import numpy as np
from . import mechanics
_P = -1
# parameters for conversion from/to cubochoric
_sc = np.pi**(1./6.)/6.**(1./6.)
_beta = np.pi**(5./6.)/6.**(1./6.)/2.
_R1 = (3.*np.pi/4.)**(1./3.)
class Rotation:
u"""
Orientation stored with functionality for conversion to different representations.
The following conventions apply:
- coordinate frames are right-handed.
- a rotation angle ω is taken to be positive for a counterclockwise rotation
when viewing from the end point of the rotation axis towards the origin.
- rotations will be interpreted in the passive sense.
- Euler angle triplets are implemented using the Bunge convention,
with the angular ranges as [0, 2π],[0, π],[0, 2π].
- the rotation angle ω is limited to the interval [0, π].
- the real part of a quaternion is positive, Re(q) > 0
- P = -1 (as default).
Examples
--------
Rotate vector "a" (defined in coordinate system "A") to
coordinates "b" expressed in system "B":
- b = Q @ a
- b = np.dot(Q.asMatrix(),a)
References
----------
D. Rowenhorst et al., Modelling and Simulation in Materials Science and Engineering 23:083501, 2015
https://doi.org/10.1088/0965-0393/23/8/083501
"""
__slots__ = ['quaternion']
def __init__(self,quaternion = np.array([1.0,0.0,0.0,0.0])):
"""
Initializes to identity unless specified.
Parameters
----------
quaternion : numpy.ndarray, optional
Unit quaternion that follows the conventions. Use .from_quaternion to perform a sanity check.
"""
self.quaternion = quaternion.copy()
@property
def shape(self):
return self.quaternion.shape[:-1]
def __copy__(self):
"""Copy."""
return self.__class__(self.quaternion)
copy = __copy__
def __repr__(self):
"""Orientation displayed as unit quaternion, rotation matrix, and Bunge-Euler angles."""
if self.quaternion.shape != (4,):
raise NotImplementedError('Support for multiple rotations missing')
return '\n'.join([
'Quaternion: (real={:.3f}, imag=<{:+.3f}, {:+.3f}, {:+.3f}>)'.format(*(self.quaternion)),
'Matrix:\n{}'.format(self.as_matrix()),
'Bunge Eulers / deg: ({:3.2f}, {:3.2f}, {:3.2f})'.format(*self.as_Eulers(degrees=True)),
])
def __matmul__(self, other):
"""
Rotation of vector, second or fourth order tensor, or rotation object.
Parameters
----------
other : numpy.ndarray or Rotation
Vector, second or fourth order tensor, or rotation object that is rotated.
Todo
----
Check rotation of 4th order tensor
"""
if isinstance(other, Rotation):
q_m = self.quaternion[...,0:1]
p_m = self.quaternion[...,1:]
q_o = other.quaternion[...,0:1]
p_o = other.quaternion[...,1:]
q = (q_m*q_o - np.einsum('...i,...i',p_m,p_o).reshape(self.shape+(1,)))
p = q_m*p_o + q_o*p_m + _P * np.cross(p_m,p_o)
return self.__class__(np.block([q,p])).standardize()
elif isinstance(other,np.ndarray):
if self.shape + (3,) == other.shape:
q_m = self.quaternion[...,0]
p_m = self.quaternion[...,1:]
A = q_m**2.0 - np.einsum('...i,...i',p_m,p_m)
B = 2.0 * np.einsum('...i,...i',p_m,other)
C = 2.0 * _P * q_m
return np.block([(A * other[...,i]).reshape(self.shape+(1,)) +
(B * p_m[...,i]).reshape(self.shape+(1,)) +
(C * ( p_m[...,(i+1)%3]*other[...,(i+2)%3]\
- p_m[...,(i+2)%3]*other[...,(i+1)%3])).reshape(self.shape+(1,))
for i in [0,1,2]])
if self.shape + (3,3) == other.shape:
R = self.as_matrix()
return np.einsum('...im,...jn,...mn',R,R,other)
if self.shape + (3,3,3,3) == other.shape:
R = self.as_matrix()
return np.einsum('...im,...jn,...ko,...lp,...mnop',R,R,R,R,other)
else:
raise ValueError('Can only rotate vectors, 2nd order tensors, and 4th order tensors')
else:
raise TypeError('Cannot rotate {}'.format(type(other)))
def inverse(self):
"""In-place inverse rotation/backward rotation."""
self.quaternion[...,1:] *= -1
return self
def inversed(self):
"""Inverse rotation/backward rotation."""
return self.copy().inverse()
def standardize(self):
"""In-place quaternion representation with positive real part."""
self.quaternion[self.quaternion[...,0] < 0.0] *= -1
return self
def standardized(self):
"""Quaternion representation with positive real part."""
return self.copy().standardize()
def misorientation(self,other):
"""
Get Misorientation.
Parameters
----------
other : Rotation
Rotation to which the misorientation is computed.
"""
return other*self.inversed()
def broadcast_to(self,shape):
if isinstance(shape,int): shape = (shape,)
if self.shape == ():
q = np.broadcast_to(self.quaternion,shape+(4,))
else:
q = np.block([np.broadcast_to(self.quaternion[...,0:1],shape).reshape(shape+(1,)),
np.broadcast_to(self.quaternion[...,1:2],shape).reshape(shape+(1,)),
np.broadcast_to(self.quaternion[...,2:3],shape).reshape(shape+(1,)),
np.broadcast_to(self.quaternion[...,3:4],shape).reshape(shape+(1,))])
return self.__class__(q)
def average(self,other):
"""
Calculate the average rotation.
Parameters
----------
other : Rotation
Rotation from which the average is rotated.
"""
if self.quaternion.shape != (4,) or other.quaternion.shape != (4,):
raise NotImplementedError('Support for multiple rotations missing')
return Rotation.fromAverage([self,other])
################################################################################################
# convert to different orientation representations (numpy arrays)
def as_quaternion(self):
"""
Unit quaternion [q, p_1, p_2, p_3].
Parameters
----------
quaternion : bool, optional
return quaternion as DAMASK object.
"""
return self.quaternion
def as_Eulers(self,
degrees = False):
"""
Bunge-Euler angles: (φ_1, ϕ, φ_2).
Parameters
----------
degrees : bool, optional
return angles in degrees.
"""
eu = Rotation._qu2eu(self.quaternion)
if degrees: eu = np.degrees(eu)
return eu
def as_axis_angle(self,
degrees = False,
pair = False):
"""
Axis angle representation [n_1, n_2, n_3, ω] unless pair == True: ([n_1, n_2, n_3], ω).
Parameters
----------
degrees : bool, optional
return rotation angle in degrees.
pair : bool, optional
return tuple of axis and angle.
"""
ax = Rotation._qu2ax(self.quaternion)
if degrees: ax[...,3] = np.degrees(ax[...,3])
return (ax[...,:3],ax[...,3]) if pair else ax
def as_matrix(self):
"""Rotation matrix."""
return Rotation._qu2om(self.quaternion)
def as_Rodrigues(self,
vector = False):
"""
Rodrigues-Frank vector representation [n_1, n_2, n_3, tan(ω/2)] unless vector == True: [n_1, n_2, n_3] * tan(ω/2).
Parameters
----------
vector : bool, optional
return as actual Rodrigues--Frank vector, i.e. rotation axis scaled by tan(ω/2).
"""
ro = Rotation._qu2ro(self.quaternion)
return ro[...,:3]*ro[...,3] if vector else ro
def as_homochoric(self):
"""Homochoric vector: (h_1, h_2, h_3)."""
return Rotation._qu2ho(self.quaternion)
def as_cubochoric(self):
"""Cubochoric vector: (c_1, c_2, c_3)."""
return Rotation._qu2cu(self.quaternion)
def M(self): # ToDo not sure about the name: as_M or M? we do not have a from_M
"""
Intermediate representation supporting quaternion averaging.
References
----------
F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
https://doi.org/10.2514/1.28949
"""
return np.einsum('...i,...j',self.quaternion,self.quaternion)
################################################################################################
# Static constructors. The input data needs to follow the conventions, options allow to
# relax the conventions.
@staticmethod
def from_quaternion(quaternion,
accept_homomorph = False,
P = -1,
acceptHomomorph = None):
if acceptHomomorph is not None:
accept_homomorph = acceptHomomorph
qu = np.array(quaternion,dtype=float)
if qu.shape[:-2:-1] != (4,):
raise ValueError('Invalid shape.')
if P > 0: qu[...,1:4] *= -1 # convert from P=1 to P=-1
if accept_homomorph:
qu[qu[...,0] < 0.0] *= -1
else:
if np.any(qu[...,0] < 0.0):
raise ValueError('Quaternion with negative first (real) component.')
if not np.all(np.isclose(np.linalg.norm(qu,axis=-1), 1.0)):
raise ValueError('Quaternion is not of unit length.')
return Rotation(qu)
@staticmethod
def from_Eulers(eulers,
degrees = False):
eu = np.array(eulers,dtype=float)
if eu.shape[:-2:-1] != (3,):
raise ValueError('Invalid shape.')
eu = np.radians(eu) if degrees else eu
if np.any(eu < 0.0) or np.any(eu > 2.0*np.pi) or np.any(eu[...,1] > np.pi): # ToDo: No separate check for PHI
raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].')
return Rotation(Rotation._eu2qu(eu))
@staticmethod
def from_axis_angle(axis_angle,
degrees = False,
normalise = False,
P = -1):
ax = np.array(axis_angle,dtype=float)
if ax.shape[:-2:-1] != (4,):
raise ValueError('Invalid shape.')
if P > 0: ax[...,0:3] *= -1 # convert from P=1 to P=-1
if degrees: ax[..., 3] = np.radians(ax[...,3])
if normalise: ax[...,0:3] /= np.linalg.norm(ax[...,0:3],axis=-1)
if np.any(ax[...,3] < 0.0) or np.any(ax[...,3] > np.pi):
raise ValueError('Axis angle rotation angle outside of [0..π].')
if not np.all(np.isclose(np.linalg.norm(ax[...,0:3],axis=-1), 1.0)):
raise ValueError('Axis angle rotation axis is not of unit length.')
return Rotation(Rotation._ax2qu(ax))
@staticmethod
def from_basis(basis,
orthonormal = True,
reciprocal = False):
om = np.array(basis,dtype=float)
if om.shape[:-3:-1] != (3,3):
raise ValueError('Invalid shape.')
if reciprocal:
om = np.linalg.inv(mechanics.transpose(om)/np.pi) # transform reciprocal basis set
orthonormal = False # contains stretch
if not orthonormal:
(U,S,Vh) = np.linalg.svd(om) # singular value decomposition
om = np.einsum('...ij,...jl->...il',U,Vh)
if not np.all(np.isclose(np.linalg.det(om),1.0)):
raise ValueError('Orientation matrix has determinant ≠ 1.')
if not np.all(np.isclose(np.einsum('...i,...i',om[...,0],om[...,1]), 0.0)) \
or not np.all(np.isclose(np.einsum('...i,...i',om[...,1],om[...,2]), 0.0)) \
or not np.all(np.isclose(np.einsum('...i,...i',om[...,2],om[...,0]), 0.0)):
raise ValueError('Orientation matrix is not orthogonal.')
return Rotation(Rotation._om2qu(om))
@staticmethod
def from_matrix(om):
return Rotation.from_basis(om)
@staticmethod
def from_Rodrigues(rodrigues,
normalise = False,
P = -1):
ro = np.array(rodrigues,dtype=float)
if ro.shape[:-2:-1] != (4,):
raise ValueError('Invalid shape.')
if P > 0: ro[...,0:3] *= -1 # convert from P=1 to P=-1
if normalise: ro[...,0:3] /= np.linalg.norm(ro[...,0:3],axis=-1)
if np.any(ro[...,3] < 0.0):
raise ValueError('Rodrigues vector rotation angle not positive.')
if not np.all(np.isclose(np.linalg.norm(ro[...,0:3],axis=-1), 1.0)):
raise ValueError('Rodrigues vector rotation axis is not of unit length.')
return Rotation(Rotation._ro2qu(ro))
@staticmethod
def from_homochoric(homochoric,
P = -1):
ho = np.array(homochoric,dtype=float)
if ho.shape[:-2:-1] != (3,):
raise ValueError('Invalid shape.')
if P > 0: ho *= -1 # convert from P=1 to P=-1
if np.any(np.linalg.norm(ho,axis=-1) >_R1+1e-9):
raise ValueError('Homochoric coordinate outside of the sphere.')
return Rotation(Rotation._ho2qu(ho))
@staticmethod
def from_cubochoric(cubochoric,
P = -1):
cu = np.array(cubochoric,dtype=float)
if cu.shape[:-2:-1] != (3,):
raise ValueError('Invalid shape.')
if np.abs(np.max(cu))>np.pi**(2./3.) * 0.5+1e-9:
raise ValueError('Cubochoric coordinate outside of the cube: {} {} {}.'.format(*cu))
ho = Rotation._cu2ho(cu)
if P > 0: ho *= -1 # convert from P=1 to P=-1
return Rotation(Rotation._ho2qu(ho))
@staticmethod
def fromAverage(rotations,weights = None):
"""
Average rotation.
References
----------
F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
https://doi.org/10.2514/1.28949
Parameters
----------
rotations : list of Rotations
Rotations to average from
weights : list of floats, optional
Weights for each rotation used for averaging
"""
if not all(isinstance(item, Rotation) for item in rotations):
raise TypeError('Only instances of Rotation can be averaged.')
N = len(rotations)
if not weights:
weights = np.ones(N,dtype='i')
for i,(r,n) in enumerate(zip(rotations,weights)):
M = r.M() * n if i == 0 \
else M + r.M() * n # noqa add (multiples) of this rotation to average noqa
eig, vec = np.linalg.eig(M/N)
return Rotation.from_quaternion(np.real(vec.T[eig.argmax()]),accept_homomorph = True)
@staticmethod
def from_random(shape=None):
if shape is None:
r = np.random.random(3)
elif hasattr(shape, '__iter__'):
r = np.random.random(tuple(shape)+(3,))
else:
r = np.random.random((shape,3))
A = np.sqrt(r[...,2])
B = np.sqrt(1.0-r[...,2])
q = np.stack([np.cos(2.0*np.pi*r[...,0])*A,
np.sin(2.0*np.pi*r[...,1])*B,
np.cos(2.0*np.pi*r[...,1])*B,
np.sin(2.0*np.pi*r[...,0])*A],axis=-1)
return Rotation(q.reshape(r.shape[:-1]+(4,)) if shape is not None else q).standardize()
# for compatibility (old names do not follow convention)
asM = M
fromQuaternion = from_quaternion
fromEulers = from_Eulers
asAxisAngle = as_axis_angle
__mul__ = __matmul__
####################################################################################################
# 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.
####################################################################################################
#---------- Quaternion ----------
@staticmethod
def _qu2om(qu):
qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2)
om = np.block([qq + 2.0*qu[...,1:2]**2,
2.0*(qu[...,2:3]*qu[...,1:2]-_P*qu[...,0:1]*qu[...,3:4]),
2.0*(qu[...,3:4]*qu[...,1:2]+_P*qu[...,0:1]*qu[...,2:3]),
2.0*(qu[...,1:2]*qu[...,2:3]+_P*qu[...,0:1]*qu[...,3:4]),
qq + 2.0*qu[...,2:3]**2,
2.0*(qu[...,3:4]*qu[...,2:3]-_P*qu[...,0:1]*qu[...,1:2]),
2.0*(qu[...,1:2]*qu[...,3:4]-_P*qu[...,0:1]*qu[...,2:3]),
2.0*(qu[...,2:3]*qu[...,3:4]+_P*qu[...,0:1]*qu[...,1:2]),
qq + 2.0*qu[...,3:4]**2,
]).reshape(qu.shape[:-1]+(3,3))
return om
@staticmethod
def _qu2eu(qu):
"""Quaternion to Bunge-Euler angles."""
q02 = qu[...,0:1]*qu[...,2:3]
q13 = qu[...,1:2]*qu[...,3:4]
q01 = qu[...,0:1]*qu[...,1:2]
q23 = qu[...,2:3]*qu[...,3:4]
q03_s = qu[...,0:1]**2+qu[...,3:4]**2
q12_s = qu[...,1:2]**2+qu[...,2:3]**2
chi = np.sqrt(q03_s*q12_s)
eu = np.where(np.abs(q12_s) < 1.0e-8,
np.block([np.arctan2(-_P*2.0*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2),
np.zeros(qu.shape[:-1]+(2,))]),
np.where(np.abs(q03_s) < 1.0e-8,
np.block([np.arctan2( 2.0*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2),
np.broadcast_to(np.pi,qu.shape[:-1]+(1,)),
np.zeros(qu.shape[:-1]+(1,))]),
np.block([np.arctan2((-_P*q02+q13)*chi, (-_P*q01-q23)*chi),
np.arctan2( 2.0*chi, q03_s-q12_s ),
np.arctan2(( _P*q02+q13)*chi, (-_P*q01+q23)*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) # needed?
return eu
@staticmethod
def _qu2ax(qu):
"""
Quaternion to axis angle pair.
Modified version of the original formulation, should be numerically more stable
"""
with np.errstate(invalid='ignore',divide='ignore'):
s = np.sign(qu[...,0:1])/np.sqrt(qu[...,1:2]**2+qu[...,2:3]**2+qu[...,3:4]**2)
omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
ax = np.where(np.broadcast_to(qu[...,0:1] < 1.0e-8,qu.shape),
np.block([qu[...,1:4],np.broadcast_to(np.pi,qu.shape[:-1]+(1,))]),
np.block([qu[...,1:4]*s,omega]))
ax[np.isclose(qu[...,0],1.,rtol=0.0)] = [0.0, 0.0, 1.0, 0.0]
return ax
@staticmethod
def _qu2ro(qu):
"""Quaternion to Rodrigues-Frank vector."""
with np.errstate(invalid='ignore',divide='ignore'):
s = np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True)
ro = np.where(np.broadcast_to(np.abs(qu[...,0:1]) < 1.0e-12,qu.shape),
np.block([qu[...,1:2], qu[...,2:3], qu[...,3:4], np.broadcast_to(np.inf,qu.shape[:-1]+(1,))]),
np.block([qu[...,1:2]/s,qu[...,2:3]/s,qu[...,3:4]/s,
np.tan(np.arccos(np.clip(qu[...,0:1],-1.0,1.0)))
])
)
ro[np.abs(s).squeeze(-1) < 1.0e-12] = [0.0,0.0,_P,0.0]
return ro
@staticmethod
def _qu2ho(qu):
"""Quaternion to homochoric vector."""
with np.errstate(invalid='ignore'):
omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
ho = np.where(np.abs(omega) < 1.0e-12,
np.zeros(3),
qu[...,1:4]/np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) \
* (0.75*(omega - np.sin(omega)))**(1./3.))
return ho
@staticmethod
def _qu2cu(qu):
"""Quaternion to cubochoric vector."""
return Rotation._ho2cu(Rotation._qu2ho(qu))
#---------- Rotation matrix ----------
@staticmethod
def _om2qu(om):
"""
Rotation matrix to quaternion.
This formulation is from www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion.
The original formulation had issues.
"""
trace = om[...,0,0:1]+om[...,1,1:2]+om[...,2,2:3]
with np.errstate(invalid='ignore',divide='ignore'):
s = [
0.5 / np.sqrt( 1.0 + trace),
2.0 * np.sqrt( 1.0 + om[...,0,0:1] - om[...,1,1:2] - om[...,2,2:3]),
2.0 * np.sqrt( 1.0 + om[...,1,1:2] - om[...,2,2:3] - om[...,0,0:1]),
2.0 * np.sqrt( 1.0 + om[...,2,2:3] - om[...,0,0:1] - om[...,1,1:2] )
]
qu= np.where(trace>0,
np.block([0.25 / s[0],
(om[...,2,1:2] - om[...,1,2:3] ) * s[0],
(om[...,0,2:3] - om[...,2,0:1] ) * s[0],
(om[...,1,0:1] - om[...,0,1:2] ) * s[0]]),
np.where(om[...,0,0:1] > np.maximum(om[...,1,1:2],om[...,2,2:3]),
np.block([(om[...,2,1:2] - om[...,1,2:3]) / s[1],
0.25 * s[1],
(om[...,0,1:2] + om[...,1,0:1]) / s[1],
(om[...,0,2:3] + om[...,2,0:1]) / s[1]]),
np.where(om[...,1,1:2] > om[...,2,2:3],
np.block([(om[...,0,2:3] - om[...,2,0:1]) / s[2],
(om[...,0,1:2] + om[...,1,0:1]) / s[2],
0.25 * s[2],
(om[...,1,2:3] + om[...,2,1:2]) / s[2]]),
np.block([(om[...,1,0:1] - om[...,0,1:2]) / s[3],
(om[...,0,2:3] + om[...,2,0:1]) / s[3],
(om[...,1,2:3] + om[...,2,1:2]) / s[3],
0.25 * s[3]]),
)
)
)*np.array([1,_P,_P,_P])
qu[qu[...,0]<0] *=-1
return qu
@staticmethod
def _om2eu(om):
"""Rotation matrix to Bunge-Euler angles."""
with np.errstate(invalid='ignore',divide='ignore'):
zeta = 1.0/np.sqrt(1.0-om[...,2,2:3]**2)
eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.0,1e-9),
np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]),
np.pi*0.5*(1-om[...,2,2:3]),
np.zeros(om.shape[:-2]+(1,)),
]),
np.block([np.arctan2(om[...,2,0:1]*zeta,-om[...,2,1:2]*zeta),
np.arccos( om[...,2,2:3]),
np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta)
])
)
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
@staticmethod
def _om2ax(om):
"""Rotation matrix to axis angle pair."""
#return Rotation._qu2ax(Rotation._om2qu(om)) # HOTFIX
diag_delta = -_P*np.block([om[...,1,2:3]-om[...,2,1:2],
om[...,2,0:1]-om[...,0,2:3],
om[...,0,1:2]-om[...,1,0:1]
])
t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.0).reshape(om.shape[:-2]+(1,))
w,vr = np.linalg.eig(om)
# mask duplicated real eigenvalues
w[np.isclose(w[...,0],1.0+0.0j),1:] = 0.
w[np.isclose(w[...,1],1.0+0.0j),2:] = 0.
vr = np.swapaxes(vr,-1,-2)
ax = np.where(np.abs(diag_delta)<1e-12,
np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,)),
np.abs(np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,))) \
*np.sign(diag_delta))
ax = np.block([ax,np.arccos(np.clip(t,-1.0,1.0))])
ax[np.abs(ax[...,3])<1.e-8] = [ 0.0, 0.0, 1.0, 0.0]
return ax
@staticmethod
def _om2ro(om):
"""Rotation matrix to Rodrigues-Frank vector."""
return Rotation._eu2ro(Rotation._om2eu(om))
@staticmethod
def _om2ho(om):
"""Rotation matrix to homochoric vector."""
return Rotation._ax2ho(Rotation._om2ax(om))
@staticmethod
def _om2cu(om):
"""Rotation matrix to cubochoric vector."""
return Rotation._ho2cu(Rotation._om2ho(om))
#---------- Bunge-Euler angles ----------
@staticmethod
def _eu2qu(eu):
"""Bunge-Euler angles to quaternion."""
ee = 0.5*eu
cPhi = np.cos(ee[...,1:2])
sPhi = np.sin(ee[...,1:2])
qu = np.block([ cPhi*np.cos(ee[...,0:1]+ee[...,2:3]),
-_P*sPhi*np.cos(ee[...,0:1]-ee[...,2:3]),
-_P*sPhi*np.sin(ee[...,0:1]-ee[...,2:3]),
-_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])])
qu[qu[...,0]<0.0]*=-1
return qu
@staticmethod
def _eu2om(eu):
"""Bunge-Euler angles to rotation matrix."""
c = np.cos(eu)
s = np.sin(eu)
om = np.block([+c[...,0:1]*c[...,2:3]-s[...,0:1]*s[...,2:3]*c[...,1:2],
+s[...,0:1]*c[...,2:3]+c[...,0:1]*s[...,2:3]*c[...,1:2],
+s[...,2:3]*s[...,1:2],
-c[...,0:1]*s[...,2:3]-s[...,0:1]*c[...,2:3]*c[...,1:2],
-s[...,0:1]*s[...,2:3]+c[...,0:1]*c[...,2:3]*c[...,1:2],
+c[...,2:3]*s[...,1:2],
+s[...,0:1]*s[...,1:2],
-c[...,0:1]*s[...,1:2],
+c[...,1:2]
]).reshape(eu.shape[:-1]+(3,3))
om[np.abs(om)<1.e-12] = 0.0
return om
@staticmethod
def _eu2ax(eu):
"""Bunge-Euler angles to axis angle pair."""
t = np.tan(eu[...,1:2]*0.5)
sigma = 0.5*(eu[...,0:1]+eu[...,2:3])
delta = 0.5*(eu[...,0:1]-eu[...,2:3])
tau = np.linalg.norm(np.block([t,np.sin(sigma)]),axis=-1,keepdims=True)
alpha = np.where(np.abs(np.cos(sigma))<1.e-12,np.pi,2.0*np.arctan(tau/np.cos(sigma)))
with np.errstate(invalid='ignore',divide='ignore'):
ax = np.where(np.broadcast_to(np.abs(alpha)<1.0e-12,eu.shape[:-1]+(4,)),
[0.0,0.0,1.0,0.0],
np.block([-_P/tau*t*np.cos(delta),
-_P/tau*t*np.sin(delta),
-_P/tau* np.sin(sigma),
alpha
]))
ax[(alpha<0.0).squeeze()] *=-1
return ax
@staticmethod
def _eu2ro(eu):
"""Bunge-Euler angles to Rodrigues-Frank vector."""
ax = Rotation._eu2ax(eu)
ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)])
ro[ax[...,3]>=np.pi,3] = np.inf
ro[np.abs(ax[...,3])<1.e-16] = [ 0.0, 0.0, _P, 0.0 ]
return ro
@staticmethod
def _eu2ho(eu):
"""Bunge-Euler angles to homochoric vector."""
return Rotation._ax2ho(Rotation._eu2ax(eu))
@staticmethod
def _eu2cu(eu):
"""Bunge-Euler angles to cubochoric vector."""
return Rotation._ho2cu(Rotation._eu2ho(eu))
#---------- Axis angle pair ----------
@staticmethod
def _ax2qu(ax):
"""Axis angle pair to quaternion."""
c = np.cos(ax[...,3:4]*.5)
s = np.sin(ax[...,3:4]*.5)
qu = np.where(np.abs(ax[...,3:4])<1.e-6,[1.0, 0.0, 0.0, 0.0],np.block([c, ax[...,:3]*s]))
return qu
@staticmethod
def _ax2om(ax):
"""Axis angle pair to rotation matrix."""
c = np.cos(ax[...,3:4])
s = np.sin(ax[...,3:4])
omc = 1. -c
om = np.block([c+omc*ax[...,0:1]**2,
omc*ax[...,0:1]*ax[...,1:2] + s*ax[...,2:3],
omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2],
omc*ax[...,0:1]*ax[...,1:2] - s*ax[...,2:3],
c+omc*ax[...,1:2]**2,
omc*ax[...,1:2]*ax[...,2:3] + s*ax[...,0:1],
omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2],
omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1],
c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3))
return om if _P < 0.0 else np.swapaxes(om,-1,-2)
@staticmethod
def _ax2eu(ax):
"""Rotation matrix to Bunge Euler angles."""
return Rotation._om2eu(Rotation._ax2om(ax))
@staticmethod
def _ax2ro(ax):
"""Axis angle pair to Rodrigues-Frank vector."""
ro = np.block([ax[...,:3],
np.where(np.isclose(ax[...,3:4],np.pi,atol=1.e-15,rtol=.0),
np.inf,
np.tan(ax[...,3:4]*0.5))
])
ro[np.abs(ax[...,3])<1.e-6] = [.0,.0,_P,.0]
return ro
@staticmethod
def _ax2ho(ax):
"""Axis angle pair to homochoric vector."""
f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1.0/3.0)
ho = ax[...,:3] * f
return ho
@staticmethod
def _ax2cu(ax):
"""Axis angle pair to cubochoric vector."""
return Rotation._ho2cu(Rotation._ax2ho(ax))
#---------- Rodrigues-Frank vector ----------
@staticmethod
def _ro2qu(ro):
"""Rodrigues-Frank vector to quaternion."""
return Rotation._ax2qu(Rotation._ro2ax(ro))
@staticmethod
def _ro2om(ro):
"""Rodgrigues-Frank vector to rotation matrix."""
return Rotation._ax2om(Rotation._ro2ax(ro))
@staticmethod
def _ro2eu(ro):
"""Rodrigues-Frank vector to Bunge-Euler angles."""
return Rotation._om2eu(Rotation._ro2om(ro))
@staticmethod
def _ro2ax(ro):
"""Rodrigues-Frank vector to axis angle pair."""
with np.errstate(invalid='ignore',divide='ignore'):
ax = np.where(np.isfinite(ro[...,3:4]),
np.block([ro[...,0:3]*np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True),2.*np.arctan(ro[...,3:4])]),
np.block([ro[...,0:3],np.broadcast_to(np.pi,ro[...,3:4].shape)]))
ax[np.abs(ro[...,3]) < 1.e-8] = np.array([ 0.0, 0.0, 1.0, 0.0 ])
return ax
@staticmethod
def _ro2ho(ro):
"""Rodrigues-Frank vector to homochoric vector."""
f = np.where(np.isfinite(ro[...,3:4]),2.0*np.arctan(ro[...,3:4]) -np.sin(2.0*np.arctan(ro[...,3:4])),np.pi)
ho = np.where(np.broadcast_to(np.sum(ro[...,0:3]**2.0,axis=-1,keepdims=True) < 1.e-8,ro[...,0:3].shape),
np.zeros(3), ro[...,0:3]* (0.75*f)**(1.0/3.0))
return ho
@staticmethod
def _ro2cu(ro):
"""Rodrigues-Frank vector to cubochoric vector."""
return Rotation._ho2cu(Rotation._ro2ho(ro))
#---------- Homochoric vector----------
@staticmethod
def _ho2qu(ho):
"""Homochoric vector to quaternion."""
return Rotation._ax2qu(Rotation._ho2ax(ho))
@staticmethod
def _ho2om(ho):
"""Homochoric vector to rotation matrix."""
return Rotation._ax2om(Rotation._ho2ax(ho))
@staticmethod
def _ho2eu(ho):
"""Homochoric vector to Bunge-Euler angles."""
return Rotation._ax2eu(Rotation._ho2ax(ho))
@staticmethod
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])
hmag_squared = np.sum(ho**2.,axis=-1,keepdims=True)
hm = hmag_squared.copy()
s = tfit[0] + tfit[1] * hmag_squared
for i in range(2,16):
hm *= hmag_squared
s += tfit[i] * hm
with np.errstate(invalid='ignore'):
ax = np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-8,ho.shape[:-1]+(4,)),
[ 0.0, 0.0, 1.0, 0.0 ],
np.block([ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))]))
return ax
@staticmethod
def _ho2ro(ho):
"""Axis angle pair to Rodrigues-Frank vector."""
return Rotation._ax2ro(Rotation._ho2ax(ho))
@staticmethod
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,axis=-1,keepdims=True)
xyz3 = np.take_along_axis(ho,Rotation._get_pyramid_order(ho,'forward'),-1)
with np.errstate(invalid='ignore',divide='ignore'):
# inverse M_3
xyz2 = xyz3[...,0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[...,2:3])) )
qxy = np.sum(xyz2**2,axis=-1,keepdims=True)
q2 = qxy + np.max(np.abs(xyz2),axis=-1,keepdims=True)**2
sq2 = np.sqrt(q2)
q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2))
tt = np.clip((np.min(np.abs(xyz2),axis=-1,keepdims=True)**2\
+np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
T_inv = np.where(np.abs(xyz2[...,1:2]) <= np.abs(xyz2[...,0:1]),
np.block([np.ones_like(tt),np.arccos(tt)/np.pi*12.0]),
np.block([np.arccos(tt)/np.pi*12.0,np.ones_like(tt)]))*q
T_inv[xyz2<0.0] *= -1.0
T_inv[np.broadcast_to(np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-12),T_inv.shape)] = 0.0
cu = np.block([T_inv, np.where(xyz3[...,2:3]<0.0,-np.ones_like(xyz3[...,2:3]),np.ones_like(xyz3[...,2:3])) \
* rs/np.sqrt(6.0/np.pi),
])/ _sc
cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
cu = np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1)
return cu
#---------- Cubochoric ----------
@staticmethod
def _cu2qu(cu):
"""Cubochoric vector to quaternion."""
return Rotation._ho2qu(Rotation._cu2ho(cu))
@staticmethod
def _cu2om(cu):
"""Cubochoric vector to rotation matrix."""
return Rotation._ho2om(Rotation._cu2ho(cu))
@staticmethod
def _cu2eu(cu):
"""Cubochoric vector to Bunge-Euler angles."""
return Rotation._ho2eu(Rotation._cu2ho(cu))
@staticmethod
def _cu2ax(cu):
"""Cubochoric vector to axis angle pair."""
return Rotation._ho2ax(Rotation._cu2ho(cu))
@staticmethod
def _cu2ro(cu):
"""Cubochoric vector to Rodrigues-Frank vector."""
return Rotation._ho2ro(Rotation._cu2ho(cu))
@staticmethod
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
"""
with np.errstate(invalid='ignore',divide='ignore'):
# get pyramide and scale by grid parameter ratio
XYZ = np.take_along_axis(cu,Rotation._get_pyramid_order(cu,'forward'),-1) * _sc
order = np.abs(XYZ[...,1:2]) <= np.abs(XYZ[...,0:1])
q = np.pi/12.0 * np.where(order,XYZ[...,1:2],XYZ[...,0:1]) \
/ np.where(order,XYZ[...,0:1],XYZ[...,1:2])
c = np.cos(q)
s = np.sin(q)
q = _R1*2.0**0.25/_beta/ np.sqrt(np.sqrt(2.0)-c) \
* np.where(order,XYZ[...,0:1],XYZ[...,1:2])
T = np.block([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
# transform to sphere grid (inverse Lambert)
c = np.sum(T**2,axis=-1,keepdims=True)
s = c * np.pi/24.0 /XYZ[...,2:3]**2
c = c * np.sqrt(np.pi/24.0)/XYZ[...,2:3]
q = np.sqrt( 1.0 - s)
ho = np.where(np.isclose(np.sum(np.abs(XYZ[...,0:2]),axis=-1,keepdims=True),0.0,rtol=0.0,atol=1.0e-16),
np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6.0/np.pi) *XYZ[...,2:3]]),
np.block([np.where(order,T[...,0:1],T[...,1:2])*q,
np.where(order,T[...,1:2],T[...,0:1])*q,
np.sqrt(6.0/np.pi) * XYZ[...,2:3] - c])
)
ho[np.isclose(np.sum(np.abs(cu),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
ho = np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1)
return ho
@staticmethod
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]])}
p = np.where(np.maximum(np.abs(xyz[...,0]),np.abs(xyz[...,1])) <= np.abs(xyz[...,2]),0,
np.where(np.maximum(np.abs(xyz[...,1]),np.abs(xyz[...,2])) <= np.abs(xyz[...,0]),1,2))
return order[direction][p]