1527 lines
57 KiB
Python
1527 lines
57 KiB
Python
import numpy as np
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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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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.
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- Euler angle triplets are implemented using the Bunge convention,
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with angular ranges of [0,2π], [0,π], [0,2π].
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- The rotation angle ω is limited to the interval [0,π].
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- The real part of a quaternion is positive, Re(q) > 0
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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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- b = Q @ a
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- b = np.dot(Q.as_matrix(),a)
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Compound rotations R1 (first) and R2 (second):
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- R = R2 * R1
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- R = Rotation.from_matrix(np.dot(R2.as_matrix(),R1.as_matrix())
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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,rotation = np.array([1.0,0.0,0.0,0.0])):
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"""
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Initialize rotation object.
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Parameters
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----------
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rotation : list, numpy.ndarray, 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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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)
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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):
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"""Represent rotation as unit quaternion(s)."""
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return f'Quaternion{" " if self.quaternion.shape == (4,) else "s of shape "+str(self.quaternion.shape)+chr(10)}'\
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+ str(self.quaternion)
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def __copy__(self,**kwargs):
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"""Create deep copy."""
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return self.__class__(rotation=kwargs['rotation'] if 'rotation' in kwargs else self.quaternion)
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copy = __copy__
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def __getitem__(self,item):
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"""Return slice according to item."""
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return self.copy() \
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if self.shape == () else \
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self.copy(rotation=self.quaternion[item+(slice(None),)] if isinstance(item,tuple) else self.quaternion[item])
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def __eq__(self,other):
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"""
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Equal to other.
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Equality is determined taking limited floating point precision into account.
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See numpy.allclose for details.
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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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s = self.quaternion
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o = other.quaternion
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if self.shape == () == other.shape:
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return np.allclose(s,o) or (np.isclose(s[0],0.0) and np.allclose(s,-1.0*o))
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else:
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return np.all(np.isclose(s,o),-1) + np.all(np.isclose(s,-1.0*o),-1) * np.isclose(s[...,0],0.0)
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def __ne__(self,other):
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"""
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Not equal to other.
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Equality is determined taking limited floating point precision into
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account. See numpy.allclose for details.
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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 np.logical_not(self==other)
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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):
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return self.quaternion[...,0].size
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@property
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def shape(self):
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return self.quaternion[...,0].shape
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def __len__(self):
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"""Length of leading/leftmost dimension of Rotation array."""
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return 0 if self.shape == () else self.shape[0]
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def __invert__(self):
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"""Inverse rotation (backward rotation)."""
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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,exp):
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"""
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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=Rotation(np.block([np.cos(exp*phi),np.sin(exp*phi)*p]))._standardize())
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def __ipow__(self,exp):
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"""
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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,other):
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"""
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Compose this rotation with other.
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Parameters
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----------
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other : Rotation of 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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q = (q_m*q_o - np.einsum('...i,...i',p_m,p_o).reshape(self.shape+(1,)))
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p = q_m*p_o + q_o*p_m + _P * np.cross(p_m,p_o)
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return 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,other):
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"""
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Compose this rotation with other (in-place).
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Parameters
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----------
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other : Rotation of 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,other):
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"""
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Compose this rotation with inverse of other.
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Parameters
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----------
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other : damask.Rotation of shape (self.shape)
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Rotation to inverse 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,other):
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"""
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Compose this rotation with inverse of other (in-place).
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Parameters
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----------
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other : Rotation of shape (self.shape)
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Rotation to inverse composition.
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"""
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return self/other
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def __matmul__(self,other):
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"""
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Rotation of vector, second order tensor, or fourth order tensor.
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Parameters
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----------
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other : numpy.ndarray of 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 of 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**2.0 - np.einsum('...i,...i',p_m,p_m)
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B = 2.0 * np.einsum('...i,...i',p_m,other)
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C = 2.0 * _P * q_m
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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, 2nd order tensors, and 4th 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):
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"""Standardize quaternion (ensure positive real hemisphere)."""
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self.quaternion[self.quaternion[...,0] < 0.0] *= -1
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return self
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def append(self,other):
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"""
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Extend rotation array along first dimension with other array(s).
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Parameters
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----------
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other : Rotation or list of Rotations.
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"""
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return self.copy(rotation=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,order = 'C'):
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"""Flatten quaternion array."""
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return self.copy(rotation=self.quaternion.reshape((-1,4),order=order))
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def reshape(self,shape,order = 'C'):
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"""Reshape quaternion array."""
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if isinstance(shape,(int,np.integer)): shape = (shape,)
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return self.copy(rotation=self.quaternion.reshape(tuple(shape)+(4,),order=order))
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def broadcast_to(self,shape,mode = 'right'):
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"""
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Broadcast quaternion array to shape.
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Parameters
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----------
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shape : tuple
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Shape of broadcasted array.
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mode : str, optional
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Where to preferentially locate missing dimensions.
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Either 'left' or 'right' (default).
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"""
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if isinstance(shape,(int,np.integer)): shape = (shape,)
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return self.copy(rotation=np.broadcast_to(self.quaternion.reshape(util.shapeshifter(self.shape,shape,mode)+(4,)),
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shape+(4,)))
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def average(self,weights = None):
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"""
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Average rotations along last dimension.
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Parameters
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----------
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weights : list of floats, optional
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Relative weight of each rotation.
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Returns
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-------
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average : Rotation
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Weighted average of original Rotation field.
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References
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----------
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F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
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https://doi.org/10.2514/1.28949
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"""
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def _M(quat):
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"""Intermediate representation supporting quaternion averaging."""
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return np.einsum('...i,...j',quat,quat)
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if weights is None:
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weights = np.ones(self.shape,dtype=float)
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eig, vec = np.linalg.eig(np.sum(_M(self.quaternion) * weights[...,np.newaxis,np.newaxis],axis=-3) \
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/np.sum( weights[...,np.newaxis,np.newaxis],axis=-3))
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return Rotation.from_quaternion(np.real(
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np.squeeze(
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np.take_along_axis(vec,
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eig.argmax(axis=-1)[...,np.newaxis,np.newaxis],
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axis=-1),
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axis=-1)),
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accept_homomorph = True)
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def misorientation(self,other):
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"""
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Calculate misorientation from self to other Rotation.
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Parameters
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----------
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other : Rotation
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Rotation to which the misorientation is computed.
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"""
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return other*~self
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################################################################################################
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# convert to different orientation representations (numpy arrays)
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def as_quaternion(self):
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"""
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Represent as unit quaternion.
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Returns
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-------
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q : numpy.ndarray of shape (...,4)
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Unit quaternion in positive real hemisphere: (q_0, q_1, q_2, q_3), ǀqǀ=1, q_0 ≥ 0.
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"""
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return self.quaternion.copy()
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def as_Euler_angles(self,
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degrees = False):
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"""
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Represent as Bunge-Euler angles.
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Parameters
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----------
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degrees : bool, optional
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Return angles in degrees.
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Returns
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-------
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phi : numpy.ndarray of shape (...,3)
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Bunge-Euler angles: (φ_1, ϕ, φ_2), φ_1 ∈ [0,2π], ϕ ∈ [0,π], φ_2 ∈ [0,2π]
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unless degrees == True: φ_1 ∈ [0,360], ϕ ∈ [0,180], φ_2 ∈ [0,360]
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"""
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eu = Rotation._qu2eu(self.quaternion)
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if degrees: eu = np.degrees(eu)
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return eu
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def as_axis_angle(self,
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degrees = False,
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pair = False):
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"""
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Represent as axis angle pair.
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Parameters
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----------
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degrees : bool, optional
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Return rotation angle in degrees. Defaults to False.
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pair : bool, optional
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Return tuple of axis and angle. Defaults to False.
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Returns
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-------
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axis_angle : numpy.ndarray of shape (...,4) unless pair == True:
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tuple containing numpy.ndarray of shapes (...,3) and (...)
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Axis angle pair: (n_1, n_2, n_3, ω), ǀnǀ = 1 and ω ∈ [0,π]
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unless degrees = True: ω ∈ [0,180].
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"""
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ax = Rotation._qu2ax(self.quaternion)
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if degrees: ax[...,3] = np.degrees(ax[...,3])
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return (ax[...,:3],ax[...,3]) if pair else ax
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def as_matrix(self):
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"""
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Represent as rotation matrix.
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Returns
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-------
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R : numpy.ndarray of shape (...,3,3)
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Rotation matrix R, det(R) = 1, R.T∙R=I.
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"""
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return Rotation._qu2om(self.quaternion)
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def as_Rodrigues_vector(self,
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compact = False):
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"""
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Represent as Rodrigues-Frank vector with separated axis and angle argument.
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Parameters
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----------
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compact : bool, optional
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Return as actual Rodrigues-Frank vector,
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i.e. axis and angle argument are not separated.
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Returns
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-------
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rho : numpy.ndarray of shape (...,4) containing
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[n_1, n_2, n_3, tan(ω/2)], ǀnǀ = 1 and ω ∈ [0,π]
|
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unless compact == True:
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numpy.ndarray of shape (...,3) containing
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tan(ω/2) [n_1, n_2, n_3], ω ∈ [0,π].
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"""
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ro = Rotation._qu2ro(self.quaternion)
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if compact:
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with np.errstate(invalid='ignore'):
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return ro[...,:3]*ro[...,3:4]
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else:
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return ro
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def as_homochoric(self):
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"""
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Represent as homochoric vector.
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Returns
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-------
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h : numpy.ndarray of shape (...,3)
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Homochoric vector: (h_1, h_2, h_3), ǀhǀ < (3/4*π)^(1/3).
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"""
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return Rotation._qu2ho(self.quaternion)
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def as_cubochoric(self):
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"""
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Represent as cubochoric vector.
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Returns
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-------
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x : numpy.ndarray of shape (...,3)
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Cubochoric vector: (x_1, x_2, x_3), max(x_i) < 1/2*π^(2/3).
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"""
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return Rotation._qu2cu(self.quaternion)
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################################################################################################
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# Static constructors. The input data needs to follow the conventions, options allow to
|
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# relax the conventions.
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@staticmethod
|
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def from_quaternion(q,
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accept_homomorph = False,
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P = -1):
|
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"""
|
||
Initialize from quaternion.
|
||
|
||
Parameters
|
||
----------
|
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q : numpy.ndarray of shape (...,4)
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Unit quaternion in positive real hemisphere: (q_0, q_1, q_2, q_3),
|
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ǀqǀ=1, q_0 ≥ 0.
|
||
accept_homomorph : boolean, optional
|
||
Allow homomorphic variants, i.e. q_0 < 0 (negative real hemisphere).
|
||
Defaults to False.
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P : int ∈ {-1,1}, optional
|
||
Convention used. Defaults to -1.
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"""
|
||
qu = np.array(q,dtype=float)
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||
if qu.shape[:-2:-1] != (4,):
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||
raise ValueError('Invalid shape.')
|
||
if abs(P) != 1:
|
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raise ValueError('P ∉ {-1,1}')
|
||
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||
qu[...,1:4] *= -P
|
||
if accept_homomorph:
|
||
qu[qu[...,0] < 0.0] *= -1
|
||
else:
|
||
if np.any(qu[...,0] < 0.0):
|
||
raise ValueError('Quaternion with negative first (real) component.')
|
||
if not np.all(np.isclose(np.linalg.norm(qu,axis=-1), 1.0)):
|
||
raise ValueError('Quaternion is not of unit length.')
|
||
|
||
return Rotation(qu)
|
||
|
||
@staticmethod
|
||
def from_Euler_angles(phi,
|
||
degrees = False):
|
||
"""
|
||
Initialize from Bunge-Euler angles.
|
||
|
||
Parameters
|
||
----------
|
||
phi : numpy.ndarray of shape (...,3)
|
||
Bunge-Euler angles: (φ_1, ϕ, φ_2), φ_1 ∈ [0,2π], ϕ ∈ [0,π], φ_2 ∈ [0,2π]
|
||
unless degrees == True: φ_1 ∈ [0,360], ϕ ∈ [0,180], φ_2 ∈ [0,360].
|
||
degrees : boolean, optional
|
||
Bunge-Euler angles are given in degrees. Defaults to False.
|
||
|
||
"""
|
||
eu = np.array(phi,dtype=float)
|
||
if eu.shape[:-2:-1] != (3,):
|
||
raise ValueError('Invalid shape.')
|
||
|
||
eu = np.radians(eu) if degrees else eu
|
||
if np.any(eu < 0.0) or np.any(eu > 2.0*np.pi) or np.any(eu[...,1] > np.pi): # ToDo: No separate check for PHI
|
||
raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].')
|
||
|
||
return Rotation(Rotation._eu2qu(eu))
|
||
|
||
@staticmethod
|
||
def from_axis_angle(axis_angle,
|
||
degrees = False,
|
||
normalize = False,
|
||
P = -1):
|
||
"""
|
||
Initialize from Axis angle pair.
|
||
|
||
Parameters
|
||
----------
|
||
axis_angle : numpy.ndarray of shape (...,4)
|
||
Axis angle pair: [n_1, n_2, n_3, ω], ǀnǀ = 1 and ω ∈ [0,π]
|
||
unless degrees = True: ω ∈ [0,180].
|
||
degrees : boolean, optional
|
||
Angle ω is given in degrees. Defaults to False.
|
||
normalize: boolean, optional
|
||
Allow ǀnǀ ≠ 1. Defaults to False.
|
||
P : int ∈ {-1,1}, optional
|
||
Convention used. Defaults to -1.
|
||
|
||
"""
|
||
ax = np.array(axis_angle,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 normalize: ax[...,0:3] /= np.linalg.norm(ax[...,0:3],axis=-1,keepdims=True)
|
||
if np.any(ax[...,3] < 0.0) or np.any(ax[...,3] > np.pi):
|
||
raise ValueError('Axis angle rotation angle outside of [0..π].')
|
||
if not np.all(np.isclose(np.linalg.norm(ax[...,0:3],axis=-1), 1.0)):
|
||
print(np.linalg.norm(ax[...,0:3],axis=-1))
|
||
raise ValueError('Axis angle rotation axis is not of unit length.')
|
||
|
||
return Rotation(Rotation._ax2qu(ax))
|
||
|
||
@staticmethod
|
||
def from_basis(basis,
|
||
orthonormal = True,
|
||
reciprocal = False):
|
||
"""
|
||
Initialize from lattice basis vectors.
|
||
|
||
Parameters
|
||
----------
|
||
basis : numpy.ndarray of shape (...,3,3)
|
||
Three three-dimensional lattice basis vectors.
|
||
orthonormal : boolean, optional
|
||
Basis is strictly orthonormal, i.e. is free of stretch components. Defaults to True.
|
||
reciprocal : boolean, optional
|
||
Basis vectors are given in reciprocal (instead of real) space. Defaults to False.
|
||
|
||
"""
|
||
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)
|
||
if not np.all(np.isclose(np.linalg.det(om),1.0)):
|
||
raise ValueError('Orientation matrix has determinant ≠ 1.')
|
||
if not np.all(np.isclose(np.einsum('...i,...i',om[...,0],om[...,1]), 0.0)) \
|
||
or not np.all(np.isclose(np.einsum('...i,...i',om[...,1],om[...,2]), 0.0)) \
|
||
or not np.all(np.isclose(np.einsum('...i,...i',om[...,2],om[...,0]), 0.0)):
|
||
raise ValueError('Orientation matrix is not orthogonal.')
|
||
|
||
return Rotation(Rotation._om2qu(om))
|
||
|
||
@staticmethod
|
||
def from_matrix(R):
|
||
"""
|
||
Initialize from rotation matrix.
|
||
|
||
Parameters
|
||
----------
|
||
R : numpy.ndarray of shape (...,3,3)
|
||
Rotation matrix: det(R) = 1, R.T∙R=I.
|
||
|
||
"""
|
||
return Rotation.from_basis(R)
|
||
|
||
@staticmethod
|
||
def from_parallel(a,b,
|
||
**kwargs):
|
||
"""
|
||
Initialize from pairs of two orthogonal lattice basis vectors.
|
||
|
||
Parameters
|
||
----------
|
||
a : numpy.ndarray of shape (...,2,3)
|
||
Two three-dimensional lattice vectors of first orthogonal basis.
|
||
b : numpy.ndarray of shape (...,2,3)
|
||
Corresponding three-dimensional lattice vectors of second basis.
|
||
|
||
"""
|
||
a_ = np.array(a)
|
||
b_ = np.array(b)
|
||
if a_.shape[-2:] != (2,3) or b_.shape[-2:] != (2,3) or a_.shape != b_.shape:
|
||
raise ValueError('Invalid shape.')
|
||
am = np.stack([ a_[...,0,:],
|
||
a_[...,1,:],
|
||
np.cross(a_[...,0,:],a_[...,1,:]) ],axis=-2)
|
||
bm = np.stack([ b_[...,0,:],
|
||
b_[...,1,:],
|
||
np.cross(b_[...,0,:],b_[...,1,:]) ],axis=-2)
|
||
|
||
return Rotation.from_basis(np.swapaxes(am/np.linalg.norm(am,axis=-1,keepdims=True),-1,-2))\
|
||
.misorientation(Rotation.from_basis(np.swapaxes(bm/np.linalg.norm(bm,axis=-1,keepdims=True),-1,-2)))
|
||
|
||
|
||
@staticmethod
|
||
def from_Rodrigues_vector(rho,
|
||
normalize = False,
|
||
P = -1):
|
||
"""
|
||
Initialize from Rodrigues-Frank vector (angle separated from axis).
|
||
|
||
Parameters
|
||
----------
|
||
rho : numpy.ndarray of shape (...,4)
|
||
Rodrigues-Frank vector. (n_1, n_2, n_3, tan(ω/2)), ǀnǀ = 1 and ω ∈ [0,π].
|
||
normalize : boolean, optional
|
||
Allow ǀnǀ ≠ 1. Defaults to False.
|
||
P : int ∈ {-1,1}, optional
|
||
Convention used. Defaults to -1.
|
||
|
||
"""
|
||
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 normalize: ro[...,0:3] /= np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True)
|
||
if np.any(ro[...,3] < 0.0):
|
||
raise ValueError('Rodrigues vector rotation angle not positive.')
|
||
if not np.all(np.isclose(np.linalg.norm(ro[...,0:3],axis=-1), 1.0)):
|
||
raise ValueError('Rodrigues vector rotation axis is not of unit length.')
|
||
|
||
return Rotation(Rotation._ro2qu(ro))
|
||
|
||
@staticmethod
|
||
def from_homochoric(h,
|
||
P = -1):
|
||
"""
|
||
Initialize from homochoric vector.
|
||
|
||
Parameters
|
||
----------
|
||
h : numpy.ndarray of shape (...,3)
|
||
Homochoric vector: (h_1, h_2, h_3), ǀhǀ < (3/4*π)^(1/3).
|
||
P : int ∈ {-1,1}, optional
|
||
Convention used. Defaults to -1.
|
||
|
||
"""
|
||
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+1e-9):
|
||
raise ValueError('Homochoric coordinate outside of the sphere.')
|
||
|
||
return Rotation(Rotation._ho2qu(ho))
|
||
|
||
@staticmethod
|
||
def from_cubochoric(x,
|
||
P = -1):
|
||
"""
|
||
Initialize from cubochoric vector.
|
||
|
||
Parameters
|
||
----------
|
||
x : numpy.ndarray of shape (...,3)
|
||
Cubochoric vector: (x_1, x_2, x_3), max(x_i) < 1/2*π^(2/3).
|
||
P : int ∈ {-1,1}, optional
|
||
Convention used. Defaults to -1.
|
||
|
||
"""
|
||
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+1e-9:
|
||
raise ValueError('Cubochoric coordinate outside of the cube.')
|
||
|
||
ho = -P * Rotation._cu2ho(cu)
|
||
|
||
return Rotation(Rotation._ho2qu(ho))
|
||
|
||
|
||
@staticmethod
|
||
def from_random(shape = None,
|
||
rng_seed = None):
|
||
"""
|
||
Draw random rotation.
|
||
|
||
Rotations are uniformly distributed.
|
||
|
||
Parameters
|
||
----------
|
||
shape : tuple of ints, optional
|
||
Shape of the sample. Defaults to None which gives a
|
||
single rotation
|
||
rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional
|
||
A seed to initialize the BitGenerator. Defaults to None.
|
||
If None, then fresh, unpredictable entropy will be pulled from the OS.
|
||
|
||
"""
|
||
rng = np.random.default_rng(rng_seed)
|
||
r = rng.random(3 if shape is None else tuple(shape)+(3,) if hasattr(shape, '__iter__') else (shape,3))
|
||
|
||
A = np.sqrt(r[...,2])
|
||
B = np.sqrt(1.0-r[...,2])
|
||
q = np.stack([np.cos(2.0*np.pi*r[...,0])*A,
|
||
np.sin(2.0*np.pi*r[...,1])*B,
|
||
np.cos(2.0*np.pi*r[...,1])*B,
|
||
np.sin(2.0*np.pi*r[...,0])*A],axis=-1)
|
||
|
||
return Rotation(q if shape is None else q.reshape(r.shape[:-1]+(4,)))._standardize()
|
||
|
||
|
||
@staticmethod
|
||
def from_ODF(weights,
|
||
phi,
|
||
N = 500,
|
||
degrees = True,
|
||
fractions = True,
|
||
rng_seed = None,
|
||
**kwargs):
|
||
"""
|
||
Sample discrete values from a binned ODF.
|
||
|
||
Parameters
|
||
----------
|
||
weights : numpy.ndarray of shape (n)
|
||
Texture intensity values (probability density or volume fraction) at Euler grid points.
|
||
phi : numpy.ndarray of shape (n,3)
|
||
Grid coordinates in Euler space at which weights are defined.
|
||
N : integer, optional
|
||
Number of discrete orientations to be sampled from the given ODF.
|
||
Defaults to 500.
|
||
degrees : boolean, optional
|
||
Euler grid values are in degrees. Defaults to True.
|
||
fractions : boolean, optional
|
||
ODF values correspond to volume fractions, not probability density.
|
||
Defaults to True.
|
||
rng_seed: {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional
|
||
A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy
|
||
will be pulled from the OS.
|
||
|
||
Returns
|
||
-------
|
||
samples : damask.Rotation of shape (N)
|
||
Array of sampled rotations closely representing the input ODF.
|
||
|
||
Notes
|
||
-----
|
||
Due to the distortion of Euler space in the vicinity of ϕ = 0, probability densities, p, defined on
|
||
grid points with ϕ = 0 will never result in reconstructed orientations as their dV/V = p dγ = p × 0.
|
||
Hence, it is recommended to transform any such dataset to cell centers that avoid grid points at ϕ = 0.
|
||
|
||
References
|
||
----------
|
||
P. Eisenlohr 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]*2.0*np.sin(delta[1]/2.0)*delta[2] / 8.0 / np.pi**2 * np.sin(np.radians(eu[:,1]) if deg else eu[:,1])
|
||
|
||
dg = 1.0 if fractions else _dg(phi,degrees)
|
||
dV_V = dg * np.maximum(0.0,weights.squeeze())
|
||
|
||
return Rotation.from_Euler_angles(phi[util.hybrid_IA(dV_V,N,rng_seed)],degrees)
|
||
|
||
|
||
@staticmethod
|
||
def from_spherical_component(center,
|
||
sigma,
|
||
N = 500,
|
||
degrees = True,
|
||
rng_seed = None):
|
||
"""
|
||
Calculate set of rotations with Gaussian distribution around center.
|
||
|
||
Parameters
|
||
----------
|
||
center : Rotation
|
||
Central Rotation.
|
||
sigma : float
|
||
Standard deviation of (Gaussian) misorientation distribution.
|
||
N : int, optional
|
||
Number of samples, defaults to 500.
|
||
degrees : boolean, optional
|
||
sigma is given in degrees.
|
||
rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional
|
||
A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy
|
||
will be pulled from the OS.
|
||
|
||
"""
|
||
rng = np.random.default_rng(rng_seed)
|
||
sigma = np.radians(sigma) if degrees else sigma
|
||
u,Theta = (rng.random((N,2)) * 2.0 * np.array([1,np.pi]) - np.array([1.0, 0])).T
|
||
omega = abs(rng.normal(scale=sigma,size=N))
|
||
p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta),
|
||
np.sqrt(1-u**2)*np.sin(Theta),
|
||
u, omega])
|
||
|
||
return Rotation.from_axis_angle(p) * center
|
||
|
||
|
||
@staticmethod
|
||
def from_fiber_component(alpha,
|
||
beta,
|
||
sigma = 0.0,
|
||
N = 500,
|
||
degrees = True,
|
||
rng_seed = None):
|
||
"""
|
||
Calculate set of rotations with Gaussian distribution around direction.
|
||
|
||
Parameters
|
||
----------
|
||
alpha : numpy.ndarray of size 2
|
||
Polar coordinates (phi from x,theta from z) of fiber direction in crystal frame.
|
||
beta : numpy.ndarray of size 2
|
||
Polar coordinates (phi from x,theta from z) of fiber direction in sample frame.
|
||
sigma : float, optional
|
||
Standard deviation of (Gaussian) misorientation distribution.
|
||
Defaults to 0.
|
||
N : int, optional
|
||
Number of samples, defaults to 500.
|
||
degrees : boolean, optional
|
||
sigma, alpha, and beta are given in degrees.
|
||
rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional
|
||
A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy
|
||
will be pulled from the OS.
|
||
|
||
"""
|
||
rng = np.random.default_rng(rng_seed)
|
||
sigma_,alpha_,beta_ = map(np.radians,(sigma,alpha,beta)) if degrees else (sigma,alpha,beta)
|
||
|
||
d_cr = np.array([np.sin(alpha_[0])*np.cos(alpha_[1]), np.sin(alpha_[0])*np.sin(alpha_[1]), np.cos(alpha_[0])])
|
||
d_lab = np.array([np.sin( beta_[0])*np.cos( beta_[1]), np.sin( beta_[0])*np.sin( beta_[1]), np.cos( beta_[0])])
|
||
ax_align = np.append(np.cross(d_lab,d_cr), np.arccos(np.dot(d_lab,d_cr)))
|
||
if np.isclose(ax_align[3],0.0): ax_align[:3] = np.array([1,0,0])
|
||
R_align = Rotation.from_axis_angle(ax_align if ax_align[3] > 0.0 else -ax_align,normalize=True) # rotate fiber axis from sample to crystal frame
|
||
|
||
u,Theta = (rng.random((N,2)) * 2.0 * np.array([1,np.pi]) - np.array([1.0, 0])).T
|
||
omega = abs(rng.normal(scale=sigma_,size=N))
|
||
p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta),
|
||
np.sqrt(1-u**2)*np.sin(Theta),
|
||
u, omega])
|
||
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)
|
||
|
||
|
||
####################################################################################################
|
||
# 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):
|
||
qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2)
|
||
om = np.block([qq + 2.0*qu[...,1:2]**2,
|
||
2.0*(qu[...,2:3]*qu[...,1:2]-_P*qu[...,0:1]*qu[...,3:4]),
|
||
2.0*(qu[...,3:4]*qu[...,1:2]+_P*qu[...,0:1]*qu[...,2:3]),
|
||
2.0*(qu[...,1:2]*qu[...,2:3]+_P*qu[...,0:1]*qu[...,3:4]),
|
||
qq + 2.0*qu[...,2:3]**2,
|
||
2.0*(qu[...,3:4]*qu[...,2:3]-_P*qu[...,0:1]*qu[...,1:2]),
|
||
2.0*(qu[...,1:2]*qu[...,3:4]-_P*qu[...,0:1]*qu[...,2:3]),
|
||
2.0*(qu[...,2:3]*qu[...,3:4]+_P*qu[...,0:1]*qu[...,1:2]),
|
||
qq + 2.0*qu[...,3:4]**2,
|
||
]).reshape(qu.shape[:-1]+(3,3))
|
||
return om
|
||
|
||
@staticmethod
|
||
def _qu2eu(qu):
|
||
"""Quaternion to Bunge-Euler angles."""
|
||
q02 = qu[...,0:1]*qu[...,2:3]
|
||
q13 = qu[...,1:2]*qu[...,3:4]
|
||
q01 = qu[...,0:1]*qu[...,1:2]
|
||
q23 = qu[...,2:3]*qu[...,3:4]
|
||
q03_s = qu[...,0:1]**2+qu[...,3:4]**2
|
||
q12_s = qu[...,1:2]**2+qu[...,2:3]**2
|
||
chi = np.sqrt(q03_s*q12_s)
|
||
|
||
eu = np.where(np.abs(q12_s) < 1.0e-8,
|
||
np.block([np.arctan2(-_P*2.0*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2),
|
||
np.zeros(qu.shape[:-1]+(2,))]),
|
||
np.where(np.abs(q03_s) < 1.0e-8,
|
||
np.block([np.arctan2( 2.0*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2),
|
||
np.broadcast_to(np.pi,qu[...,0:1].shape),
|
||
np.zeros(qu.shape[:-1]+(1,))]),
|
||
np.block([np.arctan2((-_P*q02+q13)*chi, (-_P*q01-q23)*chi),
|
||
np.arctan2( 2.0*chi, q03_s-q12_s ),
|
||
np.arctan2(( _P*q02+q13)*chi, (-_P*q01+q23)*chi)])
|
||
)
|
||
)
|
||
# reduce Euler angles to definition range
|
||
eu[np.abs(eu)<1.e-6] = 0.0
|
||
eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu) # needed?
|
||
return eu
|
||
|
||
@staticmethod
|
||
def _qu2ax(qu):
|
||
"""
|
||
Quaternion to axis angle pair.
|
||
|
||
Modified version of the original formulation, should be numerically more stable
|
||
"""
|
||
with np.errstate(invalid='ignore',divide='ignore'):
|
||
s = np.sign(qu[...,0:1])/np.sqrt(qu[...,1:2]**2+qu[...,2:3]**2+qu[...,3:4]**2)
|
||
omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
|
||
ax = np.where(np.broadcast_to(qu[...,0:1] < 1.0e-8,qu.shape),
|
||
np.block([qu[...,1:4],np.broadcast_to(np.pi,qu[...,0:1].shape)]),
|
||
np.block([qu[...,1:4]*s,omega]))
|
||
ax[np.isclose(qu[...,0],1.,rtol=0.0)] = [0.0, 0.0, 1.0, 0.0]
|
||
return ax
|
||
|
||
@staticmethod
|
||
def _qu2ro(qu):
|
||
"""Quaternion to Rodrigues-Frank vector."""
|
||
with np.errstate(invalid='ignore',divide='ignore'):
|
||
s = np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True)
|
||
ro = np.where(np.broadcast_to(np.abs(qu[...,0:1]) < 1.0e-12,qu.shape),
|
||
np.block([qu[...,1:2], qu[...,2:3], qu[...,3:4], np.broadcast_to(np.inf,qu[...,0:1].shape)]),
|
||
np.block([qu[...,1:2]/s,qu[...,2:3]/s,qu[...,3:4]/s,
|
||
np.tan(np.arccos(np.clip(qu[...,0:1],-1.0,1.0)))
|
||
])
|
||
)
|
||
ro[np.abs(s).squeeze(-1) < 1.0e-12] = [0.0,0.0,_P,0.0]
|
||
return ro
|
||
|
||
@staticmethod
|
||
def _qu2ho(qu):
|
||
"""Quaternion to homochoric vector."""
|
||
with np.errstate(invalid='ignore'):
|
||
omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
|
||
ho = np.where(np.abs(omega) < 1.0e-12,
|
||
np.zeros(3),
|
||
qu[...,1:4]/np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) \
|
||
* (0.75*(omega - np.sin(omega)))**(1./3.))
|
||
return ho
|
||
|
||
@staticmethod
|
||
def _qu2cu(qu):
|
||
"""Quaternion to cubochoric vector."""
|
||
return Rotation._ho2cu(Rotation._qu2ho(qu))
|
||
|
||
|
||
#---------- Rotation matrix ----------
|
||
@staticmethod
|
||
def _om2qu(om):
|
||
"""
|
||
Rotation matrix to quaternion.
|
||
|
||
This formulation is from www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion.
|
||
The original formulation had issues.
|
||
"""
|
||
trace = om[...,0,0:1]+om[...,1,1:2]+om[...,2,2:3]
|
||
|
||
with np.errstate(invalid='ignore',divide='ignore'):
|
||
s = [
|
||
0.5 / np.sqrt( 1.0 + trace),
|
||
2.0 * np.sqrt( 1.0 + om[...,0,0:1] - om[...,1,1:2] - om[...,2,2:3]),
|
||
2.0 * np.sqrt( 1.0 + om[...,1,1:2] - om[...,2,2:3] - om[...,0,0:1]),
|
||
2.0 * np.sqrt( 1.0 + om[...,2,2:3] - om[...,0,0:1] - om[...,1,1:2] )
|
||
]
|
||
qu= np.where(trace>0,
|
||
np.block([0.25 / s[0],
|
||
(om[...,2,1:2] - om[...,1,2:3] ) * s[0],
|
||
(om[...,0,2:3] - om[...,2,0:1] ) * s[0],
|
||
(om[...,1,0:1] - om[...,0,1:2] ) * s[0]]),
|
||
np.where(om[...,0,0:1] > np.maximum(om[...,1,1:2],om[...,2,2:3]),
|
||
np.block([(om[...,2,1:2] - om[...,1,2:3]) / s[1],
|
||
0.25 * s[1],
|
||
(om[...,0,1:2] + om[...,1,0:1]) / s[1],
|
||
(om[...,0,2:3] + om[...,2,0:1]) / s[1]]),
|
||
np.where(om[...,1,1:2] > om[...,2,2:3],
|
||
np.block([(om[...,0,2:3] - om[...,2,0:1]) / s[2],
|
||
(om[...,0,1:2] + om[...,1,0:1]) / s[2],
|
||
0.25 * s[2],
|
||
(om[...,1,2:3] + om[...,2,1:2]) / s[2]]),
|
||
np.block([(om[...,1,0:1] - om[...,0,1:2]) / s[3],
|
||
(om[...,0,2:3] + om[...,2,0:1]) / s[3],
|
||
(om[...,1,2:3] + om[...,2,1:2]) / s[3],
|
||
0.25 * s[3]]),
|
||
)
|
||
)
|
||
)*np.array([1,_P,_P,_P])
|
||
qu[qu[...,0]<0] *=-1
|
||
return qu
|
||
|
||
@staticmethod
|
||
def _om2eu(om):
|
||
"""Rotation matrix to Bunge-Euler angles."""
|
||
with np.errstate(invalid='ignore',divide='ignore'):
|
||
zeta = 1.0/np.sqrt(1.0-om[...,2,2:3]**2)
|
||
eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.0,0.0),
|
||
np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]),
|
||
np.pi*0.5*(1-om[...,2,2:3]),
|
||
np.zeros(om.shape[:-2]+(1,)),
|
||
]),
|
||
np.block([np.arctan2(om[...,2,0:1]*zeta,-om[...,2,1:2]*zeta),
|
||
np.arccos( om[...,2,2:3]),
|
||
np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta)
|
||
])
|
||
)
|
||
eu[np.abs(eu)<1.e-8] = 0.0
|
||
eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
|
||
return eu
|
||
|
||
@staticmethod
|
||
def _om2ax(om):
|
||
"""Rotation matrix to axis angle pair."""
|
||
diag_delta = -_P*np.block([om[...,1,2:3]-om[...,2,1:2],
|
||
om[...,2,0:1]-om[...,0,2:3],
|
||
om[...,0,1:2]-om[...,1,0:1]
|
||
])
|
||
t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.0).reshape(om.shape[:-2]+(1,))
|
||
w,vr = np.linalg.eig(om)
|
||
# mask duplicated real eigenvalues
|
||
w[np.isclose(w[...,0],1.0+0.0j),1:] = 0.
|
||
w[np.isclose(w[...,1],1.0+0.0j),2:] = 0.
|
||
vr = np.swapaxes(vr,-1,-2)
|
||
ax = np.where(np.abs(diag_delta)<1e-12,
|
||
np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,)),
|
||
np.abs(np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,))) \
|
||
*np.sign(diag_delta))
|
||
ax = np.block([ax,np.arccos(np.clip(t,-1.0,1.0))])
|
||
ax[np.abs(ax[...,3])<1.e-8] = [ 0.0, 0.0, 1.0, 0.0]
|
||
return ax
|
||
|
||
@staticmethod
|
||
def _om2ro(om):
|
||
"""Rotation matrix to Rodrigues-Frank vector."""
|
||
return Rotation._eu2ro(Rotation._om2eu(om))
|
||
|
||
@staticmethod
|
||
def _om2ho(om):
|
||
"""Rotation matrix to homochoric vector."""
|
||
return Rotation._ax2ho(Rotation._om2ax(om))
|
||
|
||
@staticmethod
|
||
def _om2cu(om):
|
||
"""Rotation matrix to cubochoric vector."""
|
||
return Rotation._ho2cu(Rotation._om2ho(om))
|
||
|
||
|
||
#---------- Bunge-Euler angles ----------
|
||
@staticmethod
|
||
def _eu2qu(eu):
|
||
"""Bunge-Euler angles to quaternion."""
|
||
ee = 0.5*eu
|
||
cPhi = np.cos(ee[...,1:2])
|
||
sPhi = np.sin(ee[...,1:2])
|
||
qu = np.block([ cPhi*np.cos(ee[...,0:1]+ee[...,2:3]),
|
||
-_P*sPhi*np.cos(ee[...,0:1]-ee[...,2:3]),
|
||
-_P*sPhi*np.sin(ee[...,0:1]-ee[...,2:3]),
|
||
-_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])])
|
||
qu[qu[...,0]<0.0]*=-1
|
||
return qu
|
||
|
||
@staticmethod
|
||
def _eu2om(eu):
|
||
"""Bunge-Euler angles to rotation matrix."""
|
||
c = np.cos(eu)
|
||
s = np.sin(eu)
|
||
om = np.block([+c[...,0:1]*c[...,2:3]-s[...,0:1]*s[...,2:3]*c[...,1:2],
|
||
+s[...,0:1]*c[...,2:3]+c[...,0:1]*s[...,2:3]*c[...,1:2],
|
||
+s[...,2:3]*s[...,1:2],
|
||
-c[...,0:1]*s[...,2:3]-s[...,0:1]*c[...,2:3]*c[...,1:2],
|
||
-s[...,0:1]*s[...,2:3]+c[...,0:1]*c[...,2:3]*c[...,1:2],
|
||
+c[...,2:3]*s[...,1:2],
|
||
+s[...,0:1]*s[...,1:2],
|
||
-c[...,0:1]*s[...,1:2],
|
||
+c[...,1:2]
|
||
]).reshape(eu.shape[:-1]+(3,3))
|
||
om[np.abs(om)<1.e-12] = 0.0
|
||
return om
|
||
|
||
@staticmethod
|
||
def _eu2ax(eu):
|
||
"""Bunge-Euler angles to axis angle pair."""
|
||
t = np.tan(eu[...,1:2]*0.5)
|
||
sigma = 0.5*(eu[...,0:1]+eu[...,2:3])
|
||
delta = 0.5*(eu[...,0:1]-eu[...,2:3])
|
||
tau = np.linalg.norm(np.block([t,np.sin(sigma)]),axis=-1,keepdims=True)
|
||
alpha = np.where(np.abs(np.cos(sigma))<1.e-12,np.pi,2.0*np.arctan(tau/np.cos(sigma)))
|
||
with np.errstate(invalid='ignore',divide='ignore'):
|
||
ax = np.where(np.broadcast_to(np.abs(alpha)<1.0e-12,eu.shape[:-1]+(4,)),
|
||
[0.0,0.0,1.0,0.0],
|
||
np.block([-_P/tau*t*np.cos(delta),
|
||
-_P/tau*t*np.sin(delta),
|
||
-_P/tau* np.sin(sigma),
|
||
alpha
|
||
]))
|
||
ax[(alpha<0.0).squeeze()] *=-1
|
||
return ax
|
||
|
||
@staticmethod
|
||
def _eu2ro(eu):
|
||
"""Bunge-Euler angles to Rodrigues-Frank vector."""
|
||
ax = Rotation._eu2ax(eu)
|
||
ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)])
|
||
ro[ax[...,3]>=np.pi,3] = np.inf
|
||
ro[np.abs(ax[...,3])<1.e-16] = [ 0.0, 0.0, _P, 0.0 ]
|
||
return ro
|
||
|
||
@staticmethod
|
||
def _eu2ho(eu):
|
||
"""Bunge-Euler angles to homochoric vector."""
|
||
return Rotation._ax2ho(Rotation._eu2ax(eu))
|
||
|
||
@staticmethod
|
||
def _eu2cu(eu):
|
||
"""Bunge-Euler angles to cubochoric vector."""
|
||
return Rotation._ho2cu(Rotation._eu2ho(eu))
|
||
|
||
|
||
#---------- Axis angle pair ----------
|
||
@staticmethod
|
||
def _ax2qu(ax):
|
||
"""Axis angle pair to quaternion."""
|
||
c = np.cos(ax[...,3:4]*.5)
|
||
s = np.sin(ax[...,3:4]*.5)
|
||
qu = np.where(np.abs(ax[...,3:4])<1.e-6,[1.0, 0.0, 0.0, 0.0],np.block([c, ax[...,:3]*s]))
|
||
return qu
|
||
|
||
@staticmethod
|
||
def _ax2om(ax):
|
||
"""Axis angle pair to rotation matrix."""
|
||
c = np.cos(ax[...,3:4])
|
||
s = np.sin(ax[...,3:4])
|
||
omc = 1. -c
|
||
om = np.block([c+omc*ax[...,0:1]**2,
|
||
omc*ax[...,0:1]*ax[...,1:2] + s*ax[...,2:3],
|
||
omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2],
|
||
omc*ax[...,0:1]*ax[...,1:2] - s*ax[...,2:3],
|
||
c+omc*ax[...,1:2]**2,
|
||
omc*ax[...,1:2]*ax[...,2:3] + s*ax[...,0:1],
|
||
omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2],
|
||
omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1],
|
||
c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3))
|
||
return om if _P < 0.0 else np.swapaxes(om,-1,-2)
|
||
|
||
@staticmethod
|
||
def _ax2eu(ax):
|
||
"""Rotation matrix to Bunge Euler angles."""
|
||
return Rotation._om2eu(Rotation._ax2om(ax))
|
||
|
||
@staticmethod
|
||
def _ax2ro(ax):
|
||
"""Axis angle pair to Rodrigues-Frank vector."""
|
||
ro = np.block([ax[...,:3],
|
||
np.where(np.isclose(ax[...,3:4],np.pi,atol=1.e-15,rtol=.0),
|
||
np.inf,
|
||
np.tan(ax[...,3:4]*0.5))
|
||
])
|
||
ro[np.abs(ax[...,3])<1.e-6] = [.0,.0,_P,.0]
|
||
return ro
|
||
|
||
@staticmethod
|
||
def _ax2ho(ax):
|
||
"""Axis angle pair to homochoric vector."""
|
||
f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1.0/3.0)
|
||
ho = ax[...,:3] * f
|
||
return ho
|
||
|
||
@staticmethod
|
||
def _ax2cu(ax):
|
||
"""Axis angle pair to cubochoric vector."""
|
||
return Rotation._ho2cu(Rotation._ax2ho(ax))
|
||
|
||
|
||
#---------- Rodrigues-Frank vector ----------
|
||
@staticmethod
|
||
def _ro2qu(ro):
|
||
"""Rodrigues-Frank vector to quaternion."""
|
||
return Rotation._ax2qu(Rotation._ro2ax(ro))
|
||
|
||
@staticmethod
|
||
def _ro2om(ro):
|
||
"""Rodgrigues-Frank vector to rotation matrix."""
|
||
return Rotation._ax2om(Rotation._ro2ax(ro))
|
||
|
||
@staticmethod
|
||
def _ro2eu(ro):
|
||
"""Rodrigues-Frank vector to Bunge-Euler angles."""
|
||
return Rotation._om2eu(Rotation._ro2om(ro))
|
||
|
||
@staticmethod
|
||
def _ro2ax(ro):
|
||
"""Rodrigues-Frank vector to axis angle pair."""
|
||
with np.errstate(invalid='ignore',divide='ignore'):
|
||
ax = np.where(np.isfinite(ro[...,3:4]),
|
||
np.block([ro[...,0:3]*np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True),2.*np.arctan(ro[...,3:4])]),
|
||
np.block([ro[...,0:3],np.broadcast_to(np.pi,ro[...,3:4].shape)]))
|
||
ax[np.abs(ro[...,3]) < 1.e-8] = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
||
return ax
|
||
|
||
@staticmethod
|
||
def _ro2ho(ro):
|
||
"""Rodrigues-Frank vector to homochoric vector."""
|
||
f = np.where(np.isfinite(ro[...,3:4]),2.0*np.arctan(ro[...,3:4]) -np.sin(2.0*np.arctan(ro[...,3:4])),np.pi)
|
||
ho = np.where(np.broadcast_to(np.sum(ro[...,0:3]**2.0,axis=-1,keepdims=True) < 1.e-8,ro[...,0:3].shape),
|
||
np.zeros(3), ro[...,0:3]* (0.75*f)**(1.0/3.0))
|
||
return ho
|
||
|
||
@staticmethod
|
||
def _ro2cu(ro):
|
||
"""Rodrigues-Frank vector to cubochoric vector."""
|
||
return Rotation._ho2cu(Rotation._ro2ho(ro))
|
||
|
||
|
||
#---------- Homochoric vector----------
|
||
@staticmethod
|
||
def _ho2qu(ho):
|
||
"""Homochoric vector to quaternion."""
|
||
return Rotation._ax2qu(Rotation._ho2ax(ho))
|
||
|
||
@staticmethod
|
||
def _ho2om(ho):
|
||
"""Homochoric vector to rotation matrix."""
|
||
return Rotation._ax2om(Rotation._ho2ax(ho))
|
||
|
||
@staticmethod
|
||
def _ho2eu(ho):
|
||
"""Homochoric vector to Bunge-Euler angles."""
|
||
return Rotation._ax2eu(Rotation._ho2ax(ho))
|
||
|
||
@staticmethod
|
||
def _ho2ax(ho):
|
||
"""Homochoric vector to axis angle pair."""
|
||
tfit = np.array([+1.0000000000018852, -0.5000000002194847,
|
||
-0.024999992127593126, -0.003928701544781374,
|
||
-0.0008152701535450438, -0.0002009500426119712,
|
||
-0.00002397986776071756, -0.00008202868926605841,
|
||
+0.00012448715042090092, -0.0001749114214822577,
|
||
+0.0001703481934140054, -0.00012062065004116828,
|
||
+0.000059719705868660826, -0.00001980756723965647,
|
||
+0.000003953714684212874, -0.00000036555001439719544])
|
||
hmag_squared = np.sum(ho**2.,axis=-1,keepdims=True)
|
||
hm = hmag_squared.copy()
|
||
s = tfit[0] + tfit[1] * hmag_squared
|
||
for i in range(2,16):
|
||
hm *= hmag_squared
|
||
s += tfit[i] * hm
|
||
with np.errstate(invalid='ignore'):
|
||
ax = np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-8,ho.shape[:-1]+(4,)),
|
||
[ 0.0, 0.0, 1.0, 0.0 ],
|
||
np.block([ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))]))
|
||
return ax
|
||
|
||
@staticmethod
|
||
def _ho2ro(ho):
|
||
"""Axis angle pair to Rodrigues-Frank vector."""
|
||
return Rotation._ax2ro(Rotation._ho2ax(ho))
|
||
|
||
@staticmethod
|
||
def _ho2cu(ho):
|
||
"""
|
||
Homochoric vector to cubochoric vector.
|
||
|
||
References
|
||
----------
|
||
D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
|
||
https://doi.org/10.1088/0965-0393/22/7/075013
|
||
|
||
"""
|
||
rs = np.linalg.norm(ho,axis=-1,keepdims=True)
|
||
|
||
xyz3 = np.take_along_axis(ho,Rotation._get_pyramid_order(ho,'forward'),-1)
|
||
|
||
with np.errstate(invalid='ignore',divide='ignore'):
|
||
# inverse M_3
|
||
xyz2 = xyz3[...,0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[...,2:3])) )
|
||
qxy = np.sum(xyz2**2,axis=-1,keepdims=True)
|
||
|
||
q2 = qxy + np.max(np.abs(xyz2),axis=-1,keepdims=True)**2
|
||
sq2 = np.sqrt(q2)
|
||
q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2))
|
||
tt = np.clip((np.min(np.abs(xyz2),axis=-1,keepdims=True)**2\
|
||
+np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
|
||
T_inv = np.where(np.abs(xyz2[...,1:2]) <= np.abs(xyz2[...,0:1]),
|
||
np.block([np.ones_like(tt),np.arccos(tt)/np.pi*12.0]),
|
||
np.block([np.arccos(tt)/np.pi*12.0,np.ones_like(tt)]))*q
|
||
T_inv[xyz2<0.0] *= -1.0
|
||
T_inv[np.broadcast_to(np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-12),T_inv.shape)] = 0.0
|
||
cu = np.block([T_inv, np.where(xyz3[...,2:3]<0.0,-np.ones_like(xyz3[...,2:3]),np.ones_like(xyz3[...,2:3])) \
|
||
* rs/np.sqrt(6.0/np.pi),
|
||
])/ _sc
|
||
|
||
cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
|
||
cu = np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1)
|
||
|
||
return cu
|
||
|
||
#---------- Cubochoric ----------
|
||
@staticmethod
|
||
def _cu2qu(cu):
|
||
"""Cubochoric vector to quaternion."""
|
||
return Rotation._ho2qu(Rotation._cu2ho(cu))
|
||
|
||
@staticmethod
|
||
def _cu2om(cu):
|
||
"""Cubochoric vector to rotation matrix."""
|
||
return Rotation._ho2om(Rotation._cu2ho(cu))
|
||
|
||
@staticmethod
|
||
def _cu2eu(cu):
|
||
"""Cubochoric vector to Bunge-Euler angles."""
|
||
return Rotation._ho2eu(Rotation._cu2ho(cu))
|
||
|
||
@staticmethod
|
||
def _cu2ax(cu):
|
||
"""Cubochoric vector to axis angle pair."""
|
||
return Rotation._ho2ax(Rotation._cu2ho(cu))
|
||
|
||
@staticmethod
|
||
def _cu2ro(cu):
|
||
"""Cubochoric vector to Rodrigues-Frank vector."""
|
||
return Rotation._ho2ro(Rotation._cu2ho(cu))
|
||
|
||
@staticmethod
|
||
def _cu2ho(cu):
|
||
"""
|
||
Cubochoric vector to homochoric vector.
|
||
|
||
References
|
||
----------
|
||
D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
|
||
https://doi.org/10.1088/0965-0393/22/7/075013
|
||
|
||
"""
|
||
with np.errstate(invalid='ignore',divide='ignore'):
|
||
# get pyramide and scale by grid parameter ratio
|
||
XYZ = np.take_along_axis(cu,Rotation._get_pyramid_order(cu,'forward'),-1) * _sc
|
||
order = np.abs(XYZ[...,1:2]) <= np.abs(XYZ[...,0:1])
|
||
q = np.pi/12.0 * np.where(order,XYZ[...,1:2],XYZ[...,0:1]) \
|
||
/ np.where(order,XYZ[...,0:1],XYZ[...,1:2])
|
||
c = np.cos(q)
|
||
s = np.sin(q)
|
||
q = _R1*2.0**0.25/_beta/ np.sqrt(np.sqrt(2.0)-c) \
|
||
* np.where(order,XYZ[...,0:1],XYZ[...,1:2])
|
||
|
||
T = np.block([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
|
||
|
||
# transform to sphere grid (inverse Lambert)
|
||
c = np.sum(T**2,axis=-1,keepdims=True)
|
||
s = c * np.pi/24.0 /XYZ[...,2:3]**2
|
||
c = c * np.sqrt(np.pi/24.0)/XYZ[...,2:3]
|
||
q = np.sqrt( 1.0 - s)
|
||
|
||
ho = np.where(np.isclose(np.sum(np.abs(XYZ[...,0:2]),axis=-1,keepdims=True),0.0,rtol=0.0,atol=1.0e-16),
|
||
np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6.0/np.pi) *XYZ[...,2:3]]),
|
||
np.block([np.where(order,T[...,0:1],T[...,1:2])*q,
|
||
np.where(order,T[...,1:2],T[...,0:1])*q,
|
||
np.sqrt(6.0/np.pi) * XYZ[...,2:3] - c])
|
||
)
|
||
|
||
ho[np.isclose(np.sum(np.abs(cu),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
|
||
ho = np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1)
|
||
|
||
return ho
|
||
|
||
|
||
@staticmethod
|
||
def _get_pyramid_order(xyz,direction=None):
|
||
"""
|
||
Get order of the coordinates.
|
||
|
||
Depending on the pyramid in which the point is located, the order need to be adjusted.
|
||
|
||
Parameters
|
||
----------
|
||
xyz : numpy.ndarray
|
||
coordinates of a point on a uniform refinable grid on a ball or
|
||
in a uniform refinable cubical grid.
|
||
|
||
References
|
||
----------
|
||
D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
|
||
https://doi.org/10.1088/0965-0393/22/7/075013
|
||
|
||
"""
|
||
order = {'forward': np.array([[0,1,2],[1,2,0],[2,0,1]]),
|
||
'backward':np.array([[0,1,2],[2,0,1],[1,2,0]])}
|
||
|
||
p = np.where(np.maximum(np.abs(xyz[...,0]),np.abs(xyz[...,1])) <= np.abs(xyz[...,2]),0,
|
||
np.where(np.maximum(np.abs(xyz[...,1]),np.abs(xyz[...,2])) <= np.abs(xyz[...,0]),1,2))
|
||
|
||
return order[direction][p]
|