DAMASK_EICMD/python/damask/_rotation.py

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import numpy as np
from . import tensor
from . import util
from . import grid_filters
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_P = -1
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# 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.)
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class Rotation:
u"""
Rotation with functionality for conversion between different representations.
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The following conventions apply:
- Coordinate frames are right-handed.
- A rotation angle ω is taken to be positive for a counterclockwise rotation
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when viewing from the end point of the rotation axis towards the origin.
- Rotations will be interpreted in the passive sense.
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- Euler angle triplets are implemented using the Bunge convention,
with angular ranges of [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
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- 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.as_matrix(),a)
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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,rotation = np.array([1.0,0.0,0.0,0.0])):
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"""
Initialize rotation object.
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Parameters
----------
rotation : list, numpy.ndarray, Rotation, optional
Unit quaternion in positive real hemisphere.
Use .from_quaternion to perform a sanity check.
Defaults to no rotation.
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"""
if isinstance(rotation,Rotation):
self.quaternion = rotation.quaternion.copy()
elif np.array(rotation).shape[-1] == 4:
self.quaternion = np.array(rotation)
else:
raise TypeError('"rotation" is neither a Rotation nor a quaternion')
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def __repr__(self):
"""Represent rotation as unit quaternion, rotation matrix, and Bunge-Euler angles."""
if self == Rotation():
return 'Rotation()'
else:
return f'Quaternions {self.shape}:\n'+str(self.quaternion) \
if self.quaternion.shape != (4,) else \
'\n'.join([
'Quaternion: (real={:.3f}, imag=<{:+.3f}, {:+.3f}, {:+.3f}>)'.format(*(self.quaternion)),
'Matrix:\n{}'.format(np.round(self.as_matrix(),8)),
'Bunge Eulers / deg: ({:3.2f}, {:3.2f}, {:3.2f})'.format(*self.as_Euler_angles(degrees=True)),
])
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# ToDo: Check difference __copy__ vs __deepcopy__
def __copy__(self,**kwargs):
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"""Copy."""
return self.__class__(rotation=kwargs['rotation'] if 'rotation' in kwargs else self.quaternion)
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copy = __copy__
def __getitem__(self,item):
"""Return slice according to item."""
return self.copy() \
if self.shape == () else \
self.copy(rotation=self.quaternion[item+(slice(None),)] if isinstance(item,tuple) else self.quaternion[item])
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def __eq__(self,other):
"""
Equal to other.
Equality is determined taking limited floating point precision into
account. See numpy.allclose for details.
Parameters
----------
other : Rotation
Rotation to check for equality.
"""
return np.prod(self.shape,dtype=int) == np.prod(other.shape,dtype=int) \
and np.allclose(self.quaternion,other.quaternion)
@property
def shape(self):
return self.quaternion.shape[:-1]
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def __len__(self):
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"""Length of leading/leftmost dimension of Rotation array."""
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return 0 if self.shape == () else self.shape[0]
def __invert__(self):
"""Inverse rotation (backward rotation)."""
dup = self.copy()
dup.quaternion[...,1:] *= -1
return dup
def __pow__(self,pwr):
"""
Raise quaternion to power.
Equivalent to performing the rotation 'pwr' times.
Parameters
----------
pwr : float
Power to raise quaternion to.
"""
phi = np.arccos(self.quaternion[...,0:1])
p = self.quaternion[...,1:]/np.linalg.norm(self.quaternion[...,1:],axis=-1,keepdims=True)
return self.copy(rotation=Rotation(np.block([np.cos(pwr*phi),np.sin(pwr*phi)*p]))._standardize())
def __mul__(self,other):
"""Standard multiplication is not implemented."""
raise NotImplementedError('Use "R@b", i.e. matmul, to apply rotation "R" to object "b"')
def __matmul__(self,other):
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"""
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Rotation of vector, second or fourth order tensor, or rotation object.
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Parameters
----------
other : numpy.ndarray or Rotation
Vector, second or fourth order tensor, or rotation object that is rotated.
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Returns
-------
other_rot : numpy.ndarray or Rotation
Rotated vector, second or fourth order tensor, or rotation object.
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"""
if isinstance(other,Rotation):
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q_m = self.quaternion[...,0:1]
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p_m = self.quaternion[...,1:]
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q_o = other.quaternion[...,0:1]
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p_o = other.quaternion[...,1:]
q = (q_m*q_o - np.einsum('...i,...i',p_m,p_o).reshape(self.shape+(1,)))
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p = q_m*p_o + q_o*p_m + _P * np.cross(p_m,p_o)
return Rotation(np.block([q,p]))._standardize()
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elif isinstance(other,np.ndarray):
if self.shape + (3,) == other.shape:
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q_m = self.quaternion[...,0]
p_m = self.quaternion[...,1:]
A = q_m**2.0 - np.einsum('...i,...i',p_m,p_m)
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B = 2.0 * np.einsum('...i,...i',p_m,other)
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C = 2.0 * _P * q_m
return np.block([(A * other[...,i]).reshape(self.shape+(1,)) +
(B * p_m[...,i]).reshape(self.shape+(1,)) +
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(C * ( p_m[...,(i+1)%3]*other[...,(i+2)%3]\
- p_m[...,(i+2)%3]*other[...,(i+1)%3])).reshape(self.shape+(1,))
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for i in [0,1,2]])
if self.shape + (3,3) == other.shape:
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R = self.as_matrix()
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return np.einsum('...im,...jn,...mn',R,R,other)
if self.shape + (3,3,3,3) == other.shape:
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R = self.as_matrix()
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return np.einsum('...im,...jn,...ko,...lp,...mnop',R,R,R,R,other)
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else:
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raise ValueError('Can only rotate vectors, 2nd order tensors, and 4th order tensors')
else:
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raise TypeError(f'Cannot rotate {type(other)}')
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def _standardize(self):
"""Standardize quaternion (ensure positive real hemisphere)."""
self.quaternion[self.quaternion[...,0] < 0.0] *= -1
return self
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def append(self,other):
"""Extend rotation array along first dimension with other array."""
return self.copy(rotation=np.vstack((self.quaternion,other.quaternion)))
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def flatten(self,order = 'C'):
"""Flatten quaternion array."""
return self.copy(rotation=self.quaternion.reshape((-1,4),order=order))
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def reshape(self,shape,order = 'C'):
"""Reshape quaternion array."""
if isinstance(shape,(int,np.integer)): shape = (shape,)
return self.copy(rotation=self.quaternion.reshape(tuple(shape)+(4,),order=order))
def broadcast_to(self,shape,mode = 'right'):
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"""
Broadcast quaternion array to shape.
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Parameters
----------
shape : tuple
Shape of broadcasted array.
mode : str, optional
Where to preferentially locate missing dimensions.
Either 'left' or 'right' (default).
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"""
if isinstance(shape,(int,np.integer)): shape = (shape,)
return self.copy(rotation=np.broadcast_to(self.quaternion.reshape(util.shapeshifter(self.shape,shape,mode)+(4,)),
shape+(4,)))
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def average(self,weights = None):
"""
Average rotations along last dimension.
Parameters
----------
weights : list of floats, optional
Relative weight of each rotation.
Returns
-------
average : Rotation
Weighted average of original Rotation field.
References
----------
Quaternion averaging
F. Landis Markley, Yang Cheng, John L. Crassidis, Yaakov Oshman
Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
10.2514/1.28949
"""
def _M(quat):
"""Intermediate representation supporting quaternion averaging."""
return np.einsum('...i,...j',quat,quat)
if not weights:
weights = np.ones(self.shape,dtype=float)
eig, vec = np.linalg.eig(np.sum(_M(self.quaternion) * weights[...,np.newaxis,np.newaxis],axis=-3) \
/np.sum( weights[...,np.newaxis,np.newaxis],axis=-3))
return Rotation.from_quaternion(np.real(
np.squeeze(
np.take_along_axis(vec,
eig.argmax(axis=-1)[...,np.newaxis,np.newaxis],
axis=-1),
axis=-1)),
accept_homomorph = True)
def misorientation(self,other):
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"""
Calculate misorientation from self to other Rotation.
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Parameters
----------
other : Rotation
Rotation to which the misorientation is computed.
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"""
return other@~self
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################################################################################################
# convert to different orientation representations (numpy arrays)
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def as_quaternion(self):
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"""
Represent as unit quaternion.
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Returns
-------
q : numpy.ndarray of shape (...,4)
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Unit quaternion in positive real hemisphere: (q_0, q_1, q_2, q_3), ǀqǀ=1, q_0 0.
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"""
return self.quaternion.copy()
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def as_Euler_angles(self,
degrees = False):
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"""
Represent as Bunge-Euler angles.
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Parameters
----------
degrees : bool, optional
Return angles in degrees.
Returns
-------
phi : numpy.ndarray of shape (...,3)
Bunge-Euler angles: (φ_1, ϕ, φ_2), φ_1 [0,2π], ϕ [0,π], φ_2 [0,2π]
unless degrees == True: φ_1 [0,360], ϕ [0,180], φ_2 [0,360]
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"""
eu = Rotation._qu2eu(self.quaternion)
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if degrees: eu = np.degrees(eu)
return eu
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def as_axis_angle(self,
degrees = False,
pair = False):
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"""
Represent as axis angle pair.
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Parameters
----------
degrees : bool, optional
Return rotation angle in degrees. Defaults to False.
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pair : bool, optional
Return tuple of axis and angle. Defaults to False.
Returns
-------
axis_angle : numpy.ndarray of shape (...,4) unless pair == True:
tuple containing numpy.ndarray of shapes (...,3) and (...)
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Axis angle pair: (n_1, n_2, n_3, ω), ǀnǀ = 1 and ω [0,π]
unless degrees = True: ω [0,180].
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"""
ax = Rotation._qu2ax(self.quaternion)
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if degrees: ax[...,3] = np.degrees(ax[...,3])
return (ax[...,:3],ax[...,3]) if pair else ax
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def as_matrix(self):
"""
Represent as rotation matrix.
Returns
-------
R : numpy.ndarray of shape (...,3,3)
Rotation matrix R, det(R) = 1, R.TR=I.
"""
return Rotation._qu2om(self.quaternion)
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def as_Rodrigues_vector(self,
compact = False):
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"""
Represent as Rodrigues-Frank vector with separated axis and angle argument.
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Parameters
----------
vector : bool, optional
Return as actual Rodrigues-Frank vector, i.e. axis
and angle argument are not separated.
Returns
-------
rho : numpy.ndarray of shape (...,4) unless vector == True:
numpy.ndarray of shape (...,3)
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Rodrigues-Frank vector: [n_1, n_2, n_3, tan(ω/2)], ǀnǀ = 1 and ω [0,π].
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"""
ro = Rotation._qu2ro(self.quaternion)
if compact:
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with np.errstate(invalid='ignore'):
return ro[...,:3]*ro[...,3:4]
else:
return ro
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def as_homochoric(self):
"""
Represent as homochoric vector.
Returns
-------
h : numpy.ndarray of shape (...,3)
Homochoric vector: (h_1, h_2, h_3), ǀhǀ < (3/4*π)^(1/3).
"""
return Rotation._qu2ho(self.quaternion)
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def as_cubochoric(self):
"""
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Represent as cubochoric vector.
Returns
-------
c : numpy.ndarray of shape (...,3)
Cubochoric vector: (c_1, c_2, c_3), max(c_i) < 1/2*π^(2/3).
"""
return Rotation._qu2cu(self.quaternion)
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################################################################################################
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# Static constructors. The input data needs to follow the conventions, options allow to
# relax the conventions.
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@staticmethod
def from_quaternion(q,
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accept_homomorph = False,
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P = -1,
**kwargs):
"""
Initialize from quaternion.
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Parameters
----------
q : numpy.ndarray of shape (...,4)
Unit quaternion in positive real hemisphere: (q_0, q_1, q_2, q_3),
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ǀqǀ=1, q_0 0.
accept_homomorph : boolean, optional
Allow homomorphic variants, i.e. q_0 < 0 (negative real hemisphere).
Defaults to False.
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P : int {-1,1}, optional
Convention used. Defaults to -1.
"""
qu = np.array(q,dtype=float)
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if qu.shape[:-2:-1] != (4,):
raise ValueError('Invalid shape.')
if abs(P) != 1:
raise ValueError('P ∉ {-1,1}')
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qu[...,1:4] *= -P
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if accept_homomorph:
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qu[qu[...,0] < 0.0] *= -1
else:
if np.any(qu[...,0] < 0.0):
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raise ValueError('Quaternion with negative first (real) component.')
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if not np.all(np.isclose(np.linalg.norm(qu,axis=-1), 1.0)):
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raise ValueError('Quaternion is not of unit length.')
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return Rotation(qu)
@staticmethod
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def from_Euler_angles(phi,
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degrees = False,
**kwargs):
"""
Initialize from Bunge-Euler angles.
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Parameters
----------
phi : numpy.ndarray of shape (...,3)
Bunge-Euler angles: (φ_1, ϕ, φ_2), φ_1 [0,2π], ϕ [0,π], φ_2 [0,2π]
unless degrees == True: φ_1 [0,360], ϕ [0,180], φ_2 [0,360].
degrees : boolean, optional
Bunge-Euler angles are given in degrees. Defaults to False.
"""
eu = np.array(phi,dtype=float)
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if eu.shape[:-2:-1] != (3,):
raise ValueError('Invalid shape.')
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eu = np.radians(eu) if degrees else eu
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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
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raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].')
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return Rotation(Rotation._eu2qu(eu))
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@staticmethod
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def from_axis_angle(axis_angle,
degrees = False,
normalize = False,
P = -1,
**kwargs):
"""
Initialize from Axis angle pair.
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Parameters
----------
axis_angle : numpy.ndarray of shape (...,4)
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Axis angle pair: [n_1, n_2, n_3, ω], ǀnǀ = 1 and ω [0,π]
unless degrees = True: ω [0,180].
degrees : boolean, optional
Angle ω is given in degrees. Defaults to False.
normalize: boolean, optional
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Allow ǀnǀ 1. Defaults to False.
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P : int {-1,1}, optional
Convention used. Defaults to -1.
"""
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ax = np.array(axis_angle,dtype=float)
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if ax.shape[:-2:-1] != (4,):
raise ValueError('Invalid shape.')
if abs(P) != 1:
raise ValueError('P ∉ {-1,1}')
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ax[...,0:3] *= -P
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if degrees: ax[..., 3] = np.radians(ax[...,3])
if normalize: ax[...,0:3] /= np.linalg.norm(ax[...,0:3],axis=-1,keepdims=True)
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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)):
print(np.linalg.norm(ax[...,0:3],axis=-1))
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raise ValueError('Axis angle rotation axis is not of unit length.')
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return Rotation(Rotation._ax2qu(ax))
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@staticmethod
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def from_basis(basis,
orthonormal = True,
reciprocal = False,
**kwargs):
"""
Initialize from lattice basis vectors.
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Parameters
----------
basis : numpy.ndarray of shape (...,3,3)
Three three-dimensional lattice basis vectors.
orthonormal : boolean, optional
Basis is strictly orthonormal, i.e. is free of stretch components. Defaults to True.
reciprocal : boolean, optional
Basis vectors are given in reciprocal (instead of real) space. Defaults to False.
"""
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om = np.array(basis,dtype=float)
if om.shape[-2:] != (3,3):
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raise ValueError('Invalid shape.')
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if reciprocal:
om = np.linalg.inv(tensor.transpose(om)/np.pi) # transform reciprocal basis set
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orthonormal = False # contains stretch
if not orthonormal:
(U,S,Vh) = np.linalg.svd(om) # singular value decomposition
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om = np.einsum('...ij,...jl',U,Vh)
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if not np.all(np.isclose(np.linalg.det(om),1.0)):
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raise ValueError('Orientation matrix has determinant ≠ 1.')
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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)):
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raise ValueError('Orientation matrix is not orthogonal.')
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return Rotation(Rotation._om2qu(om))
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@staticmethod
def from_matrix(R,**kwargs):
"""
Initialize from rotation matrix.
Parameters
----------
R : numpy.ndarray of shape (...,3,3)
Rotation matrix: det(R) = 1, R.TR=I.
"""
return Rotation.from_basis(R)
@staticmethod
def from_parallel(a,b,
**kwargs):
"""
Initialize from pairs of two orthogonal lattice basis vectors.
Parameters
----------
a : numpy.ndarray of shape (...,2,3)
Two three-dimensional lattice vectors of first orthogonal basis.
b : numpy.ndarray of shape (...,2,3)
Corresponding three-dimensional lattice vectors of second basis.
"""
a_ = np.array(a)
b_ = np.array(b)
if a_.shape[-2:] != (2,3) or b_.shape[-2:] != (2,3) or a_.shape != b_.shape:
raise ValueError('Invalid shape.')
am = np.stack([ a_[...,0,:],
a_[...,1,:],
np.cross(a_[...,0,:],a_[...,1,:]) ],axis=-2)
bm = np.stack([ b_[...,0,:],
b_[...,1,:],
np.cross(b_[...,0,:],b_[...,1,:]) ],axis=-2)
return Rotation.from_basis(np.swapaxes(am/np.linalg.norm(am,axis=-1,keepdims=True),-1,-2))\
.misorientation(Rotation.from_basis(np.swapaxes(bm/np.linalg.norm(bm,axis=-1,keepdims=True),-1,-2)))
@staticmethod
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def from_Rodrigues_vector(rho,
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normalize = False,
P = -1,
**kwargs):
"""
Initialize from Rodrigues-Frank vector (angle separated from axis).
Parameters
----------
rho : numpy.ndarray of shape (...,4)
Rodrigues-Frank vector. (n_1, n_2, n_3, tan(ω/2)), ǀnǀ = 1 and ω [0,π].
normalize : boolean, optional
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Allow ǀnǀ 1. Defaults to False.
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P : int {-1,1}, optional
Convention used. Defaults to -1.
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"""
ro = np.array(rho,dtype=float)
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if ro.shape[:-2:-1] != (4,):
raise ValueError('Invalid shape.')
if abs(P) != 1:
raise ValueError('P ∉ {-1,1}')
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ro[...,0:3] *= -P
if normalize: ro[...,0:3] /= np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True)
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if np.any(ro[...,3] < 0.0):
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raise ValueError('Rodrigues vector rotation angle not positive.')
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if not np.all(np.isclose(np.linalg.norm(ro[...,0:3],axis=-1), 1.0)):
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raise ValueError('Rodrigues vector rotation axis is not of unit length.')
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return Rotation(Rotation._ro2qu(ro))
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@staticmethod
def from_homochoric(h,
P = -1,
**kwargs):
"""
Initialize from homochoric vector.
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Parameters
----------
h : numpy.ndarray of shape (...,3)
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Homochoric vector: (h_1, h_2, h_3), ǀhǀ < (3/4*π)^(1/3).
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P : int {-1,1}, optional
Convention used. Defaults to -1.
"""
ho = np.array(h,dtype=float)
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if ho.shape[:-2:-1] != (3,):
raise ValueError('Invalid shape.')
if abs(P) != 1:
raise ValueError('P ∉ {-1,1}')
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ho *= -P
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if np.any(np.linalg.norm(ho,axis=-1) >_R1+1e-9):
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raise ValueError('Homochoric coordinate outside of the sphere.')
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return Rotation(Rotation._ho2qu(ho))
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@staticmethod
def from_cubochoric(c,
P = -1,
**kwargs):
"""
Initialize from cubochoric vector.
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Parameters
----------
c : numpy.ndarray of shape (...,3)
Cubochoric vector: (c_1, c_2, c_3), max(c_i) < 1/2*π^(2/3).
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P : int {-1,1}, optional
Convention used. Defaults to -1.
"""
cu = np.array(c,dtype=float)
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if cu.shape[:-2:-1] != (3,):
raise ValueError('Invalid shape.')
if abs(P) != 1:
raise ValueError('P ∉ {-1,1}')
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if np.abs(np.max(cu)) > np.pi**(2./3.) * 0.5+1e-9:
raise ValueError('Cubochoric coordinate outside of the cube.')
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ho = -P * Rotation._cu2ho(cu)
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return Rotation(Rotation._ho2qu(ho))
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@staticmethod
def from_random(shape = None,
rng_seed = None,
**kwargs):
"""
Draw random rotation.
Rotations are uniformly distributed.
Parameters
----------
shape : tuple of ints, optional
Shape of the sample. Defaults to None which gives a
single rotation
rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional
A seed to initialize the BitGenerator. Defaults to None.
If None, then fresh, unpredictable entropy will be pulled from the OS.
"""
rng = np.random.default_rng(rng_seed)
r = rng.random(3 if shape is None else tuple(shape)+(3,) if hasattr(shape, '__iter__') else (shape,3))
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A = np.sqrt(r[...,2])
B = np.sqrt(1.0-r[...,2])
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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 if shape is None else q.reshape(r.shape[:-1]+(4,)))._standardize()
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@staticmethod
def from_ODF(weights,
phi,
N = 500,
degrees = True,
fractions = True,
rng_seed = None,
**kwargs):
"""
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Sample discrete values from a binned ODF.
Parameters
----------
weights : numpy.ndarray of shape (n)
Texture intensity values (probability density or volume fraction) at Euler grid points.
phi : numpy.ndarray of shape (n,3)
Grid coordinates in Euler space at which weights are defined.
N : integer, optional
Number of discrete orientations to be sampled from the given ODF.
Defaults to 500.
degrees : boolean, optional
Euler grid values are in degrees. Defaults to True.
fractions : boolean, optional
ODF values correspond to volume fractions, not probability density.
Defaults to True.
rng_seed: {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional
A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy
will be pulled from the OS.
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Returns
-------
samples : damask.Rotation of shape (N)
Array of sampled rotations closely representing the input ODF.
Notes
-----
Due to the distortion of Euler space in the vicinity of ϕ = 0, probability densities, p, defined on
grid points with ϕ = 0 will never result in reconstructed orientations as their dV/V = p dγ = p × 0.
Hence, it is recommended to transform any such dataset to cell centers that avoid grid points at ϕ = 0.
References
----------
P. Eisenlohr, F. Roters, Computational Materials Science 42(4), 670-678, 2008
https://doi.org/10.1016/j.commatsci.2007.09.015
"""
def _dg(eu,deg):
"""Return infinitesimal Euler space volume of bin(s)."""
phi_sorted = eu[np.lexsort((eu[:,0],eu[:,1],eu[:,2]))]
steps,size,_ = grid_filters.cellsSizeOrigin_coordinates0_point(phi_sorted)
delta = np.radians(size/steps) if deg else size/steps
return delta[0]*2.0*np.sin(delta[1]/2.0)*delta[2] / 8.0 / np.pi**2 * np.sin(np.radians(eu[:,1]) if deg else eu[:,1])
dg = 1.0 if fractions else _dg(phi,degrees)
dV_V = dg * np.maximum(0.0,weights.squeeze())
return Rotation.from_Euler_angles(phi[util.hybrid_IA(dV_V,N,rng_seed)],degrees)
@staticmethod
def from_spherical_component(center,
sigma,
N = 500,
degrees = True,
rng_seed = None,
**kwargs):
"""
Calculate set of rotations with Gaussian distribution around center.
Parameters
----------
center : Rotation
Central Rotation.
sigma : float
Standard deviation of (Gaussian) misorientation distribution.
N : int, optional
Number of samples, defaults to 500.
degrees : boolean, optional
sigma is given in degrees.
rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional
A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy
will be pulled from the OS.
"""
rng = np.random.default_rng(rng_seed)
sigma = np.radians(sigma) if degrees else sigma
u,Theta = (rng.random((N,2)) * 2.0 * np.array([1,np.pi]) - np.array([1.0, 0])).T
omega = abs(rng.normal(scale=sigma,size=N))
p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta),
np.sqrt(1-u**2)*np.sin(Theta),
u, omega])
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return Rotation.from_axis_angle(p) @ center
@staticmethod
def from_fiber_component(alpha,
beta,
sigma = 0.0,
N = 500,
degrees = True,
rng_seed = None,
**kwargs):
"""
Calculate set of rotations with Gaussian distribution around direction.
Parameters
----------
alpha : numpy.ndarray of size 2
Polar coordinates (phi from x,theta from z) of fiber direction in crystal frame.
beta : numpy.ndarray of size 2
Polar coordinates (phi from x,theta from z) of fiber direction in sample frame.
sigma : float, optional
Standard deviation of (Gaussian) misorientation distribution.
Defaults to 0.
N : int, optional
Number of samples, defaults to 500.
degrees : boolean, optional
sigma, alpha, and beta are given in degrees.
rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional
A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy
will be pulled from the OS.
"""
rng = np.random.default_rng(rng_seed)
sigma_,alpha_,beta_ = map(np.radians,(sigma,alpha,beta)) if degrees else (sigma,alpha,beta)
d_cr = np.array([np.sin(alpha_[0])*np.cos(alpha_[1]), np.sin(alpha_[0])*np.sin(alpha_[1]), np.cos(alpha_[0])])
d_lab = np.array([np.sin( beta_[0])*np.cos( beta_[1]), np.sin( beta_[0])*np.sin( beta_[1]), np.cos( beta_[0])])
ax_align = np.append(np.cross(d_lab,d_cr), np.arccos(np.dot(d_lab,d_cr)))
if np.isclose(ax_align[3],0.0): ax_align[:3] = np.array([1,0,0])
R_align = Rotation.from_axis_angle(ax_align if ax_align[3] > 0.0 else -ax_align,normalize=True) # rotate fiber axis from sample to crystal frame
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u,Theta = (rng.random((N,2)) * 2.0 * np.array([1,np.pi]) - np.array([1.0, 0])).T
omega = abs(rng.normal(scale=sigma_,size=N))
p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta),
np.sqrt(1-u**2)*np.sin(Theta),
u, omega])
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p[:,:3] = np.einsum('ij,...j',np.eye(3)-np.outer(d_lab,d_lab),p[:,:3]) # remove component along fiber axis
f = np.column_stack((np.broadcast_to(d_lab,(N,3)),rng.random(N)*np.pi))
f[::2,:3] *= -1 # flip half the rotation axes to negative sense
return R_align.broadcast_to(N) \
@ Rotation.from_axis_angle(p,normalize=True) \
@ Rotation.from_axis_angle(f)
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####################################################################################################
# Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations
####################################################################################################
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# Copyright (c) 2017-2020, Martin Diehl/Max-Planck-Institut für Eisenforschung GmbH
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# 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,
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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,
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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
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@staticmethod
def _qu2eu(qu):
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"""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),
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np.broadcast_to(np.pi,qu[...,0:1].shape),
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)])
)
)
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# reduce Euler angles to definition range
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eu[np.abs(eu)<1.e-6] = 0.0
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eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu) # needed?
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return eu
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@staticmethod
def _qu2ax(qu):
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"""
Quaternion to axis angle pair.
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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),
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np.block([qu[...,1:4],np.broadcast_to(np.pi,qu[...,0:1].shape)]),
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):
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"""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),
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np.block([qu[...,1:2], qu[...,2:3], qu[...,3:4], np.broadcast_to(np.inf,qu[...,0:1].shape)]),
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
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@staticmethod
def _qu2ho(qu):
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"""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.))
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return ho
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@staticmethod
def _qu2cu(qu):
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"""Quaternion to cubochoric vector."""
return Rotation._ho2cu(Rotation._qu2ho(qu))
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#---------- Rotation matrix ----------
@staticmethod
def _om2qu(om):
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"""
Rotation matrix to quaternion.
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This formulation is from www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion.
The original formulation had issues.
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"""
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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
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return qu
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@staticmethod
def _om2eu(om):
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"""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]),1.0,0.0),
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
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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
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@staticmethod
def _om2ax(om):
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"""Rotation matrix to axis angle pair."""
diag_delta = -_P*np.block([om[...,1,2:3]-om[...,2,1:2],
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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]
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return ax
@staticmethod
def _om2ro(om):
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"""Rotation matrix to Rodrigues-Frank vector."""
return Rotation._eu2ro(Rotation._om2eu(om))
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@staticmethod
def _om2ho(om):
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"""Rotation matrix to homochoric vector."""
return Rotation._ax2ho(Rotation._om2ax(om))
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@staticmethod
def _om2cu(om):
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"""Rotation matrix to cubochoric vector."""
return Rotation._ho2cu(Rotation._om2ho(om))
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#---------- Bunge-Euler angles ----------
@staticmethod
def _eu2qu(eu):
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"""Bunge-Euler angles to quaternion."""
ee = 0.5*eu
cPhi = np.cos(ee[...,1:2])
sPhi = np.sin(ee[...,1:2])
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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
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return qu
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@staticmethod
def _eu2om(eu):
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"""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
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return om
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@staticmethod
def _eu2ax(eu):
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"""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
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return ax
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@staticmethod
def _eu2ro(eu):
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"""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 ]
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return ro
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@staticmethod
def _eu2ho(eu):
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"""Bunge-Euler angles to homochoric vector."""
return Rotation._ax2ho(Rotation._eu2ax(eu))
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@staticmethod
def _eu2cu(eu):
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"""Bunge-Euler angles to cubochoric vector."""
return Rotation._ho2cu(Rotation._eu2ho(eu))
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#---------- Axis angle pair ----------
@staticmethod
def _ax2qu(ax):
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"""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]))
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return qu
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@staticmethod
def _ax2om(ax):
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"""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))
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return om if _P < 0.0 else np.swapaxes(om,-1,-2)
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@staticmethod
def _ax2eu(ax):
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"""Rotation matrix to Bunge Euler angles."""
return Rotation._om2eu(Rotation._ax2om(ax))
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@staticmethod
def _ax2ro(ax):
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"""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]
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return ro
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@staticmethod
def _ax2ho(ax):
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"""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
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return ho
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@staticmethod
def _ax2cu(ax):
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"""Axis angle pair to cubochoric vector."""
return Rotation._ho2cu(Rotation._ax2ho(ax))
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#---------- Rodrigues-Frank vector ----------
@staticmethod
def _ro2qu(ro):
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"""Rodrigues-Frank vector to quaternion."""
return Rotation._ax2qu(Rotation._ro2ax(ro))
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@staticmethod
def _ro2om(ro):
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"""Rodgrigues-Frank vector to rotation matrix."""
return Rotation._ax2om(Rotation._ro2ax(ro))
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@staticmethod
def _ro2eu(ro):
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"""Rodrigues-Frank vector to Bunge-Euler angles."""
return Rotation._om2eu(Rotation._ro2om(ro))
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@staticmethod
def _ro2ax(ro):
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"""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)]))
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ax[np.abs(ro[...,3]) < 1.e-8] = np.array([ 0.0, 0.0, 1.0, 0.0 ])
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return ax
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@staticmethod
def _ro2ho(ro):
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"""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)
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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))
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return ho
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@staticmethod
def _ro2cu(ro):
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"""Rodrigues-Frank vector to cubochoric vector."""
return Rotation._ho2cu(Rotation._ro2ho(ro))
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#---------- Homochoric vector----------
@staticmethod
def _ho2qu(ho):
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"""Homochoric vector to quaternion."""
return Rotation._ax2qu(Rotation._ho2ax(ho))
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@staticmethod
def _ho2om(ho):
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"""Homochoric vector to rotation matrix."""
return Rotation._ax2om(Rotation._ho2ax(ho))
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@staticmethod
def _ho2eu(ho):
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"""Homochoric vector to Bunge-Euler angles."""
return Rotation._ax2eu(Rotation._ho2ax(ho))
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@staticmethod
def _ho2ax(ho):
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"""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'):
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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))]))
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return ax
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@staticmethod
def _ho2ro(ho):
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"""Axis angle pair to Rodrigues-Frank vector."""
return Rotation._ax2ro(Rotation._ho2ax(ho))
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@staticmethod
def _ho2cu(ho):
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"""
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)
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return cu
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#---------- Cubochoric ----------
@staticmethod
def _cu2qu(cu):
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"""Cubochoric vector to quaternion."""
return Rotation._ho2qu(Rotation._cu2ho(cu))
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@staticmethod
def _cu2om(cu):
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"""Cubochoric vector to rotation matrix."""
return Rotation._ho2om(Rotation._cu2ho(cu))
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@staticmethod
def _cu2eu(cu):
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"""Cubochoric vector to Bunge-Euler angles."""
return Rotation._ho2eu(Rotation._cu2ho(cu))
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@staticmethod
def _cu2ax(cu):
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"""Cubochoric vector to axis angle pair."""
return Rotation._ho2ax(Rotation._cu2ho(cu))
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@staticmethod
def _cu2ro(cu):
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"""Cubochoric vector to Rodrigues-Frank vector."""
return Rotation._ho2ro(Rotation._cu2ho(cu))
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@staticmethod
def _cu2ho(cu):
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"""
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)
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return ho
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@staticmethod
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def _get_pyramid_order(xyz,direction=None):
"""
Get order of the coordinates.
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Depending on the pyramid in which the point is located, the order need to be adjusted.
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Parameters
----------
xyz : numpy.ndarray
coordinates of a point on a uniform refinable grid on a ball or
in a uniform refinable cubical grid.
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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
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"""
order = {'forward': np.array([[0,1,2],[1,2,0],[2,0,1]]),
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'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]