        a,b = (Rotation.from_random(),Rotation.from_random())
        c = a * b
        a *= b
        assert c == a

    def test_composition_invalid(self):
        with pytest.raises(TypeError):
            Rotation()*np.ones(3)

    def test_composition_inverse(self):
        a,b = (Rotation.from_random(),Rotation.from_random())
        c = a / b
        a /= b
        assert c == a

    def test_composition_inverse_invalid(self):
        with pytest.raises(TypeError):
            Rotation()/np.ones(3)

    def test_power(self):
        a = Rotation.from_random()
        r = (np.random.rand()-.5)*4
        b = a**r
        a **= r
        assert a == b

    def test_invariant(self):
        R = Rotation.from_random()
        assert (R/R).isclose(R*R**(-1)) and (R/R).isclose(Rotation())

    @pytest.mark.parametrize('item',[np.ones(3),np.ones((3,3)), np.ones((3,3,3,3))])
    def test_apply(self,item):
        r = Rotation.from_random()
        assert (r.apply(item) == r@item).all()

    @pytest.mark.parametrize('angle',[10,20,30,40,50,60,70,80,90,100,120])
    def test_average(self,angle):
        R = Rotation.from_axis_angle([[0,0,1,10],[0,0,1,angle]],degrees=True)
        avg_angle = R.average().as_axis_angle(degrees=True,pair=True)[1]
        assert np.isclose(avg_angle,10+(angle-10)/2.)


    @pytest.mark.parametrize('sigma',[5,10,15,20])
    @pytest.mark.parametrize('shape',[1000,10000,100000,(10,100)])
    def test_spherical_component(self,sigma,shape):
        p = []
        for run in range(5):
            c = Rotation.from_random()
            o = Rotation.from_spherical_component(c,sigma,shape,degrees=True)
            _, angles = c.misorientation(o).as_axis_angle(pair=True,degrees=True)
            angles[::2] *= -1                                                                       # flip angle for every second to symmetrize distribution

            p.append(stats.normaltest(angles)[1])

        sigma_out = np.std(angles)
        p = np.average(p)
        assert (.9 < sigma/sigma_out < 1.1) and p > 1e-2, f'{sigma/sigma_out},{p}'


    @pytest.mark.parametrize('sigma',[5,10,15,20])
    @pytest.mark.parametrize('shape',[1000,10000,100000,(10,100)])
    def test_from_fiber_component(self,sigma,shape):

        def astuple(a):
            return tuple(a) if hasattr(a,'__len__') else (a,)

        p = []
        for run in range(9):
            alpha = np.arccos(np.random.random()),np.random.random()*2*np.pi
            beta  = np.arccos(np.random.random()),np.random.random()*2*np.pi

            f_in_C = np.array([np.sin(alpha[0])*np.cos(alpha[1]), np.sin(alpha[0])*np.sin(alpha[1]), np.cos(alpha[0])])
            f_in_S = np.array([np.sin( beta[0])*np.cos( beta[1]), np.sin( beta[0])*np.sin( beta[1]), np.cos( beta[0])])
            ax = np.append(np.cross(f_in_C,f_in_S), - np.arccos(np.dot(f_in_C,f_in_S)))
            n = Rotation.from_axis_angle(ax if ax[3] > 0.0 else -ax,normalize=True)           # rotation to align fiber axis in crystal and sample system
            o = Rotation.from_fiber_component(alpha,beta,np.radians(sigma),shape,False)
            angles = np.arccos(np.clip(np.dot(o@np.broadcast_to(f_in_S,astuple(shape)+(3,)),n@f_in_S),-1,1))
            dist   = np.array(angles) * (np.random.randint(0,2,shape)*2-1)

            p.append(stats.normaltest(dist)[1])

        sigma_out = np.degrees(np.std(dist))
        p = np.average(p)
        assert (.9 < sigma/sigma_out < 1.1) and p > 1e-2, f'{sigma/sigma_out},{p}'


    @pytest.mark.parametrize('fractions',[True,False])
    @pytest.mark.parametrize('degrees',[True,False])
    @pytest.mark.parametrize('shape',[2**13,2**14,2**15,(2**8,2**6)])
    def test_ODF_cell(self,ref_path,fractions,degrees,shape):
        steps = np.array([144,36,36])
        limits = np.array([360.,90.,90.])
        rng = tuple(zip(np.zeros(3),limits))

        weights = Table.load(ref_path/'ODF_experimental_cell.txt').get('intensity').flatten()
        Eulers = grid_filters.coordinates0_point(steps,limits)
        Eulers = np.radians(Eulers) if not degrees else Eulers

        Eulers_r = Rotation.from_ODF(weights,Eulers.reshape(-1,3,order='F'),shape,degrees,fractions).as_Euler_angles(True)
        weights_r = np.histogramdd(Eulers_r.reshape(-1,3),steps,rng)[0].flatten(order='F')/np.prod(shape) * np.sum(weights)

        if fractions: assert np.sqrt(((weights_r - weights) ** 2).mean()) < 4

    @pytest.mark.parametrize('degrees',[True,False])
    @pytest.mark.parametrize('shape',[2**13,2**14,2**15,(2**8,2**6)])
    def test_ODF_node(self,ref_path,degrees,shape):
        steps = np.array([144,36,36])
        limits = np.array([360.,90.,90.])
        rng = tuple(zip(-limits/steps*.5,limits-limits/steps*.5))

        weights = Table.load(ref_path/'ODF_experimental.txt').get('intensity')
        weights = weights.reshape(steps+1,order='F')[:-1,:-1,:-1].reshape(-1,order='F')

        Eulers = grid_filters.coordinates0_node(steps,limits)[:-1,:-1,:-1]
        Eulers = np.radians(Eulers) if not degrees else Eulers

        Eulers_r = Rotation.from_ODF(weights,Eulers.reshape(-1,3,order='F'),shape,degrees).as_Euler_angles(True)
        weights_r = np.histogramdd(Eulers_r.reshape(-1,3),steps,rng)[0].flatten(order='F')/np.prod(shape) * np.sum(weights)

        assert np.sqrt(((weights_r - weights) ** 2).mean()) < 5

    def test_mul_invalid(self):
        with pytest.raises(TypeError):
            Rotation.from_random()*np.ones(3)