import copy
from typing import Optional, Union, Sequence, Tuple, Literal, List, TypeVar
import numpy as np
from ._typehints import FloatSequence, IntSequence, NumpyRngSeed
from . import tensor
from . import util
from . import grid_filters
_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.)
MyType = TypeVar('MyType', bound='Rotation')
[docs]class Rotation:
u"""
Rotation with functionality for conversion between 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, i.e. as rotation of
the coordinate frame.
- P = -1 (as default).
Examples
--------
Rotate vector 'a' (defined in coordinate system 'A') to
coordinates 'b' expressed in system 'B':
>>> import numpy as np
>>> import damask
>>> Q = damask.Rotation.from_random()
>>> a = np.random.rand(3)
>>> b = Q @ a
>>> np.allclose(np.dot(Q.as_matrix(),a),b)
True
Compound rotations R1 (first) and R2 (second):
>>> import numpy as np
>>> import damask
>>> R1 = damask.Rotation.from_random()
>>> R2 = damask.Rotation.from_random()
>>> R = R2 * R1
>>> np.allclose(R.as_matrix(), np.dot(R2.as_matrix(),R1.as_matrix()))
True
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: Union[FloatSequence, 'Rotation'] = np.array([1.,0.,0.,0.])):
"""
New rotation.
Parameters
----------
rotation : list, numpy.ndarray, or Rotation, optional
Unit quaternion in positive real hemisphere.
Use .from_quaternion to perform a sanity check.
Defaults to no rotation.
"""
self.quaternion: np.ndarray
if isinstance(rotation,Rotation):
self.quaternion = rotation.quaternion.copy()
elif np.array(rotation).shape[-1] == 4:
self.quaternion = np.array(rotation,dtype=float)
else:
raise TypeError('"rotation" is neither a Rotation nor a quaternion')
def __repr__(self) -> str:
"""
Return repr(self).
Give short, human-readable summary.
"""
return f'Quaternion{" " if self.quaternion.shape == (4,) else "s of shape "+str(self.quaternion.shape[:-1])+chr(10)}'\
+ str(self.quaternion)
def __copy__(self: MyType,
rotation: Union[None, FloatSequence, 'Rotation'] = None) -> MyType:
"""
Return deepcopy(self).
Create deep copy.
"""
dup = copy.deepcopy(self)
if rotation is not None:
dup.quaternion = Rotation(rotation).quaternion
return dup
copy = __copy__
def __getitem__(self,
item: Union[Tuple[Union[None, int, slice]], int, bool, np.bool_, np.ndarray]):
"""
Return self[item].
Return slice according to item.
"""
return self.copy() if self.shape == () else \
self.copy(self.quaternion[item+(slice(None),)] if isinstance(item,tuple) else self.quaternion[item])
def __eq__(self,
other: object) -> bool:
"""
Return self==other.
Test equality of other.
Parameters
----------
other : Rotation
Rotation to check for equality.
"""
return NotImplemented if not isinstance(other, Rotation) else \
np.logical_or(np.all(self.quaternion == other.quaternion,axis=-1),
np.all(self.quaternion == -1.*other.quaternion,axis=-1))
def __ne__(self,
other: object) -> bool:
"""
Return self!=other.
Test inequality of other.
Parameters
----------
other : Rotation
Rotation to check for inequality.
"""
return np.logical_not(self==other) if isinstance(other, Rotation) else NotImplemented
[docs] def isclose(self: MyType,
other: MyType,
rtol: float = 1.e-5,
atol: float = 1.e-8,
equal_nan: bool = True) -> bool:
"""
Report where values are approximately equal to corresponding ones of other Rotation.
Parameters
----------
other : Rotation
Rotation to compare against.
rtol : float, optional
Relative tolerance of equality.
atol : float, optional
Absolute tolerance of equality.
equal_nan : bool, optional
Consider matching NaN values as equal. Defaults to True.
Returns
-------
mask : numpy.ndarray of bool, shape (self.shape)
Mask indicating where corresponding rotations are close.
"""
s = self.quaternion
o = other.quaternion
return np.logical_or(np.all(np.isclose(s, o,rtol,atol,equal_nan),axis=-1),
np.all(np.isclose(s,-1.*o,rtol,atol,equal_nan),axis=-1))
[docs] def allclose(self: MyType,
other: MyType,
rtol: float = 1.e-5,
atol: float = 1.e-8,
equal_nan: bool = True) -> Union[np.bool_, bool]:
"""
Test whether all values are approximately equal to corresponding ones of other Rotation.
Parameters
----------
other : Rotation
Rotation to compare against.
rtol : float, optional
Relative tolerance of equality.
atol : float, optional
Absolute tolerance of equality.
equal_nan : bool, optional
Consider matching NaN values as equal. Defaults to True.
Returns
-------
answer : bool
Whether all values are close between both rotations.
"""
return np.all(self.isclose(other,rtol,atol,equal_nan))
def __array__(self):
"""Initializer for numpy."""
return self.quaternion
@property
def size(self) -> int:
return self.quaternion[...,0].size
@property
def shape(self) -> Tuple[int, ...]:
return self.quaternion[...,0].shape
def __len__(self) -> int:
"""
Return len(self).
Length of leading/leftmost dimension of array.
"""
return 0 if self.shape == () else self.shape[0]
def __invert__(self: MyType) -> MyType:
"""
Return ~self.
Inverse rotation (backward rotation).
"""
dup = self.copy()
dup.quaternion[...,1:] *= -1.
return dup
def __pow__(self: MyType,
exp: Union[float, int]) -> MyType:
"""
Return self**exp.
Perform the rotation 'exp' times.
Parameters
----------
exp : float
Exponent.
"""
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(np.block([np.cos(exp*phi),np.sin(exp*phi)*p]))._standardize())
def __ipow__(self: MyType,
exp: Union[float, int]) -> MyType:
"""
Return self**=exp.
Perform the rotation 'exp' times (in-place).
Parameters
----------
exp : float
Exponent.
"""
return self**exp
def __mul__(self: MyType,
other: MyType) -> MyType:
"""
Return self*other.
Compose with other.
Parameters
----------
other : Rotation, shape (self.shape)
Rotation for composition.
Compatible innermost dimensions will blend.
Returns
-------
composition : Rotation
Compound rotation self*other, i.e. first other then self rotation.
"""
if isinstance(other,Rotation):
blend = util.shapeblender( self.shape,other.shape)
s_m = util.shapeshifter( self.shape,blend,mode='right')
s_o = util.shapeshifter(other.shape,blend,mode='left')
q_m = self.broadcast_to(s_m).quaternion[...,0:1]
p_m = self.broadcast_to(s_m).quaternion[...,1:]
q_o = other.broadcast_to(s_o).quaternion[...,0:1]
p_o = other.broadcast_to(s_o).quaternion[...,1:]
qmo = q_m*q_o
q = (qmo - np.einsum('...i,...i',p_m,p_o).reshape(qmo.shape))
p = q_m*p_o + q_o*p_m + _P * np.cross(p_m,p_o)
return self.copy(Rotation(np.block([q,p]))._standardize())
else:
raise TypeError('use "R@b", i.e. matmul, to apply rotation "R" to object "b"')
def __imul__(self: MyType,
other: MyType) -> MyType:
"""
Return self*=other.
Compose with other (in-place).
Parameters
----------
other : Rotation, shape (self.shape)
Rotation for composition.
Compatible innermost dimensions will blend.
"""
return self*other
def __truediv__(self: MyType,
other: MyType) -> MyType:
"""
Return self/other.
Compose with inverse of other.
Parameters
----------
other : damask.Rotation, shape (self.shape)
Rotation to invert for composition.
Compatible innermost dimensions will blend.
Returns
-------
composition : Rotation
Compound rotation self*(~other), i.e. first inverse of other then self rotation.
"""
if isinstance(other,Rotation):
return self*~other
else:
raise TypeError('use "R@b", i.e. matmul, to apply rotation "R" to object "b"')
def __itruediv__(self: MyType,
other: MyType) -> MyType:
"""
Return self/=other.
Compose with inverse of other (in-place).
Parameters
----------
other : Rotation, shape (self.shape)
Rotation to invert for composition.
"""
return self/other
def __matmul__(self,
other: np.ndarray) -> np.ndarray:
"""
Return self@other.
Rotate vector, second-order tensor, or fourth-order tensor.
`other` is interpreted as an array of tensor quantities with the highest-possible order
considering the shape of `self`. Compatible innermost dimensions will blend.
For instance, shapes of (2,) and (3,3) for `self` and `other` prompt interpretation of
`other` as a second-rank tensor and result in (2,) rotated tensors, whereas
shapes of (2,1) and (3,3) for `self` and `other` result in (2,3) rotated vectors.
Parameters
----------
other : numpy.ndarray, shape (...,3), (...,3,3), or (...,3,3,3,3)
Vector or tensor on which to apply the rotation.
Returns
-------
rotated : numpy.ndarray, shape (...,3), (...,3,3), or (...,3,3,3,3)
Rotated vector or tensor, i.e. transformed to frame defined by rotation.
Examples
--------
All below examples rely on imported modules:
>>> import numpy as np
>>> import damask
Application of twelve (random) rotations to a set of five vectors.
>>> r = damask.Rotation.from_random(shape=(12))
>>> o = np.ones((5,3))
>>> (r@o).shape # (12) @ (5, 3)
(12,5, 3)
Application of a (random) rotation to all twelve second-rank tensors.
>>> r = damask.Rotation.from_random()
>>> o = np.ones((12,3,3))
>>> (r@o).shape # (1) @ (12, 3,3)
(12,3,3)
Application of twelve (random) rotations to the corresponding twelve second-rank tensors.
>>> r = damask.Rotation.from_random(shape=(12))
>>> o = np.ones((12,3,3))
>>> (r@o).shape # (12) @ (3,3)
(12,3,3)
Application of each of three (random) rotations to all three vectors.
>>> r = damask.Rotation.from_random(shape=(3))
>>> o = np.ones((3,3))
>>> (r[...,np.newaxis]@o[np.newaxis,...]).shape # (3,1) @ (1,3, 3)
(3,3,3)
Application of twelve (random) rotations to all twelve second-rank tensors.
>>> r = damask.Rotation.from_random(shape=(12))
>>> o = np.ones((12,3,3))
>>> (r@o[np.newaxis,...]).shape # (12) @ (1,12, 3,3)
(12,3,3,3)
"""
if isinstance(other, np.ndarray):
obs = util.shapeblender(self.shape,other.shape)[len(self.shape):]
for l in [4,2,1]:
if obs[-l:] == l*(3,):
bs = util.shapeblender(self.shape,other.shape[:-l],False)
self_ = self.broadcast_to(bs) if self.shape != bs else self
if l==1:
q_m = self_.quaternion[...,0]
p_m = self_.quaternion[...,1:]
A = q_m**2 - np.einsum('...i,...i',p_m,p_m)
B = 2. * np.einsum('...i,...i',p_m,other)
C = 2. * _P * q_m
return np.block([(A * other[...,i]) +
(B * p_m[...,i]) +
(C * ( p_m[...,(i+1)%3]*other[...,(i+2)%3]
- p_m[...,(i+2)%3]*other[...,(i+1)%3]))
for i in [0,1,2]]).reshape(bs+(3,),order='F')
else:
return np.einsum({2: '...im,...jn,...mn',
4: '...im,...jn,...ko,...lp,...mnop'}[l],
*l*[self_.as_matrix()],
other)
raise ValueError('can only rotate vectors, second-order tensors, and fourth-order tensors')
elif isinstance(other, Rotation):
raise TypeError('use "R2*R1", i.e. multiplication, to compose rotations "R1" and "R2"')
else:
raise TypeError(f'cannot rotate "{type(other)}"')
apply = __matmul__
def _standardize(self: MyType) -> MyType:
"""Standardize quaternion (ensure positive real hemisphere)."""
self.quaternion[self.quaternion[...,0] < 0.] *= -1.
return self
[docs] def append(self: MyType,
other: Union[MyType, List[MyType]]) -> MyType:
"""
Extend array along first dimension with other array(s).
Parameters
----------
other : (list of) damask.Rotation
"""
return self.copy(np.vstack(tuple(map(lambda x:x.quaternion,
[self]+other if isinstance(other,list) else [self,other]))))
[docs] def flatten(self: MyType,
order: Literal['C','F','A'] = 'C') -> MyType:
"""
Flatten array.
Parameters
----------
order : {'C', 'F', 'A'}, optional
'C' flattens in row-major (C-style) order.
'F' flattens in column-major (Fortran-style) order.
'A' flattens in column-major order if object is Fortran contiguous in memory,
row-major order otherwise.
Defaults to 'C'.
Returns
-------
flattened : damask.Rotation
Rotation flattened to single dimension.
"""
return self.copy(self.quaternion.reshape((-1,4),order=order))
[docs] def reshape(self: MyType,
shape: Union[int, IntSequence],
order: Literal['C','F','A'] = 'C') -> MyType:
"""
Reshape array.
Parameters
----------
shape : (sequence of) int
New shape, number of elements needs to match the original shape.
If an integer is supplied, then the result will be a 1-D array of that length.
order : {'C', 'F', 'A'}, optional
'C' flattens in row-major (C-style) order.
'F' flattens in column-major (Fortran-style) order.
'A' flattens in column-major order if object is Fortran contiguous in memory,
row-major order otherwise.
Defaults to 'C'.
Returns
-------
reshaped : damask.Rotation
Rotation of given shape.
"""
if isinstance(shape,(int,np.integer)): shape = (shape,)
return self.copy(self.quaternion.reshape(tuple(shape)+(4,),order=order))
[docs] def broadcast_to(self: MyType,
shape: Union[int, IntSequence],
mode: Literal['left', 'right'] = 'right') -> MyType:
"""
Broadcast array.
Parameters
----------
shape : (sequence of) int
Shape of broadcasted array, needs to be compatible with the original shape.
mode : str, optional
Where to preferentially locate missing dimensions.
Either 'left' or 'right' (default).
Returns
-------
broadcasted : damask.Rotation
Rotation broadcasted to given shape.
"""
shape_ = (shape,) if isinstance(shape,(int,np.integer)) else tuple(shape)
return self.copy(np.broadcast_to(self.quaternion.reshape(util.shapeshifter(self.shape,shape_,mode)+(4,)),
shape_+(4,)))
[docs] def average(self: MyType,
weights: Optional[FloatSequence] = None) -> MyType:
"""
Average along last array dimension.
Parameters
----------
weights : numpy.ndarray, shape (self.shape), optional
Relative weight of each rotation.
Returns
-------
average : damask.Rotation
Weighted average of original Rotation field.
References
----------
F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
https://doi.org/10.2514/1.28949
"""
def _M(quat):
"""Intermediate representation supporting quaternion averaging."""
return np.einsum('...i,...j',quat,quat)
weights_ = np.ones(self.shape,dtype=float) if weights is None else np.array(weights,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 self.copy(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))
[docs] def misorientation(self: MyType,
other: MyType) -> MyType:
"""
Calculate misorientation to other Rotation.
Parameters
----------
other : damask.Rotation
Rotation to which the misorientation is computed.
Returns
-------
g : damask.Rotation
Misorientation.
"""
return other/self
################################################################################################
# convert to different orientation representations (numpy arrays)
[docs] def as_quaternion(self) -> np.ndarray:
"""
Represent as unit quaternion.
Returns
-------
q : numpy.ndarray, shape (...,4)
Unit quaternion (q_0, q_1, q_2, q_3) in positive real hemisphere, i.e. ǀqǀ = 1, q_0 ≥ 0.
"""
return self.quaternion.copy()
[docs] def as_Euler_angles(self,
degrees: bool = False) -> np.ndarray:
"""
Represent as Bunge Euler angles.
Parameters
----------
degrees : bool, optional
Return angles in degrees. Defaults to False.
Returns
-------
phi : numpy.ndarray, shape (...,3)
Bunge Euler angles (φ_1 ∈ [0,2π], ϕ ∈ [0,π], φ_2 ∈ [0,2π])
or (φ_1 ∈ [0,360], ϕ ∈ [0,180], φ_2 ∈ [0,360]) if degrees == True.
Notes
-----
Bunge Euler angles correspond to a rotation axis sequence of z–x'–z''.
Examples
--------
Cube orientation as Bunge Euler angles.
>>> import damask
>>> damask.Rotation([1,0,0,0]).as_Euler_angles()
array([0., 0., 0.])
"""
eu = Rotation._qu2eu(self.quaternion)
return np.degrees(eu) if degrees else eu
[docs] def as_axis_angle(self,
degrees: bool = False,
pair: bool = False) -> Union[Tuple[np.ndarray, np.ndarray], np.ndarray]:
"""
Represent as axis–angle pair.
Parameters
----------
degrees : bool, optional
Return rotation angle in degrees. Defaults to False.
pair : bool, optional
Return tuple of axis and angle. Defaults to False.
Returns
-------
n_omega : numpy.ndarray, shape (...,4) or tuple ((...,3), (...)) if pair == True
Axis and angle [n_1, n_2, n_3, ω] with ǀnǀ = 1 and ω ∈ [0,π]
or ω ∈ [0,180] if degrees == True.
Examples
--------
Cube orientation as axis–angle pair.
>>> import damask
>>> damask.Rotation([1,0,0,0]).as_axis_angle(pair=True)
(array([0., 0., 1.]), array(0.))
"""
ax: np.ndarray = Rotation._qu2ax(self.quaternion)
if degrees: ax[...,3] = np.degrees(ax[...,3])
return (ax[...,:3],ax[...,3]) if pair else ax
[docs] def as_matrix(self) -> np.ndarray:
"""
Represent as rotation matrix.
Returns
-------
R : numpy.ndarray, shape (...,3,3)
Rotation matrix R with det(R) = 1, R.T ∙ R = I.
Examples
--------
Cube orientation as rotation matrix.
>>> import damask
>>> damask.Rotation([1,0,0,0]).as_matrix()
array([[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.]])
"""
return Rotation._qu2om(self.quaternion)
[docs] def as_Rodrigues_vector(self,
compact: bool = False) -> np.ndarray:
"""
Represent as Rodrigues–Frank vector with separate axis and angle argument.
Parameters
----------
compact : bool, optional
Return three-component Rodrigues–Frank vector,
i.e. axis and angle argument are not separated.
Returns
-------
rho : numpy.ndarray, shape (...,4) or (...,3) if compact == True
Rodrigues–Frank vector [n_1, n_2, n_3, tan(ω/2)] with ǀnǀ = 1 and ω ∈ [0,π]
or [n_1, n_2, n_3] with ǀnǀ = tan(ω/2) and ω ∈ [0,π] if compact == True.
Examples
--------
Cube orientation as three-component Rodrigues–Frank vector.
>>> import damask
>>> damask.Rotation([1,0,0,0]).as_Rodrigues_vector(compact=True)
array([ 0., 0., 0.])
"""
ro = Rotation._qu2ro(self.quaternion)
if compact:
with np.errstate(invalid='ignore'):
return ro[...,:3]*ro[...,3:4]
else:
return ro
[docs] def as_homochoric(self) -> np.ndarray:
"""
Represent as homochoric vector.
Returns
-------
h : numpy.ndarray, shape (...,3)
Homochoric vector (h_1, h_2, h_3) with ǀhǀ < (3/4*π)^(1/3).
Examples
--------
Cube orientation as homochoric vector.
>>> import damask
>>> damask.Rotation([1,0,0,0]).as_homochoric()
array([0., 0., 0.])
"""
return Rotation._qu2ho(self.quaternion)
[docs] def as_cubochoric(self) -> np.ndarray:
"""
Represent as cubochoric vector.
Returns
-------
x : numpy.ndarray, shape (...,3)
Cubochoric vector (x_1, x_2, x_3) with max(x_i) < 1/2*π^(2/3).
Examples
--------
Cube orientation as cubochoric vector.
>>> import damask
>>> damask.Rotation([1,0,0,0]).as_cubochoric()
array([0., 0., 0.])
"""
return Rotation._qu2cu(self.quaternion)
################################################################################################
# Static constructors. The input data needs to follow the conventions, options allow to
# relax the conventions.
[docs] @staticmethod
def from_quaternion(q: Union[Sequence[FloatSequence], np.ndarray],
accept_homomorph: bool = False,
normalize: bool = False,
P: Literal[1, -1] = -1) -> 'Rotation':
"""
Initialize from quaternion.
Parameters
----------
q : numpy.ndarray, shape (...,4)
Unit quaternion (q_0, q_1, q_2, q_3) in positive real hemisphere, i.e. ǀqǀ = 1 and q_0 ≥ 0.
accept_homomorph : bool, optional
Allow homomorphic variants, i.e. q_0 < 0 (negative real hemisphere).
Defaults to False.
normalize: bool, optional
Allow ǀqǀ ≠ 1. Defaults to False.
P : int ∈ {-1,1}, optional
Sign convention. Defaults to -1.
Returns
-------
new : damask.Rotation
"""
qu = np.array(q,dtype=float)
if qu.shape[:-2:-1] != (4,): raise ValueError('invalid shape')
if abs(P) != 1: raise ValueError('P ∉ {-1,1}')
qu[...,1:4] *= -P
if accept_homomorph:
qu[qu[...,0]<0.] *= -1.
elif np.any(qu[...,0] < 0.):
raise ValueError('quaternion with negative first (real) component')
if normalize:
qu /= np.linalg.norm(qu,axis=-1,keepdims=True)
elif not np.allclose(np.linalg.norm(qu,axis=-1),1.,rtol=1.e-8):
raise ValueError('quaternion is not of unit length')
return Rotation(qu)
[docs] @staticmethod
def from_Euler_angles(phi: np.ndarray,
degrees: bool = False) -> 'Rotation':
"""
Initialize from Bunge Euler angles.
Parameters
----------
phi : numpy.ndarray, shape (...,3)
Euler angles (φ_1 ∈ [0,2π], ϕ ∈ [0,π], φ_2 ∈ [0,2π])
or (φ_1 ∈ [0,360], ϕ ∈ [0,180], φ_2 ∈ [0,360]) if degrees == True.
degrees : bool, optional
Euler angles are given in degrees. Defaults to False.
Returns
-------
new : damask.Rotation
Notes
-----
Bunge Euler angles correspond to a rotation axis sequence of z–x'–z''.
"""
eu = np.array(phi,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.) or np.any(eu > np.pi*np.array([2.,1.,2.])):
raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π]')
return Rotation(Rotation._eu2qu(eu))
[docs] @staticmethod
def from_axis_angle(n_omega: np.ndarray,
degrees: bool = False,
normalize: bool = False,
P: Literal[1, -1] = -1) -> 'Rotation':
"""
Initialize from axis–angle pair.
Parameters
----------
n_omega : numpy.ndarray, shape (...,4)
Axis and angle (n_1, n_2, n_3, ω) with ǀnǀ = 1 and ω ∈ [0,π]
or ω ∈ [0,180] if degrees == True.
degrees : bool, optional
Angle ω is given in degrees. Defaults to False.
normalize: bool, optional
Allow ǀnǀ ≠ 1. Defaults to False.
P : int ∈ {-1,1}, optional
Sign convention. Defaults to -1.
Returns
-------
new : damask.Rotation
"""
ax = np.array(n_omega,dtype=float)
if ax.shape[:-2:-1] != (4,): raise ValueError('invalid shape')
if abs(P) != 1: raise ValueError('P ∉ {-1,1}')
ax[...,0:3] *= -P
if degrees: ax[..., 3] = np.radians(ax[...,3])
if np.any(ax[...,3] < 0.) or np.any(ax[...,3] > np.pi):
raise ValueError('axis–angle rotation angle outside of [0..π]')
if normalize:
ax[...,0:3] /= np.linalg.norm(ax[...,0:3],axis=-1,keepdims=True)
elif not np.allclose(np.linalg.norm(ax[...,0:3],axis=-1),1.):
raise ValueError('axis–angle rotation axis is not of unit length')
return Rotation(Rotation._ax2qu(ax))
[docs] @staticmethod
def from_basis(basis: np.ndarray,
orthonormal: bool = True,
reciprocal: bool = False) -> 'Rotation':
"""
Initialize from basis vector triplet.
Parameters
----------
basis : numpy.ndarray, shape (...,3,3)
Three three-dimensional basis vectors.
orthonormal : bool, optional
Basis is strictly orthonormal, i.e. is free of stretch components. Defaults to True.
reciprocal : bool, optional
Basis vectors are given in reciprocal (instead of real) space. Defaults to False.
Returns
-------
new : damask.Rotation
"""
om = np.array(basis,dtype=float)
if om.shape[-2:] != (3,3): raise ValueError('invalid shape')
if reciprocal:
om = np.linalg.inv(tensor.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',U,Vh)
elif not np.allclose(np.einsum('...i,...i',om[...,0],om[...,1]),0.) \
or not np.allclose(np.einsum('...i,...i',om[...,1],om[...,2]),0.) \
or not np.allclose(np.einsum('...i,...i',om[...,2],om[...,0]),0.):
raise ValueError('orientation matrix is not orthogonal')
if not np.allclose(np.linalg.det(om),1.):
raise ValueError('orientation matrix has determinant ≠ 1')
return Rotation(Rotation._om2qu(om))
[docs] @staticmethod
def from_matrix(R: np.ndarray,
normalize: bool = False) -> 'Rotation':
"""
Initialize from rotation matrix.
Parameters
----------
R : numpy.ndarray, shape (...,3,3)
Rotation matrix with det(R) = 1 and R.T ∙ R = I.
normalize : bool, optional
Rescales rotation matrix to unit determinant. Defaults to False.
Returns
-------
new : damask.Rotation
"""
return Rotation.from_basis(np.array(R,dtype=float) * (np.linalg.det(R)**(-1./3.))[...,np.newaxis,np.newaxis]
if normalize else
R)
[docs] @staticmethod
def from_parallel(a: np.ndarray,
b: np.ndarray ) -> 'Rotation':
"""
Initialize from pairs of two orthogonal basis vectors.
Parameters
----------
a : numpy.ndarray, shape (...,2,3)
Two three-dimensional vectors of first orthogonal basis.
b : numpy.ndarray, shape (...,2,3)
Corresponding three-dimensional vectors of second basis.
Returns
-------
new : damask.Rotation
"""
a_ = np.array(a,dtype=float)
b_ = np.array(b,dtype=float)
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)))
[docs] @staticmethod
def from_Rodrigues_vector(rho: np.ndarray,
normalize: bool = False,
P: Literal[1, -1] = -1) -> 'Rotation':
"""
Initialize from Rodrigues–Frank vector (with angle separated from axis).
Parameters
----------
rho : numpy.ndarray, shape (...,4)
Rodrigues–Frank vector (n_1, n_2, n_3, tan(ω/2)) with ǀnǀ = 1 and ω ∈ [0,π].
normalize : bool, optional
Allow ǀnǀ ≠ 1. Defaults to False.
P : int ∈ {-1,1}, optional
Sign convention. Defaults to -1.
Returns
-------
new : damask.Rotation
"""
ro = np.array(rho,dtype=float)
if ro.shape[:-2:-1] != (4,): raise ValueError('invalid shape')
if abs(P) != 1: raise ValueError('P ∉ {-1,1}')
ro[...,0:3] *= -P
if np.any(ro[...,3] < 0.): raise ValueError('Rodrigues vector rotation angle is negative')
if normalize:
ro[...,0:3] /= np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True)
elif not np.allclose(np.linalg.norm(ro[...,0:3],axis=-1),1.):
raise ValueError('Rodrigues vector rotation axis is not of unit length')
return Rotation(Rotation._ro2qu(ro))
[docs] @staticmethod
def from_homochoric(h: np.ndarray,
P: Literal[1, -1] = -1) -> 'Rotation':
"""
Initialize from homochoric vector.
Parameters
----------
h : numpy.ndarray, shape (...,3)
Homochoric vector (h_1, h_2, h_3) with ǀhǀ < (3/4*π)^(1/3).
P : int ∈ {-1,1}, optional
Sign convention. Defaults to -1.
Returns
-------
new : damask.Rotation
"""
ho = np.array(h,dtype=float)
if ho.shape[:-2:-1] != (3,): raise ValueError('invalid shape')
if abs(P) != 1: raise ValueError('P ∉ {-1,1}')
ho *= -P
if np.any(np.linalg.norm(ho,axis=-1) > _R1+1.e-9):
raise ValueError('homochoric coordinate outside of the sphere')
return Rotation(Rotation._ho2qu(ho))
[docs] @staticmethod
def from_cubochoric(x: np.ndarray,
P: Literal[1, -1] = -1) -> 'Rotation':
"""
Initialize from cubochoric vector.
Parameters
----------
x : numpy.ndarray, shape (...,3)
Cubochoric vector (x_1, x_2, x_3) with max(x_i) < 1/2*π^(2/3).
P : int ∈ {-1,1}, optional
Sign convention. Defaults to -1.
Returns
-------
new : damask.Rotation
"""
cu = np.array(x,dtype=float)
if cu.shape[:-2:-1] != (3,): raise ValueError('invalid shape')
if abs(P) != 1: raise ValueError('P ∉ {-1,1}')
if np.abs(np.max(cu)) > np.pi**(2./3.) * 0.5+1.e-9:
raise ValueError('cubochoric coordinate outside of the cube')
ho = -P * Rotation._cu2ho(cu)
return Rotation(Rotation._ho2qu(ho))
[docs] @staticmethod
def from_random(shape: Union[None, int, IntSequence] = None,
rng_seed: Optional[NumpyRngSeed] = None) -> 'Rotation':
"""
Initialize with samples from a uniform distribution.
Parameters
----------
shape : (sequence of) int, optional
Shape of the returned array. Defaults to None, which gives a scalar.
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.
Returns
-------
new : damask.Rotation
"""
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)) # type: ignore
A = np.sqrt(r[...,2])
B = np.sqrt(1.-r[...,2])
q = np.stack([np.cos(2.*np.pi*r[...,0])*A,
np.sin(2.*np.pi*r[...,1])*B,
np.cos(2.*np.pi*r[...,1])*B,
np.sin(2.*np.pi*r[...,0])*A],axis=-1)
return Rotation(q if shape is None else q.reshape(r.shape[:-1]+(4,)))._standardize()
[docs] @staticmethod
def from_ODF(weights: np.ndarray,
phi: np.ndarray,
shape: Union[None, int, IntSequence] = None,
degrees: bool = False,
fractions: bool = True,
rng_seed: Optional[NumpyRngSeed] = None) -> 'Rotation':
"""
Initialize with samples from a binned orientation distribution function (ODF).
Parameters
----------
weights : numpy.ndarray, shape (n)
Texture intensity values (probability density or volume fraction) at Euler space grid points.
phi : numpy.ndarray, shape (n,3)
Grid coordinates in Euler space at which weights are defined.
shape : (sequence of) int, optional
Shape of the returned array. Defaults to None, which gives a scalar.
degrees : bool, optional
Euler space grid coordinates are in degrees. Defaults to True.
fractions : bool, optional
ODF values correspond to volume fractions, not probability densities.
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.
Returns
-------
new : damask.Rotation
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 a cell-centered version, which avoids grid points at ϕ = 0.
References
----------
P. Eisenlohr and 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.*np.sin(delta[1]/2.)*delta[2] / 8. / np.pi**2 * np.sin(np.radians(eu[:,1]) if deg else eu[:,1])
dg = 1. if fractions else _dg(phi,degrees)
dV_V = dg * np.maximum(0.,weights.squeeze())
N = 1 if shape is None else np.prod(shape).astype(int)
return Rotation.from_Euler_angles(phi[util.hybrid_IA(dV_V,N,rng_seed)],degrees).reshape(() if shape is None else shape)
[docs] @staticmethod
def from_spherical_component(center: 'Rotation',
sigma: float,
shape: Union[None, int, IntSequence] = None,
degrees: bool = False,
rng_seed: Optional[NumpyRngSeed] = None) -> 'Rotation':
"""
Initialize with samples from a Gaussian distribution around a given center.
Parameters
----------
center : Rotation or Orientation
Central rotation.
sigma : float
Standard deviation of (Gaussian) misorientation distribution.
shape : (sequence of) int, optional
Shape of the returned array. Defaults to None, which gives a scalar.
degrees : bool, optional
sigma is given in degrees. Defaults to False.
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.
Examples
--------
Create a brass texture consisting of
200 orientations:
>>> import damask
>>> center = damask.Rotation.from_Euler_angles([35.,45.,0.],degrees=True)
>>> brass = damask.Rotation.from_spherical_component(center=center,sigma=3.,shape=200,degrees=True)
Create a Goss texture consisting of
100 orientations:
>>> import damask
>>> center = damask.Rotation.from_Euler_angles([0.,45.,0.],degrees=True)
>>> goss = damask.Rotation.from_spherical_component(center=center,sigma=3.,shape=100,degrees=True)
"""
rng = np.random.default_rng(rng_seed)
sigma = np.radians(sigma) if degrees else sigma
N = 1 if shape is None else np.prod(shape)
u,Theta = (rng.random((N,2)) * 2. * np.array([1.,np.pi]) - np.array([1.,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])
return Rotation.from_axis_angle(p).reshape(() if shape is None else shape) * center
[docs] @staticmethod
def from_fiber_component(crystal: IntSequence,
sample: IntSequence,
sigma: float = 0.,
shape: Union[None, int, IntSequence] = None,
degrees: bool = False,
rng_seed: Optional[NumpyRngSeed] = None) -> 'Rotation':
"""
Initialize with samples from a Gaussian distribution around a given direction.
Parameters
----------
crystal : numpy.ndarray, shape (2)
Polar coordinates (polar angle θ from [0 0 1], azimuthal angle φ from [1 0 0])
of fiber direction in crystal frame.
sample : numpy.ndarray, shape (2)
Polar coordinates (polar angle θ from z, azimuthal angle φ from x)
of fiber direction in sample frame.
sigma : float, optional
Standard deviation of (Gaussian) misorientation distribution.
Defaults to 0.
shape : (sequence of) int, optional
Shape of the returned array. Defaults to None, which gives a scalar.
degrees : bool, optional
sigma and polar coordinates are given in degrees. Defaults to False.
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.
Returns
-------
new : damask.Rotation
Notes
-----
The crystal direction for (θ=0,φ=0) is [0 0 1],
the sample direction for (θ=0,φ=0) is z.
Polar coordinates follow the ISO 80000-2:2019 convention
typically used in physics.
See https://en.wikipedia.org/wiki/Spherical_coordinate_system.
Ranges 0≤θ≤π and 0≤φ≤2π give a unique set of coordinates.
Examples
--------
Create an ideal α-fiber texture (<1 1 0> ǀǀ RD=x) consisting of
200 orientations:
>>> import damask
>>> import numpy as np
>>> alpha = damask.Rotation.from_fiber_component([np.pi/4.,0.],[np.pi/2.,0.],shape=200)
Create an ideal γ-fiber texture (<1 1 1> ǀǀ ND=z) consisting of
100 orientations:
>>> import damask
>>> gamma = damask.Rotation.from_fiber_component([54.7,45.],[0.,0.],shape=100,degrees=True)
"""
rng = np.random.default_rng(rng_seed)
sigma_,alpha,beta = (np.radians(coordinate) for coordinate in (sigma,crystal,sample)) if degrees else \
map(np.array, (sigma,crystal,sample))
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.): ax_align[:3] = np.array([1.,0.,0.])
R_align = Rotation.from_axis_angle(ax_align if ax_align[3] > 0. else -ax_align,normalize=True) # rotate fiber axis from sample to crystal frame
N = 1 if shape is None else np.prod(shape).astype(int)
u,Theta = (rng.random((N,2)) * 2. * np.array([1.,np.pi]) - np.array([1.,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])
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)).reshape(() if shape is None else shape)
####################################################################################################
# Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations
####################################################################################################
# Copyright (c) 2017-2020, 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: np.ndarray) -> np.ndarray:
qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2)
om = np.block([qq + 2.*qu[...,1:2]**2,
2.*(qu[...,2:3]*qu[...,1:2]-_P*qu[...,0:1]*qu[...,3:4]),
2.*(qu[...,3:4]*qu[...,1:2]+_P*qu[...,0:1]*qu[...,2:3]),
2.*(qu[...,1:2]*qu[...,2:3]+_P*qu[...,0:1]*qu[...,3:4]),
qq + 2.*qu[...,2:3]**2,
2.*(qu[...,3:4]*qu[...,2:3]-_P*qu[...,0:1]*qu[...,1:2]),
2.*(qu[...,1:2]*qu[...,3:4]-_P*qu[...,0:1]*qu[...,2:3]),
2.*(qu[...,2:3]*qu[...,3:4]+_P*qu[...,0:1]*qu[...,1:2]),
qq + 2.*qu[...,3:4]**2,
]).reshape(qu.shape[:-1]+(3,3))
return om
@staticmethod
def _qu2eu(qu: np.ndarray) -> np.ndarray:
"""
Quaternion to Bunge Euler angles.
References
----------
E. Bernardes and S. Viollet, PLoS ONE 17(11):e0276302, 2022
https://doi.org/10.1371/journal.pone.0276302
"""
a = qu[...,0:1]
b = -_P*qu[...,3:4]
c = -_P*qu[...,1:2]
d = -_P*qu[...,2:3]
eu = np.block([
np.arctan2(b,a),
np.arccos(2*(a**2+b**2)/(a**2+b**2+c**2+d**2)-1),
np.arctan2(-d,c),
])
eu_sum = eu[...,0] + eu[...,2]
eu_diff = eu[...,0] - eu[...,2]
is_zero = np.isclose(eu[...,1],0.0)
is_pi = np.isclose(eu[...,1],np.pi)
is_ok = ~np.logical_or(is_zero,is_pi)
eu[...,0][is_zero] = 2*eu[...,0][is_zero]
eu[...,0][is_pi] = -2*eu[...,2][is_pi]
eu[...,2][~is_ok] = 0.0
eu[...,0][is_ok] = eu_diff[is_ok]
eu[...,2][is_ok] = eu_sum [is_ok]
eu[np.logical_or(np.abs(eu) < 1.e-6,
np.abs(eu-2*np.pi) < 1.e-6)] = 0.
return np.where(eu < 0., eu%(np.pi*np.array([2.,1.,2.])),eu)
@staticmethod
def _qu2ax(qu: np.ndarray) -> np.ndarray:
"""
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. * np.arccos(np.clip(qu[...,0:1],-1.,1.))
ax = np.where(np.broadcast_to(qu[...,0:1] < 1.e-8,qu.shape),
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.)] = np.array([0.,0.,1.,0.])
return ax
@staticmethod
def _qu2ro(qu: np.ndarray) -> np.ndarray:
"""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.e-12,qu.shape),
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.,1.)))
])
)
ro[np.abs(s).squeeze(-1) < 1.e-12] = np.array([0.,0.,_P,0.])
return ro
@staticmethod
def _qu2ho(qu: np.ndarray) -> np.ndarray:
"""Quaternion to homochoric vector."""
with np.errstate(invalid='ignore'):
omega = 2. * np.arccos(np.clip(qu[...,0:1],-1.,1.))
ho = np.where(np.abs(omega) < 1.e-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: np.ndarray) -> np.ndarray:
"""Quaternion to cubochoric vector."""
return Rotation._ho2cu(Rotation._qu2ho(qu))
#---------- Rotation matrix ----------
@staticmethod
def _om2qu(om: np.ndarray) -> np.ndarray:
"""
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 = np.array([
0.5 / np.sqrt( 1. + trace),
2. * np.sqrt( 1. + om[...,0,0:1] - om[...,1,1:2] - om[...,2,2:3]),
2. * np.sqrt( 1. + om[...,1,1:2] - om[...,2,2:3] - om[...,0,0:1]),
2. * np.sqrt( 1. + 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: np.ndarray) -> np.ndarray:
"""Rotation matrix to Bunge Euler angles."""
with np.errstate(invalid='ignore',divide='ignore'):
zeta = 1./np.sqrt(1.-om[...,2,2:3]**2)
eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.,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
return np.where(eu < 0., eu%(np.pi*np.array([2.,1.,2.])),eu)
@staticmethod
def _om2ax(om: np.ndarray) -> np.ndarray:
"""Rotation matrix to axis–angle pair."""
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.).reshape(om.shape[:-2]+(1,))
w,vr = np.linalg.eig(om)
# mask duplicated real eigenvalues
w[np.isclose(w[...,0],1.+0.j),1:] = 0.
w[np.isclose(w[...,1],1.+0.j),2:] = 0.
vr = np.swapaxes(vr,-1,-2)
ax = np.where(np.abs(diag_delta)<1.e-13,
np.real(vr[np.isclose(w,1.+0.j)]).reshape(om.shape[:-2]+(3,)),
np.abs(np.real(vr[np.isclose(w,1.+0.j)]).reshape(om.shape[:-2]+(3,)))
*np.sign(diag_delta))
ax = np.block([ax,np.arccos(np.clip(t,-1.,1.))])
ax[np.abs(ax[...,3]) < 1.e-8] = np.array([0.,0.,1.,0.])
return ax
@staticmethod
def _om2ro(om: np.ndarray) -> np.ndarray:
"""Rotation matrix to Rodrigues–Frank vector."""
return Rotation._eu2ro(Rotation._om2eu(om))
@staticmethod
def _om2ho(om: np.ndarray) -> np.ndarray:
"""Rotation matrix to homochoric vector."""
return Rotation._ax2ho(Rotation._om2ax(om))
@staticmethod
def _om2cu(om: np.ndarray) -> np.ndarray:
"""Rotation matrix to cubochoric vector."""
return Rotation._ho2cu(Rotation._om2ho(om))
#---------- Bunge Euler angles ----------
@staticmethod
def _eu2qu(eu: np.ndarray) -> np.ndarray:
"""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.] *= -1.
return qu
@staticmethod
def _eu2om(eu: np.ndarray) -> np.ndarray:
"""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.
return om
@staticmethod
def _eu2ax(eu: np.ndarray) -> np.ndarray:
"""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.*np.arctan(tau/np.cos(sigma)))
with np.errstate(invalid='ignore',divide='ignore'):
ax = np.where(np.broadcast_to(np.abs(alpha)<1.e-12,eu.shape[:-1]+(4,)),
[0.,0.,1.,0.],
np.block([-_P/tau*t*np.cos(delta),
-_P/tau*t*np.sin(delta),
-_P/tau* np.sin(sigma),
alpha
]))
ax[(alpha<0.).squeeze()] *= -1.
return ax
@staticmethod
def _eu2ro(eu: np.ndarray) -> np.ndarray:
"""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] = np.array([0.,0.,_P,0.])
return ro
@staticmethod
def _eu2ho(eu: np.ndarray) -> np.ndarray:
"""Bunge Euler angles to homochoric vector."""
return Rotation._ax2ho(Rotation._eu2ax(eu))
@staticmethod
def _eu2cu(eu: np.ndarray) -> np.ndarray:
"""Bunge Euler angles to cubochoric vector."""
return Rotation._ho2cu(Rotation._eu2ho(eu))
#---------- Axis angle pair ----------
@staticmethod
def _ax2qu(ax: np.ndarray) -> np.ndarray:
"""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.],np.block([c,ax[...,:3]*s]))
return qu
@staticmethod
def _ax2om(ax: np.ndarray) -> np.ndarray:
"""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. else np.swapaxes(om,-1,-2)
@staticmethod
def _ax2eu(ax: np.ndarray) -> np.ndarray:
"""Rotation matrix to Bunge Euler angles."""
return Rotation._om2eu(Rotation._ax2om(ax))
@staticmethod
def _ax2ro(ax: np.ndarray) -> np.ndarray:
"""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] = np.array([.0,.0,_P,.0])
return ro
@staticmethod
def _ax2ho(ax: np.ndarray) -> np.ndarray:
"""Axis–angle pair to homochoric vector."""
f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1./3.)
return ax[...,:3] * f
@staticmethod
def _ax2cu(ax: np.ndarray) -> np.ndarray:
"""Axis–angle pair to cubochoric vector."""
return Rotation._ho2cu(Rotation._ax2ho(ax))
#---------- Rodrigues-Frank vector ----------
@staticmethod
def _ro2qu(ro: np.ndarray) -> np.ndarray:
"""Rodrigues–Frank vector to quaternion."""
return Rotation._ax2qu(Rotation._ro2ax(ro))
@staticmethod
def _ro2om(ro: np.ndarray) -> np.ndarray:
"""Rodgrigues–Frank vector to rotation matrix."""
return Rotation._ax2om(Rotation._ro2ax(ro))
@staticmethod
def _ro2eu(ro: np.ndarray) -> np.ndarray:
"""Rodrigues–Frank vector to Bunge Euler angles."""
return Rotation._om2eu(Rotation._ro2om(ro))
@staticmethod
def _ro2ax(ro: np.ndarray) -> np.ndarray:
"""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.,1.,0.])
return ax
@staticmethod
def _ro2ho(ro: np.ndarray) -> np.ndarray:
"""Rodrigues–Frank vector to homochoric vector."""
f = np.where(np.isfinite(ro[...,3:4]),2.*np.arctan(ro[...,3:4]) -np.sin(2.*np.arctan(ro[...,3:4])),np.pi)
return np.where(np.broadcast_to(np.sum(ro[...,0:3]**2,axis=-1,keepdims=True) < 1.e-8,ro[...,0:3].shape),
np.zeros(3), ro[...,0:3]* (0.75*f)**(1./3.))
@staticmethod
def _ro2cu(ro: np.ndarray) -> np.ndarray:
"""Rodrigues–Frank vector to cubochoric vector."""
return Rotation._ho2cu(Rotation._ro2ho(ro))
#---------- Homochoric vector----------
@staticmethod
def _ho2qu(ho: np.ndarray) -> np.ndarray:
"""Homochoric vector to quaternion."""
return Rotation._ax2qu(Rotation._ho2ax(ho))
@staticmethod
def _ho2om(ho: np.ndarray) -> np.ndarray:
"""Homochoric vector to rotation matrix."""
return Rotation._ax2om(Rotation._ho2ax(ho))
@staticmethod
def _ho2eu(ho: np.ndarray) -> np.ndarray:
"""Homochoric vector to Bunge Euler angles."""
return Rotation._ax2eu(Rotation._ho2ax(ho))
@staticmethod
def _ho2ax(ho: np.ndarray) -> np.ndarray:
"""Homochoric vector to axis–angle pair."""
tfit = np.array([+0.9999999999999968, -0.49999999999986866, -0.025000000000632055,
-0.003928571496460683, -0.0008164666077062752, -0.00019411896443261646,
-0.00004985822229871769, -0.000014164962366386031, -1.9000248160936107e-6,
-5.72184549898506e-6, +7.772149920658778e-6, -0.00001053483452909705,
+9.528014229335313e-6, -5.660288876265125e-6, +1.2844901692764126e-6,
+1.1255185726258763e-6, -1.3834391419956455e-6, +7.513691751164847e-7,
-2.401996891720091e-7, +4.386887017466388e-8, -3.5917775353564864e-9])
hmag_squared = np.sum(ho**2,axis=-1,keepdims=True)
s = np.sum(tfit*hmag_squared**np.arange(len(tfit)),axis=-1,keepdims=True)
with np.errstate(invalid='ignore'):
return np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-8,ho.shape[:-1]+(4,)),
[0.,0.,1.,0.],
np.block([ho/np.sqrt(hmag_squared),2.*np.arccos(np.clip(s,-1.,1.))]))
@staticmethod
def _ho2ro(ho: np.ndarray) -> np.ndarray:
"""Axis–angle pair to Rodrigues–Frank vector."""
return Rotation._ax2ro(Rotation._ho2ax(ho))
@staticmethod
def _ho2cu(ho: np.ndarray) -> np.ndarray:
"""
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.*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.)/_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.)/qxy,-1.,1.)
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.]),
np.block([np.arccos(tt)/np.pi*12.,np.ones_like(tt)]))*q
T_inv[xyz2<0.] *= -1.
T_inv[np.broadcast_to(np.isclose(qxy,0.,rtol=0.,atol=1.e-12),T_inv.shape)] = 0.
cu = np.block([T_inv, np.where(xyz3[...,2:3]<0.,-np.ones_like(xyz3[...,2:3]),np.ones_like(xyz3[...,2:3])) \
* rs/np.sqrt(6./np.pi),
])/ _sc
cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.,rtol=0.,atol=1.e-16)] = 0.
return np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1)
#---------- Cubochoric ----------
@staticmethod
def _cu2qu(cu: np.ndarray) -> np.ndarray:
"""Cubochoric vector to quaternion."""
return Rotation._ho2qu(Rotation._cu2ho(cu))
@staticmethod
def _cu2om(cu: np.ndarray) -> np.ndarray:
"""Cubochoric vector to rotation matrix."""
return Rotation._ho2om(Rotation._cu2ho(cu))
@staticmethod
def _cu2eu(cu: np.ndarray) -> np.ndarray:
"""Cubochoric vector to Bunge Euler angles."""
return Rotation._ho2eu(Rotation._cu2ho(cu))
@staticmethod
def _cu2ax(cu: np.ndarray) -> np.ndarray:
"""Cubochoric vector to axis–angle pair."""
return Rotation._ho2ax(Rotation._cu2ho(cu))
@staticmethod
def _cu2ro(cu: np.ndarray) -> np.ndarray:
"""Cubochoric vector to Rodrigues–Frank vector."""
return Rotation._ho2ro(Rotation._cu2ho(cu))
@staticmethod
def _cu2ho(cu: np.ndarray) -> np.ndarray:
"""
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 pyramid 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. * 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.**0.25/_beta/ np.sqrt(np.sqrt(2.)-c) \
* np.where(order,XYZ[...,0:1],XYZ[...,1:2])
T = np.block([(np.sqrt(2.)*c - 1.), np.sqrt(2.) * s]) * q
# transform to sphere grid (inverse Lambert)
c = np.sum(T**2,axis=-1,keepdims=True)
s = c * np.pi/24. /XYZ[...,2:3]**2
c = c * np.sqrt(np.pi/24.)/XYZ[...,2:3]
q = np.sqrt( 1. - s)
ho = np.where(np.isclose(np.sum(np.abs(XYZ[...,0:2]),axis=-1,keepdims=True),0.,rtol=0.,atol=1.e-16),
np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6./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./np.pi) * XYZ[...,2:3] - c])
)
ho[np.isclose(np.sum(np.abs(cu),axis=-1),0.,rtol=0.,atol=1.e-16)] = 0.
return np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1)
@staticmethod
def _get_pyramid_order(xyz: np.ndarray,
direction: Literal['forward', 'backward']) -> np.ndarray:
"""
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]