1842 lines
68 KiB
Python
1842 lines
68 KiB
Python
import copy
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from typing import Optional, Union, Sequence, Tuple, Literal, List, TypeVar
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import numpy as np
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from ._typehints import FloatSequence, IntSequence, NumpyRngSeed
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from . import tensor
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from . import util
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from . import grid_filters
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_P = -1
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# parameters for conversion from/to cubochoric
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_sc = np.pi**(1./6.)/6.**(1./6.)
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_beta = np.pi**(5./6.)/6.**(1./6.)/2.
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_R1 = (3.*np.pi/4.)**(1./3.)
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MyType = TypeVar('MyType', bound='Rotation')
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class Rotation:
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u"""
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Rotation with functionality for conversion between different representations.
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The following conventions apply:
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- Coordinate frames are right-handed.
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- 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.
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- Rotations will be interpreted in the passive sense, i.e. as rotation of
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the coordinate frame.
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- P = -1 (as default).
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Examples
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--------
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Rotate vector 'a' (defined in coordinate system 'A') to
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coordinates 'b' expressed in system 'B':
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>>> import numpy as np
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>>> import damask
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>>> Q = damask.Rotation.from_random()
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>>> a = np.random.rand(3)
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>>> b = Q @ a
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>>> np.allclose(np.dot(Q.as_matrix(),a),b)
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True
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Compound rotations R1 (first) and R2 (second):
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>>> import numpy as np
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>>> import damask
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>>> R1 = damask.Rotation.from_random()
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>>> R2 = damask.Rotation.from_random()
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>>> R = R2 * R1
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>>> np.allclose(R.as_matrix(), np.dot(R2.as_matrix(),R1.as_matrix()))
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True
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References
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----------
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D. Rowenhorst et al., Modelling and Simulation in Materials Science and Engineering 23:083501, 2015
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https://doi.org/10.1088/0965-0393/23/8/083501
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"""
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__slots__ = ['quaternion']
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def __init__(self,
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rotation: Union[FloatSequence, 'Rotation'] = np.array([1.,0.,0.,0.])):
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"""
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New rotation.
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Parameters
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----------
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rotation : list, numpy.ndarray, or Rotation, optional
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Unit quaternion in positive real hemisphere.
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Use .from_quaternion to perform a sanity check.
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Defaults to no rotation.
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"""
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self.quaternion: np.ndarray
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if isinstance(rotation,Rotation):
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self.quaternion = rotation.quaternion.copy()
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elif np.array(rotation).shape[-1] == 4:
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self.quaternion = np.array(rotation,dtype=float)
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else:
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raise TypeError('"rotation" is neither a Rotation nor a quaternion')
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def __repr__(self) -> str:
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"""
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Return repr(self).
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Give short, human-readable summary.
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"""
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return f'Quaternion{" " if self.quaternion.shape == (4,) else "s of shape "+str(self.quaternion.shape[:-1])+chr(10)}'\
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+ str(self.quaternion)
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def __copy__(self: MyType,
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rotation: Union[None, FloatSequence, 'Rotation'] = None) -> MyType:
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"""
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Return deepcopy(self).
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Create deep copy.
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"""
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dup = copy.deepcopy(self)
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if rotation is not None:
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dup.quaternion = Rotation(rotation).quaternion
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return dup
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copy = __copy__
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def __getitem__(self,
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item: Union[Tuple[int], int, bool, np.bool_, np.ndarray]):
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"""
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Return self[item].
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Return slice according to item.
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"""
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return self.copy() if self.shape == () else \
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self.copy(self.quaternion[item+(slice(None),)] if isinstance(item,tuple) else self.quaternion[item])
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def __eq__(self,
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other: object) -> bool:
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"""
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Return self==other.
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Test equality of other.
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Parameters
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----------
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other : Rotation
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Rotation to check for equality.
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"""
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return NotImplemented if not isinstance(other, Rotation) else \
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np.logical_or(np.all(self.quaternion == other.quaternion,axis=-1),
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np.all(self.quaternion == -1.*other.quaternion,axis=-1))
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def __ne__(self,
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other: object) -> bool:
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"""
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Return self!=other.
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Test inequality of other.
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Parameters
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----------
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other : Rotation
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Rotation to check for inequality.
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"""
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return np.logical_not(self==other) if isinstance(other, Rotation) else NotImplemented
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def isclose(self: MyType,
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other: MyType,
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rtol: float = 1.e-5,
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atol: float = 1.e-8,
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equal_nan: bool = True) -> bool:
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"""
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Report where values are approximately equal to corresponding ones of other Rotation.
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Parameters
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----------
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other : Rotation
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Rotation to compare against.
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rtol : float, optional
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Relative tolerance of equality.
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atol : float, optional
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Absolute tolerance of equality.
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equal_nan : bool, optional
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Consider matching NaN values as equal. Defaults to True.
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Returns
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-------
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mask : numpy.ndarray of bool, shape (self.shape)
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Mask indicating where corresponding rotations are close.
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"""
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s = self.quaternion
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o = other.quaternion
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return np.logical_or(np.all(np.isclose(s, o,rtol,atol,equal_nan),axis=-1),
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np.all(np.isclose(s,-1.*o,rtol,atol,equal_nan),axis=-1))
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def allclose(self: MyType,
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other: MyType,
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rtol: float = 1.e-5,
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atol: float = 1.e-8,
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equal_nan: bool = True) -> Union[np.bool_, bool]:
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"""
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Test whether all values are approximately equal to corresponding ones of other Rotation.
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Parameters
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----------
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other : Rotation
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Rotation to compare against.
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rtol : float, optional
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Relative tolerance of equality.
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atol : float, optional
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Absolute tolerance of equality.
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equal_nan : bool, optional
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Consider matching NaN values as equal. Defaults to True.
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Returns
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-------
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answer : bool
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Whether all values are close between both rotations.
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"""
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return np.all(self.isclose(other,rtol,atol,equal_nan))
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def __array__(self):
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"""Initializer for numpy."""
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return self.quaternion
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@property
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def size(self) -> int:
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return self.quaternion[...,0].size
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@property
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def shape(self) -> Tuple[int, ...]:
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return self.quaternion[...,0].shape
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def __len__(self) -> int:
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"""
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Return len(self).
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Length of leading/leftmost dimension of array.
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"""
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return 0 if self.shape == () else self.shape[0]
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def __invert__(self: MyType) -> MyType:
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"""
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Return ~self.
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Inverse rotation (backward rotation).
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"""
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dup = self.copy()
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dup.quaternion[...,1:] *= -1.
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return dup
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def __pow__(self: MyType,
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exp: Union[float, int]) -> MyType:
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"""
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Return self**exp.
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Perform the rotation 'exp' times.
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Parameters
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----------
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exp : float
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Exponent.
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"""
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phi = np.arccos(self.quaternion[...,0:1])
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p = self.quaternion[...,1:]/np.linalg.norm(self.quaternion[...,1:],axis=-1,keepdims=True)
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return self.copy(Rotation(np.block([np.cos(exp*phi),np.sin(exp*phi)*p]))._standardize())
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def __ipow__(self: MyType,
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exp: Union[float, int]) -> MyType:
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"""
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Return self**=exp.
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Perform the rotation 'exp' times (in-place).
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Parameters
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----------
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exp : float
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Exponent.
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"""
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return self**exp
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def __mul__(self: MyType,
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other: MyType) -> MyType:
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"""
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Return self*other.
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Compose with other.
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Parameters
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----------
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other : Rotation, shape (self.shape)
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Rotation for composition.
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Returns
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-------
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composition : Rotation
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Compound rotation self*other, i.e. first other then self rotation.
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"""
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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:]
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qmo = q_m*q_o
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q = (qmo - np.einsum('...i,...i',p_m,p_o).reshape(qmo.shape))
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p = q_m*p_o + q_o*p_m + _P * np.cross(p_m,p_o)
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return self.copy(Rotation(np.block([q,p]))._standardize())
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else:
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raise TypeError('use "R@b", i.e. matmul, to apply rotation "R" to object "b"')
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def __imul__(self: MyType,
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other: MyType) -> MyType:
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"""
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Return self*=other.
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Compose with other (in-place).
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Parameters
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----------
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other : Rotation, shape (self.shape)
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Rotation for composition.
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"""
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return self*other
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def __truediv__(self: MyType,
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other: MyType) -> MyType:
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"""
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Return self/other.
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Compose with inverse of other.
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Parameters
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----------
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other : damask.Rotation, shape (self.shape)
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Rotation to invert for composition.
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Returns
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-------
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composition : Rotation
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Compound rotation self*(~other), i.e. first inverse of other then self rotation.
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"""
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if isinstance(other,Rotation):
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return self*~other
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else:
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raise TypeError('use "R@b", i.e. matmul, to apply rotation "R" to object "b"')
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def __itruediv__(self: MyType,
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other: MyType) -> MyType:
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"""
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Return self/=other.
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Compose with inverse of other (in-place).
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Parameters
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----------
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other : Rotation, shape (self.shape)
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Rotation to invert for composition.
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"""
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return self/other
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def __matmul__(self,
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other: np.ndarray) -> np.ndarray:
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"""
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Return self@other.
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Rotate vector, second-order tensor, or fourth-order tensor.
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Parameters
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----------
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other : numpy.ndarray, shape (...,3), (...,3,3), or (...,3,3,3,3)
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Vector or tensor on which to apply the rotation.
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Returns
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-------
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rotated : numpy.ndarray, shape (...,3), (...,3,3), or (...,3,3,3,3)
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Rotated vector or tensor, i.e. transformed to frame defined by rotation.
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"""
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if isinstance(other, np.ndarray):
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if self.shape + (3,) == other.shape:
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q_m = self.quaternion[...,0]
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p_m = self.quaternion[...,1:]
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A = q_m**2 - np.einsum('...i,...i',p_m,p_m)
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B = 2. * np.einsum('...i,...i',p_m,other)
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C = 2. * _P * q_m
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return np.block([(A * other[...,i]).reshape(self.shape+(1,)) +
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(B * p_m[...,i]).reshape(self.shape+(1,)) +
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(C * ( p_m[...,(i+1)%3]*other[...,(i+2)%3]\
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- p_m[...,(i+2)%3]*other[...,(i+1)%3])).reshape(self.shape+(1,))
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for i in [0,1,2]])
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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)
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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, second-order tensors, and fourth-order tensors')
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elif isinstance(other, Rotation):
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raise TypeError('use "R1*R2", i.e. multiplication, to compose rotations "R1" and "R2"')
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else:
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raise TypeError(f'cannot rotate "{type(other)}"')
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apply = __matmul__
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def _standardize(self: MyType) -> MyType:
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"""Standardize quaternion (ensure positive real hemisphere)."""
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self.quaternion[self.quaternion[...,0] < 0.] *= -1.
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return self
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def append(self: MyType,
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other: Union[MyType, List[MyType]]) -> MyType:
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"""
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Extend array along first dimension with other array(s).
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Parameters
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||
----------
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other : (list of) damask.Rotation
|
||
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"""
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return self.copy(np.vstack(tuple(map(lambda x:x.quaternion,
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[self]+other if isinstance(other,list) else [self,other]))))
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def flatten(self: MyType,
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order: Literal['C','F','A'] = 'C') -> MyType:
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"""
|
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Flatten array.
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||
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Parameters
|
||
----------
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order : {'C', 'F', 'A'}, optional
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'C' flattens in row-major (C-style) order.
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'F' flattens in column-major (Fortran-style) order.
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'A' flattens in column-major order if object is Fortran contiguous in memory,
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||
row-major order otherwise.
|
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Defaults to 'C'.
|
||
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Returns
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||
-------
|
||
flattened : damask.Rotation
|
||
Rotation flattened to single dimension.
|
||
|
||
"""
|
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return self.copy(self.quaternion.reshape((-1,4),order=order))
|
||
|
||
|
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def reshape(self: MyType,
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shape: Union[int, IntSequence],
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||
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))
|
||
|
||
|
||
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,)))
|
||
|
||
|
||
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))
|
||
|
||
|
||
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)
|
||
|
||
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()
|
||
|
||
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
|
||
|
||
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
|
||
|
||
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)
|
||
|
||
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
|
||
|
||
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)
|
||
|
||
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.
|
||
@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)
|
||
|
||
@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))
|
||
|
||
@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))
|
||
|
||
@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))
|
||
|
||
@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)
|
||
|
||
@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)))
|
||
|
||
|
||
@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))
|
||
|
||
@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))
|
||
|
||
@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))
|
||
|
||
|
||
@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()
|
||
|
||
|
||
@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)
|
||
|
||
|
||
@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
|
||
|
||
|
||
@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]
|