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DAMASK_EICMD/python/tests/test_Rotation.py

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import os
import pytest
import numpy as np
from damask import Rotation
from damask import _rotation
n = 1000
atol=1.e-4
@pytest.fixture
def reference_dir(reference_dir_base):
"""Directory containing reference results."""
return os.path.join(reference_dir_base,'Rotation')
@pytest.fixture
def set_of_rotations(set_of_quaternions):
return [Rotation.from_quaternion(s) for s in set_of_quaternions]
####################################################################################################
# Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations
####################################################################################################
# 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(a):
trace = a[0,0] + a[1,1] + a[2,2]
if trace > 0:
s = 0.5 / np.sqrt(trace+ 1.0)
qu = np.array([0.25 / s,( a[2,1] - a[1,2] ) * s,( a[0,2] - a[2,0] ) * s,( a[1,0] - a[0,1] ) * s])
else:
if ( a[0,0] > a[1,1] and a[0,0] > a[2,2] ):
s = 2.0 * np.sqrt( 1.0 + a[0,0] - a[1,1] - a[2,2])
qu = np.array([ (a[2,1] - a[1,2]) / s,0.25 * s,(a[0,1] + a[1,0]) / s,(a[0,2] + a[2,0]) / s])
elif (a[1,1] > a[2,2]):
s = 2.0 * np.sqrt( 1.0 + a[1,1] - a[0,0] - a[2,2])
qu = np.array([ (a[0,2] - a[2,0]) / s,(a[0,1] + a[1,0]) / s,0.25 * s,(a[1,2] + a[2,1]) / s])
else:
s = 2.0 * np.sqrt( 1.0 + a[2,2] - a[0,0] - a[1,1] )
qu = np.array([ (a[1,0] - a[0,1]) / s,(a[0,2] + a[2,0]) / s,(a[1,2] + a[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]),1.0,1.e-9):
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."""
#return qu2ax(om2qu(om)) # HOTFIX
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([+1.0000000000018852, -0.5000000002194847,
-0.024999992127593126, -0.003928701544781374,
-0.0008152701535450438, -0.0002009500426119712,
-0.00002397986776071756, -0.00008202868926605841,
+0.00012448715042090092, -0.0001749114214822577,
+0.0001703481934140054, -0.00012062065004116828,
+0.000059719705868660826, -0.00001980756723965647,
+0.000003953714684212874, -0.00000036555001439719544])
# 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 = hmag_squared
# convert the magnitude to the rotation angle
s = tfit[0] + tfit[1] * hmag_squared
for i in range(2,16):
hm *= hmag_squared
s += tfit[i] * hm
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.
Todo
----
Document details active/passive)
consider rotation of (3,3,3,3)-matrix
"""
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 ternsors, and 4th order tensors')
else:
raise TypeError('Cannot rotate {}'.format(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 = ok or np.allclose(m*-1.,o,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and np.isclose(np.linalg.norm(o),1.0)
@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)
print(m,o,rot.as_quaternion())
assert ok and np.isclose(np.linalg.det(o),1.0)
@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_Eulers()
o = backward(forward(m))
u = np.array([np.pi*2,np.pi,np.pi*2])
ok = np.allclose(m,o,atol=atol)
ok = ok or 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 = ok or np.isclose(sum_phi[0],sum_phi[1],atol=atol)
print(m,o,rot.as_quaternion())
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()
@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 = ok or np.allclose(m*np.array([-1.,-1.,-1.,1.]),o,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and np.isclose(np.linalg.norm(o[:3]),1.0) and o[3]<=np.pi+1.e-9
@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-4))
for rot in set_of_rotations:
m = rot.as_Rodrigues()
o = backward(forward(m))
ok = np.allclose(np.clip(m,None,cutoff),np.clip(o,None,cutoff),atol=atol)
ok = ok or np.isclose(m[3],0.0,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and np.isclose(np.linalg.norm(o[:3]),1.0)
@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)
print(m,o,rot.as_quaternion())
assert ok and np.linalg.norm(o) < _R1 + 1.e-9
@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 = ok or np.allclose(m*-1.,o,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and np.max(np.abs(o)) < np.pi**(2./3.) * 0.5 + 1.e-9
@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):
print(q,c)
assert np.allclose(single(q),c) and np.allclose(single(q),vectorized(q))
@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):
print(o,c)
assert np.allclose(single(o),c) and np.allclose(single(o),vectorized(o))
@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_Eulers() 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):
print(e,c)
assert np.allclose(single(e),c) and np.allclose(single(e),vectorized(e))
@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):
print(a,c)
assert np.allclose(single(a),c) and np.allclose(single(a),vectorized(a))
@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() 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):
print(r,c)
assert np.allclose(single(r),c) and np.allclose(single(r),vectorized(r))
@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):
print(h,c)
assert np.allclose(single(h),c) and np.allclose(single(h),vectorized(h))
@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):
print(u,c)
assert np.allclose(single(u),c) and np.allclose(single(u),vectorized(u))
@pytest.mark.parametrize('degrees',[True,False])
def test_Eulers(self,set_of_rotations,degrees):
for rot in set_of_rotations:
m = rot.as_quaternion()
o = Rotation.from_Eulers(rot.as_Eulers(degrees),degrees).as_quaternion()
ok = np.allclose(m,o,atol=atol)
if np.isclose(rot.as_quaternion()[0],0.0,atol=atol):
ok = ok or np.allclose(m*-1.,o,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and np.isclose(np.linalg.norm(o),1.0)
@pytest.mark.parametrize('P',[1,-1])
@pytest.mark.parametrize('normalise',[True,False])
@pytest.mark.parametrize('degrees',[True,False])
def test_axis_angle(self,set_of_rotations,degrees,normalise,P):
c = np.array([P*-1,P*-1,P*-1,1.])
for rot in set_of_rotations:
m = rot.as_Eulers()
o = Rotation.from_axis_angle(rot.as_axis_angle(degrees)*c,degrees,normalise,P).as_Eulers()
u = np.array([np.pi*2,np.pi,np.pi*2])
ok = np.allclose(m,o,atol=atol)
ok = ok or 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 = ok or np.isclose(sum_phi[0],sum_phi[1],atol=atol)
print(m,o,rot.as_quaternion())
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()
def test_matrix(self,set_of_rotations):
for rot in set_of_rotations:
m = rot.as_axis_angle()
o = Rotation.from_axis_angle(rot.as_axis_angle()).as_axis_angle()
ok = np.allclose(m,o,atol=atol)
if np.isclose(m[3],np.pi,atol=atol):
ok = ok or np.allclose(m*np.array([-1.,-1.,-1.,1.]),o,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and np.isclose(np.linalg.norm(o[:3]),1.0) and o[3]<=np.pi+1.e-9
@pytest.mark.parametrize('P',[1,-1])
@pytest.mark.parametrize('normalise',[True,False])
def test_Rodrigues(self,set_of_rotations,normalise,P):
c = np.array([P*-1,P*-1,P*-1,1.])
for rot in set_of_rotations:
m = rot.as_matrix()
o = Rotation.from_Rodrigues(rot.as_Rodrigues()*c,normalise,P).as_matrix()
ok = np.allclose(m,o,atol=atol)
print(m,o)
assert ok and np.isclose(np.linalg.det(o),1.0)
@pytest.mark.parametrize('P',[1,-1])
def test_homochoric(self,set_of_rotations,P):
cutoff = np.tan(np.pi*.5*(1.-1e-4))
for rot in set_of_rotations:
m = rot.as_Rodrigues()
o = Rotation.from_homochoric(rot.as_homochoric()*P*-1,P).as_Rodrigues()
ok = np.allclose(np.clip(m,None,cutoff),np.clip(o,None,cutoff),atol=atol)
ok = ok or np.isclose(m[3],0.0,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and np.isclose(np.linalg.norm(o[:3]),1.0)
@pytest.mark.parametrize('P',[1,-1])
def test_cubochoric(self,set_of_rotations,P):
for rot in set_of_rotations:
m = rot.as_homochoric()
o = Rotation.from_cubochoric(rot.as_cubochoric()*P*-1,P).as_homochoric()
ok = np.allclose(m,o,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and np.linalg.norm(o) < (3.*np.pi/4.)**(1./3.) + 1.e-9
@pytest.mark.parametrize('P',[1,-1])
@pytest.mark.parametrize('accept_homomorph',[True,False])
def test_quaternion(self,set_of_rotations,P,accept_homomorph):
c = np.array([1,P*-1,P*-1,P*-1]) * (-1 if accept_homomorph else 1)
for rot in set_of_rotations:
m = rot.as_cubochoric()
o = Rotation.from_quaternion(rot.as_quaternion()*c,accept_homomorph,P).as_cubochoric()
ok = np.allclose(m,o,atol=atol)
if np.count_nonzero(np.isclose(np.abs(o),np.pi**(2./3.)*.5)):
ok = ok or np.allclose(m*-1.,o,atol=atol)
print(m,o,rot.as_quaternion())
assert ok and o.max() < np.pi**(2./3.)*0.5+1.e-9
@pytest.mark.parametrize('reciprocal',[True,False])
def test_basis(self,set_of_rotations,reciprocal):
for rot in set_of_rotations:
om = rot.as_matrix() + 0.1*np.eye(3)
rot = Rotation.from_basis(om,False,reciprocal=reciprocal)
assert np.isclose(np.linalg.det(rot.as_matrix()),1.0)
@pytest.mark.parametrize('function',[Rotation.from_quaternion,
Rotation.from_Eulers,
Rotation.from_axis_angle,
Rotation.from_matrix,
Rotation.from_Rodrigues,
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)
@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_Eulers, np.array([1,4,0])),
(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, np.array([1,0,0,-1])),
(Rotation.from_Rodrigues, 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 np.all(o_i==Rotation._get_pyramid_order(p[i],direction))
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.all(np.take_along_axis(np.take_along_axis(a,f,-1),b,-1) == a)
@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]):
print(i-data[i])
assert np.allclose(mul(rot,data[i]),v[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()
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_Eulers(np.array([phi_1,0.,0.]))
R_2 = Rotation.from_Eulers(np.array([0.,0.,phi_2]))
assert np.allclose(data,R_2*(R_1*data))
@pytest.mark.parametrize('data',[np.random.rand(3),
np.random.rand(3,3),
np.random.rand(3,3,3,3)])
def test_rotate_inverse(self,data):
R = Rotation.from_random()
assert np.allclose(data,R.inversed()*(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
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