1201 lines
46 KiB
Python
1201 lines
46 KiB
Python
import numpy as np
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from . import mechanics
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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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Orientation stored with functionality for conversion to 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 the angular ranges as [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.asMatrix(),a)
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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,quaternion = np.array([1.0,0.0,0.0,0.0])):
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"""
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Initializes to identity unless specified.
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Parameters
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----------
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quaternion : numpy.ndarray, 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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"""
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self.quaternion = quaternion.copy()
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@property
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def shape(self):
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return self.quaternion.shape[:-1]
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def __copy__(self):
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"""Copy."""
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return self.__class__(self.quaternion)
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copy = __copy__
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def __repr__(self):
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"""Orientation displayed as unit quaternion, rotation matrix, and Bunge-Euler angles."""
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if self.quaternion.shape != (4,):
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raise NotImplementedError('Support for multiple rotations missing')
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return '\n'.join([
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'Quaternion: (real={:.3f}, imag=<{:+.3f}, {:+.3f}, {:+.3f}>)'.format(*(self.quaternion)),
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'Matrix:\n{}'.format(np.round(self.as_matrix(),8)),
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'Bunge Eulers / deg: ({:3.2f}, {:3.2f}, {:3.2f})'.format(*self.as_Eulers(degrees=True)),
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])
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def __matmul__(self, other):
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"""
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Rotation of vector, second or fourth order tensor, or rotation object.
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Parameters
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----------
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other : numpy.ndarray or Rotation
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Vector, second or fourth order tensor, or rotation object that is rotated.
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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 self.__class__(np.block([q,p]))._standardize()
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elif 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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else:
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raise TypeError('Cannot rotate {}'.format(type(other)))
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def _standardize(self):
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"""Standardize (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 inverse(self):
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"""In-place inverse rotation (backward rotation)."""
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self.quaternion[...,1:] *= -1
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return self
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def __invert__(self):
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"""Inverse rotation (backward rotation)."""
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return self.copy().inverse()
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def inversed(self):
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"""Inverse rotation (backward rotation)."""
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return ~ self
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def misorientation(self,other):
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"""
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Get Misorientation.
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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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def broadcast_to(self,shape):
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if isinstance(shape,int): shape = (shape,)
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if self.shape == ():
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q = np.broadcast_to(self.quaternion,shape+(4,))
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else:
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q = np.block([np.broadcast_to(self.quaternion[...,0:1],shape).reshape(shape+(1,)),
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np.broadcast_to(self.quaternion[...,1:2],shape).reshape(shape+(1,)),
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np.broadcast_to(self.quaternion[...,2:3],shape).reshape(shape+(1,)),
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np.broadcast_to(self.quaternion[...,3:4],shape).reshape(shape+(1,))])
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return self.__class__(q)
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def average(self,other): #ToDo: discuss calling for vectors
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"""
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Calculate the average rotation.
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Parameters
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----------
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other : Rotation
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Rotation from which the average is rotated.
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"""
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if self.quaternion.shape != (4,) or other.quaternion.shape != (4,):
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raise NotImplementedError('Support for multiple rotations missing')
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return Rotation.fromAverage([self,other])
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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), |q|=1, q_0 ≥ 0.
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"""
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return self.quaternion.copy()
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def as_Eulers(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), |n| = 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(self,
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vector = 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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vector : bool, optional
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Return as actual Rodrigues-Frank vector, i.e. axis
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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) unless vector == True:
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numpy.ndarray of shape (...,3)
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Rodrigues-Frank vector: [n_1, n_2, n_3, tan(ω/2)], |n| = 1 and ω ∈ [0,π].
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"""
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ro = Rotation._qu2ro(self.quaternion)
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return ro[...,:3]*ro[...,3] if vector else 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), |h| < 1/2*π^(2/3).
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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 cubocoric vector.
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Returns
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-------
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c : numpy.ndarray of shape (...,3)
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Cubochoric vector: (c_1, c_2, c_3), max(c_i) < 1/2*π^(2/3).
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"""
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return Rotation._qu2cu(self.quaternion)
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def M(self): # ToDo not sure about the name: as_M or M? we do not have a from_M
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"""
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Intermediate representation supporting quaternion averaging.
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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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return np.einsum('...i,...j',self.quaternion,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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acceptHomomorph = None): # old name (for compatibility)
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"""
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Initialize from quaternion.
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Parameters
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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),
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|q|=1, q_0 ≥ 0.
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accept_homomorph: boolean, optional
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Allow homomorphic variants, i.e. q_0 < 0 (negative real hemisphere).
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Defaults to False.
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P: integer ∈ {-1,1}, optional
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Convention used. Defaults to -1.
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"""
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if acceptHomomorph is not None:
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accept_homomorph = acceptHomomorph # for compatibility
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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.')
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if abs(P) != 1:
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raise ValueError('P ∉ {-1,1}')
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if P == 1: qu[...,1:4] *= -1
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if accept_homomorph:
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qu[qu[...,0] < 0.0] *= -1
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else:
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if np.any(qu[...,0] < 0.0):
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raise ValueError('Quaternion with negative first (real) component.')
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if not np.all(np.isclose(np.linalg.norm(qu,axis=-1), 1.0)):
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raise ValueError('Quaternion is not of unit length.')
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return Rotation(qu)
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@staticmethod
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def from_Eulers(phi,
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degrees = False):
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"""
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Initialize from Bunge-Euler angles.
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Parameters
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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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degrees: boolean, optional
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Bunge-Euler angles are given in degrees. Defaults to False.
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"""
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eu = np.array(phi,dtype=float)
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if eu.shape[:-2:-1] != (3,):
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raise ValueError('Invalid shape.')
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eu = np.radians(eu) if degrees else eu
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if np.any(eu < 0.0) or np.any(eu > 2.0*np.pi) or np.any(eu[...,1] > np.pi): # ToDo: No separate check for PHI
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raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].')
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return Rotation(Rotation._eu2qu(eu))
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@staticmethod
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def from_axis_angle(axis_angle,
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degrees = False,
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normalise = False,
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P = -1):
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"""
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Initialize from Axis angle pair.
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Parameters
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----------
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axis_angle: numpy.ndarray of shape (...,4)
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Axis angle pair: [n_1, n_2, n_3, ω], |n| = 1 and ω ∈ [0,π]
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unless degrees = True: ω ∈ [0,180].
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degrees: boolean, optional
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Angle ω is given in degrees. Defaults to False.
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normalize: boolean, optional
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Allow |n| ≠ 1. Defaults to False.
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P: integer ∈ {-1,1}, optional
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Convention used. Defaults to -1.
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"""
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ax = np.array(axis_angle,dtype=float)
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if ax.shape[:-2:-1] != (4,):
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raise ValueError('Invalid shape.')
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if abs(P) != 1:
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raise ValueError('P ∉ {-1,1}')
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if P == 1: ax[...,0:3] *= -1
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if degrees: ax[..., 3] = np.radians(ax[...,3])
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if normalise: ax[...,0:3] /= np.linalg.norm(ax[...,0:3],axis=-1)
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if np.any(ax[...,3] < 0.0) or np.any(ax[...,3] > np.pi):
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raise ValueError('Axis angle rotation angle outside of [0..π].')
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if not np.all(np.isclose(np.linalg.norm(ax[...,0:3],axis=-1), 1.0)):
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raise ValueError('Axis angle rotation axis is not of unit length.')
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return Rotation(Rotation._ax2qu(ax))
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@staticmethod
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def from_basis(basis,
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orthonormal = True,
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reciprocal = False):
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"""
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Initialize from tbd.
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Parameters
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----------
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basis: numpy.ndarray of shape (...,3,3)
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tbd
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orthonormal: boolean, optional
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tbd. Defaults to True.
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reciprocal: boolean, optional
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tbd. Defaults to False.
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"""
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om = np.array(basis,dtype=float)
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if om.shape[:-3:-1] != (3,3):
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raise ValueError('Invalid shape.')
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if reciprocal:
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om = np.linalg.inv(mechanics.transpose(om)/np.pi) # transform reciprocal basis set
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orthonormal = False # contains stretch
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if not orthonormal:
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(U,S,Vh) = np.linalg.svd(om) # singular value decomposition
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om = np.einsum('...ij,...jl->...il',U,Vh)
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if not np.all(np.isclose(np.linalg.det(om),1.0)):
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raise ValueError('Orientation matrix has determinant ≠ 1.')
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if not np.all(np.isclose(np.einsum('...i,...i',om[...,0],om[...,1]), 0.0)) \
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or not np.all(np.isclose(np.einsum('...i,...i',om[...,1],om[...,2]), 0.0)) \
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or not np.all(np.isclose(np.einsum('...i,...i',om[...,2],om[...,0]), 0.0)):
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raise ValueError('Orientation matrix is not orthogonal.')
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return Rotation(Rotation._om2qu(om))
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@staticmethod
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def from_directions(hkl,uvw):
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"""
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Initialize from pair of directions/planes.
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Parameters
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----------
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hkl: numpy.ndarray of shape (...,3)
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Direction parallel to z direction, i.e. (h k l) || (0,0,1).
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uvw: numpy.ndarray of shape (...,3)
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Direction parallel to x direction, i.e. <u v w> || (1,0,0).
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"""
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hkl_ = hkl/np.linalg.norm(hkl,axis=-1,keepdims=True)
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uvw_ = uvw/np.linalg.norm(uvw,axis=-1,keepdims=True)
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v_1 = np.block([uvw_,np.cross(hkl_,uvw_),hkl_]).reshape(hkl_.shape+(3,))
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v_2 = np.block([uvw_,np.cross(uvw_,hkl_),hkl_]).reshape(hkl_.shape+(3,))
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R = np.where(np.broadcast_to(np.expand_dims(np.expand_dims(np.linalg.det(v_1)>0,-1),-1),v_1.shape),
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v_1,v_2)
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return Rotation.from_basis(np.swapaxes(R,axis2=-2,axis1=-1))
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@staticmethod
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def from_matrix(R):
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"""
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Initialize from rotation matrix.
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Parameters
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----------
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R: numpy.ndarray of shape (...,3,3)
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Rotation matrix: det(R) = 1, R.T∙R=I.
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"""
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return Rotation.from_basis(R)
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@staticmethod
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def from_Rodrigues(rho,
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normalise = False,
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P = -1):
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"""
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Initialize from Rodrigues-Frank vector.
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Parameters
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----------
|
|
rho: numpy.ndarray of shape (...,4)
|
|
Rodrigues-Frank vector (angle separated from axis).
|
|
(n_1, n_2, n_3, tan(ω/2)), |n| = 1 and ω ∈ [0,π].
|
|
normalize: boolean, optional
|
|
Allow |n| ≠ 1. Defaults to False.
|
|
P: integer ∈ {-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}')
|
|
|
|
if P == 1: ro[...,0:3] *= -1
|
|
if normalise: ro[...,0:3] /= np.linalg.norm(ro[...,0:3],axis=-1)
|
|
if np.any(ro[...,3] < 0.0):
|
|
raise ValueError('Rodrigues vector rotation angle not positive.')
|
|
if not np.all(np.isclose(np.linalg.norm(ro[...,0:3],axis=-1), 1.0)):
|
|
raise ValueError('Rodrigues vector rotation axis is not of unit length.')
|
|
|
|
return Rotation(Rotation._ro2qu(ro))
|
|
|
|
@staticmethod
|
|
def from_homochoric(h,
|
|
P = -1):
|
|
"""
|
|
Initialize from homochoric vector.
|
|
|
|
Parameters
|
|
----------
|
|
h: numpy.ndarray of shape (...,3)
|
|
Homochoric vector: (h_1, h_2, h_3), |h| < (3/4*π)^(1/3).
|
|
P: integer ∈ {-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}')
|
|
|
|
if P == 1: ho *= -1
|
|
|
|
if np.any(np.linalg.norm(ho,axis=-1) >_R1+1e-9):
|
|
raise ValueError('Homochoric coordinate outside of the sphere.')
|
|
|
|
return Rotation(Rotation._ho2qu(ho))
|
|
|
|
@staticmethod
|
|
def from_cubochoric(c,
|
|
P = -1):
|
|
"""
|
|
Initialize from cubochoric vector.
|
|
|
|
Parameters
|
|
----------
|
|
c: numpy.ndarray of shape (...,3)
|
|
Cubochoric vector: (c_1, c_2, c_3), max(c_i) < 1/2*π^(2/3).
|
|
P: integer ∈ {-1,1}, optional
|
|
Convention used. Defaults to -1.
|
|
|
|
"""
|
|
cu = np.array(c,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 = Rotation._cu2ho(cu)
|
|
if P == 1: ho *= -1
|
|
|
|
return Rotation(Rotation._ho2qu(ho))
|
|
|
|
|
|
@staticmethod
|
|
def fromAverage(rotations,weights = None):
|
|
"""
|
|
Average rotation.
|
|
|
|
References
|
|
----------
|
|
F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
|
|
https://doi.org/10.2514/1.28949
|
|
|
|
Parameters
|
|
----------
|
|
rotations : list of Rotations
|
|
Rotations to average from
|
|
weights : list of floats, optional
|
|
Weights for each rotation used for averaging
|
|
|
|
"""
|
|
if not all(isinstance(item, Rotation) for item in rotations):
|
|
raise TypeError('Only instances of Rotation can be averaged.')
|
|
|
|
N = len(rotations)
|
|
if not weights:
|
|
weights = np.ones(N,dtype='i')
|
|
|
|
for i,(r,n) in enumerate(zip(rotations,weights)):
|
|
M = r.M() * n if i == 0 \
|
|
else M + r.M() * n # noqa add (multiples) of this rotation to average noqa
|
|
eig, vec = np.linalg.eig(M/N)
|
|
|
|
return Rotation.from_quaternion(np.real(vec.T[eig.argmax()]),accept_homomorph = True)
|
|
|
|
@staticmethod
|
|
def from_random(shape=None):
|
|
if shape is None:
|
|
r = np.random.random(3)
|
|
elif hasattr(shape, '__iter__'):
|
|
r = np.random.random(tuple(shape)+(3,))
|
|
else:
|
|
r = np.random.random((shape,3))
|
|
|
|
A = np.sqrt(r[...,2])
|
|
B = np.sqrt(1.0-r[...,2])
|
|
q = np.stack([np.cos(2.0*np.pi*r[...,0])*A,
|
|
np.sin(2.0*np.pi*r[...,1])*B,
|
|
np.cos(2.0*np.pi*r[...,1])*B,
|
|
np.sin(2.0*np.pi*r[...,0])*A],axis=-1)
|
|
|
|
return Rotation(q.reshape(r.shape[:-1]+(4,)) if shape is not None else q)._standardize()
|
|
|
|
|
|
# for compatibility (old names do not follow convention)
|
|
asM = M
|
|
fromQuaternion = from_quaternion
|
|
fromEulers = from_Eulers
|
|
asAxisAngle = as_axis_angle
|
|
__mul__ = __matmul__
|
|
|
|
####################################################################################################
|
|
# Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations
|
|
####################################################################################################
|
|
# Copyright (c) 2017-2019, Martin Diehl/Max-Planck-Institut für Eisenforschung GmbH
|
|
# Copyright (c) 2013-2014, Marc De Graef/Carnegie Mellon University
|
|
# All rights reserved.
|
|
#
|
|
# Redistribution and use in source and binary forms, with or without modification, are
|
|
# permitted provided that the following conditions are met:
|
|
#
|
|
# - Redistributions of source code must retain the above copyright notice, this list
|
|
# of conditions and the following disclaimer.
|
|
# - Redistributions in binary form must reproduce the above copyright notice, this
|
|
# list of conditions and the following disclaimer in the documentation and/or
|
|
# other materials provided with the distribution.
|
|
# - Neither the names of Marc De Graef, Carnegie Mellon University nor the names
|
|
# of its contributors may be used to endorse or promote products derived from
|
|
# this software without specific prior written permission.
|
|
#
|
|
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
|
|
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
|
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
|
# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
|
|
# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
|
# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
|
|
# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
|
|
# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
|
|
# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
|
|
# USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
|
####################################################################################################
|
|
#---------- Quaternion ----------
|
|
@staticmethod
|
|
def _qu2om(qu):
|
|
qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2)
|
|
om = np.block([qq + 2.0*qu[...,1:2]**2,
|
|
2.0*(qu[...,2:3]*qu[...,1:2]-_P*qu[...,0:1]*qu[...,3:4]),
|
|
2.0*(qu[...,3:4]*qu[...,1:2]+_P*qu[...,0:1]*qu[...,2:3]),
|
|
2.0*(qu[...,1:2]*qu[...,2:3]+_P*qu[...,0:1]*qu[...,3:4]),
|
|
qq + 2.0*qu[...,2:3]**2,
|
|
2.0*(qu[...,3:4]*qu[...,2:3]-_P*qu[...,0:1]*qu[...,1:2]),
|
|
2.0*(qu[...,1:2]*qu[...,3:4]-_P*qu[...,0:1]*qu[...,2:3]),
|
|
2.0*(qu[...,2:3]*qu[...,3:4]+_P*qu[...,0:1]*qu[...,1:2]),
|
|
qq + 2.0*qu[...,3:4]**2,
|
|
]).reshape(qu.shape[:-1]+(3,3))
|
|
return om
|
|
|
|
@staticmethod
|
|
def _qu2eu(qu):
|
|
"""Quaternion to Bunge-Euler angles."""
|
|
q02 = qu[...,0:1]*qu[...,2:3]
|
|
q13 = qu[...,1:2]*qu[...,3:4]
|
|
q01 = qu[...,0:1]*qu[...,1:2]
|
|
q23 = qu[...,2:3]*qu[...,3:4]
|
|
q03_s = qu[...,0:1]**2+qu[...,3:4]**2
|
|
q12_s = qu[...,1:2]**2+qu[...,2:3]**2
|
|
chi = np.sqrt(q03_s*q12_s)
|
|
|
|
eu = np.where(np.abs(q12_s) < 1.0e-8,
|
|
np.block([np.arctan2(-_P*2.0*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2),
|
|
np.zeros(qu.shape[:-1]+(2,))]),
|
|
np.where(np.abs(q03_s) < 1.0e-8,
|
|
np.block([np.arctan2( 2.0*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2),
|
|
np.broadcast_to(np.pi,qu[...,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]),1.0,1e-9),
|
|
np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]),
|
|
np.pi*0.5*(1-om[...,2,2:3]),
|
|
np.zeros(om.shape[:-2]+(1,)),
|
|
]),
|
|
np.block([np.arctan2(om[...,2,0:1]*zeta,-om[...,2,1:2]*zeta),
|
|
np.arccos( om[...,2,2:3]),
|
|
np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta)
|
|
])
|
|
)
|
|
eu[np.abs(eu)<1.e-8] = 0.0
|
|
eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
|
|
return eu
|
|
|
|
@staticmethod
|
|
def _om2ax(om):
|
|
"""Rotation matrix to axis angle pair."""
|
|
#return Rotation._qu2ax(Rotation._om2qu(om)) # HOTFIX
|
|
diag_delta = -_P*np.block([om[...,1,2:3]-om[...,2,1:2],
|
|
om[...,2,0:1]-om[...,0,2:3],
|
|
om[...,0,1:2]-om[...,1,0:1]
|
|
])
|
|
t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.0).reshape(om.shape[:-2]+(1,))
|
|
w,vr = np.linalg.eig(om)
|
|
# mask duplicated real eigenvalues
|
|
w[np.isclose(w[...,0],1.0+0.0j),1:] = 0.
|
|
w[np.isclose(w[...,1],1.0+0.0j),2:] = 0.
|
|
vr = np.swapaxes(vr,-1,-2)
|
|
ax = np.where(np.abs(diag_delta)<1e-12,
|
|
np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,)),
|
|
np.abs(np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,))) \
|
|
*np.sign(diag_delta))
|
|
ax = np.block([ax,np.arccos(np.clip(t,-1.0,1.0))])
|
|
ax[np.abs(ax[...,3])<1.e-8] = [ 0.0, 0.0, 1.0, 0.0]
|
|
return ax
|
|
|
|
@staticmethod
|
|
def _om2ro(om):
|
|
"""Rotation matrix to Rodrigues-Frank vector."""
|
|
return Rotation._eu2ro(Rotation._om2eu(om))
|
|
|
|
@staticmethod
|
|
def _om2ho(om):
|
|
"""Rotation matrix to homochoric vector."""
|
|
return Rotation._ax2ho(Rotation._om2ax(om))
|
|
|
|
@staticmethod
|
|
def _om2cu(om):
|
|
"""Rotation matrix to cubochoric vector."""
|
|
return Rotation._ho2cu(Rotation._om2ho(om))
|
|
|
|
|
|
#---------- Bunge-Euler angles ----------
|
|
@staticmethod
|
|
def _eu2qu(eu):
|
|
"""Bunge-Euler angles to quaternion."""
|
|
ee = 0.5*eu
|
|
cPhi = np.cos(ee[...,1:2])
|
|
sPhi = np.sin(ee[...,1:2])
|
|
qu = np.block([ cPhi*np.cos(ee[...,0:1]+ee[...,2:3]),
|
|
-_P*sPhi*np.cos(ee[...,0:1]-ee[...,2:3]),
|
|
-_P*sPhi*np.sin(ee[...,0:1]-ee[...,2:3]),
|
|
-_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])])
|
|
qu[qu[...,0]<0.0]*=-1
|
|
return qu
|
|
|
|
@staticmethod
|
|
def _eu2om(eu):
|
|
"""Bunge-Euler angles to rotation matrix."""
|
|
c = np.cos(eu)
|
|
s = np.sin(eu)
|
|
om = np.block([+c[...,0:1]*c[...,2:3]-s[...,0:1]*s[...,2:3]*c[...,1:2],
|
|
+s[...,0:1]*c[...,2:3]+c[...,0:1]*s[...,2:3]*c[...,1:2],
|
|
+s[...,2:3]*s[...,1:2],
|
|
-c[...,0:1]*s[...,2:3]-s[...,0:1]*c[...,2:3]*c[...,1:2],
|
|
-s[...,0:1]*s[...,2:3]+c[...,0:1]*c[...,2:3]*c[...,1:2],
|
|
+c[...,2:3]*s[...,1:2],
|
|
+s[...,0:1]*s[...,1:2],
|
|
-c[...,0:1]*s[...,1:2],
|
|
+c[...,1:2]
|
|
]).reshape(eu.shape[:-1]+(3,3))
|
|
om[np.abs(om)<1.e-12] = 0.0
|
|
return om
|
|
|
|
@staticmethod
|
|
def _eu2ax(eu):
|
|
"""Bunge-Euler angles to axis angle pair."""
|
|
t = np.tan(eu[...,1:2]*0.5)
|
|
sigma = 0.5*(eu[...,0:1]+eu[...,2:3])
|
|
delta = 0.5*(eu[...,0:1]-eu[...,2:3])
|
|
tau = np.linalg.norm(np.block([t,np.sin(sigma)]),axis=-1,keepdims=True)
|
|
alpha = np.where(np.abs(np.cos(sigma))<1.e-12,np.pi,2.0*np.arctan(tau/np.cos(sigma)))
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
ax = np.where(np.broadcast_to(np.abs(alpha)<1.0e-12,eu.shape[:-1]+(4,)),
|
|
[0.0,0.0,1.0,0.0],
|
|
np.block([-_P/tau*t*np.cos(delta),
|
|
-_P/tau*t*np.sin(delta),
|
|
-_P/tau* np.sin(sigma),
|
|
alpha
|
|
]))
|
|
ax[(alpha<0.0).squeeze()] *=-1
|
|
return ax
|
|
|
|
@staticmethod
|
|
def _eu2ro(eu):
|
|
"""Bunge-Euler angles to Rodrigues-Frank vector."""
|
|
ax = Rotation._eu2ax(eu)
|
|
ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)])
|
|
ro[ax[...,3]>=np.pi,3] = np.inf
|
|
ro[np.abs(ax[...,3])<1.e-16] = [ 0.0, 0.0, _P, 0.0 ]
|
|
return ro
|
|
|
|
@staticmethod
|
|
def _eu2ho(eu):
|
|
"""Bunge-Euler angles to homochoric vector."""
|
|
return Rotation._ax2ho(Rotation._eu2ax(eu))
|
|
|
|
@staticmethod
|
|
def _eu2cu(eu):
|
|
"""Bunge-Euler angles to cubochoric vector."""
|
|
return Rotation._ho2cu(Rotation._eu2ho(eu))
|
|
|
|
|
|
#---------- Axis angle pair ----------
|
|
@staticmethod
|
|
def _ax2qu(ax):
|
|
"""Axis angle pair to quaternion."""
|
|
c = np.cos(ax[...,3:4]*.5)
|
|
s = np.sin(ax[...,3:4]*.5)
|
|
qu = np.where(np.abs(ax[...,3:4])<1.e-6,[1.0, 0.0, 0.0, 0.0],np.block([c, ax[...,:3]*s]))
|
|
return qu
|
|
|
|
@staticmethod
|
|
def _ax2om(ax):
|
|
"""Axis angle pair to rotation matrix."""
|
|
c = np.cos(ax[...,3:4])
|
|
s = np.sin(ax[...,3:4])
|
|
omc = 1. -c
|
|
om = np.block([c+omc*ax[...,0:1]**2,
|
|
omc*ax[...,0:1]*ax[...,1:2] + s*ax[...,2:3],
|
|
omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2],
|
|
omc*ax[...,0:1]*ax[...,1:2] - s*ax[...,2:3],
|
|
c+omc*ax[...,1:2]**2,
|
|
omc*ax[...,1:2]*ax[...,2:3] + s*ax[...,0:1],
|
|
omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2],
|
|
omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1],
|
|
c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3))
|
|
return om if _P < 0.0 else np.swapaxes(om,-1,-2)
|
|
|
|
@staticmethod
|
|
def _ax2eu(ax):
|
|
"""Rotation matrix to Bunge Euler angles."""
|
|
return Rotation._om2eu(Rotation._ax2om(ax))
|
|
|
|
@staticmethod
|
|
def _ax2ro(ax):
|
|
"""Axis angle pair to Rodrigues-Frank vector."""
|
|
ro = np.block([ax[...,:3],
|
|
np.where(np.isclose(ax[...,3:4],np.pi,atol=1.e-15,rtol=.0),
|
|
np.inf,
|
|
np.tan(ax[...,3:4]*0.5))
|
|
])
|
|
ro[np.abs(ax[...,3])<1.e-6] = [.0,.0,_P,.0]
|
|
return ro
|
|
|
|
@staticmethod
|
|
def _ax2ho(ax):
|
|
"""Axis angle pair to homochoric vector."""
|
|
f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1.0/3.0)
|
|
ho = ax[...,:3] * f
|
|
return ho
|
|
|
|
@staticmethod
|
|
def _ax2cu(ax):
|
|
"""Axis angle pair to cubochoric vector."""
|
|
return Rotation._ho2cu(Rotation._ax2ho(ax))
|
|
|
|
|
|
#---------- Rodrigues-Frank vector ----------
|
|
@staticmethod
|
|
def _ro2qu(ro):
|
|
"""Rodrigues-Frank vector to quaternion."""
|
|
return Rotation._ax2qu(Rotation._ro2ax(ro))
|
|
|
|
@staticmethod
|
|
def _ro2om(ro):
|
|
"""Rodgrigues-Frank vector to rotation matrix."""
|
|
return Rotation._ax2om(Rotation._ro2ax(ro))
|
|
|
|
@staticmethod
|
|
def _ro2eu(ro):
|
|
"""Rodrigues-Frank vector to Bunge-Euler angles."""
|
|
return Rotation._om2eu(Rotation._ro2om(ro))
|
|
|
|
@staticmethod
|
|
def _ro2ax(ro):
|
|
"""Rodrigues-Frank vector to axis angle pair."""
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
ax = np.where(np.isfinite(ro[...,3:4]),
|
|
np.block([ro[...,0:3]*np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True),2.*np.arctan(ro[...,3:4])]),
|
|
np.block([ro[...,0:3],np.broadcast_to(np.pi,ro[...,3:4].shape)]))
|
|
ax[np.abs(ro[...,3]) < 1.e-8] = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
|
return ax
|
|
|
|
@staticmethod
|
|
def _ro2ho(ro):
|
|
"""Rodrigues-Frank vector to homochoric vector."""
|
|
f = np.where(np.isfinite(ro[...,3:4]),2.0*np.arctan(ro[...,3:4]) -np.sin(2.0*np.arctan(ro[...,3:4])),np.pi)
|
|
ho = np.where(np.broadcast_to(np.sum(ro[...,0:3]**2.0,axis=-1,keepdims=True) < 1.e-8,ro[...,0:3].shape),
|
|
np.zeros(3), ro[...,0:3]* (0.75*f)**(1.0/3.0))
|
|
return ho
|
|
|
|
@staticmethod
|
|
def _ro2cu(ro):
|
|
"""Rodrigues-Frank vector to cubochoric vector."""
|
|
return Rotation._ho2cu(Rotation._ro2ho(ro))
|
|
|
|
|
|
#---------- Homochoric vector----------
|
|
@staticmethod
|
|
def _ho2qu(ho):
|
|
"""Homochoric vector to quaternion."""
|
|
return Rotation._ax2qu(Rotation._ho2ax(ho))
|
|
|
|
@staticmethod
|
|
def _ho2om(ho):
|
|
"""Homochoric vector to rotation matrix."""
|
|
return Rotation._ax2om(Rotation._ho2ax(ho))
|
|
|
|
@staticmethod
|
|
def _ho2eu(ho):
|
|
"""Homochoric vector to Bunge-Euler angles."""
|
|
return Rotation._ax2eu(Rotation._ho2ax(ho))
|
|
|
|
@staticmethod
|
|
def _ho2ax(ho):
|
|
"""Homochoric vector to axis angle pair."""
|
|
tfit = np.array([+1.0000000000018852, -0.5000000002194847,
|
|
-0.024999992127593126, -0.003928701544781374,
|
|
-0.0008152701535450438, -0.0002009500426119712,
|
|
-0.00002397986776071756, -0.00008202868926605841,
|
|
+0.00012448715042090092, -0.0001749114214822577,
|
|
+0.0001703481934140054, -0.00012062065004116828,
|
|
+0.000059719705868660826, -0.00001980756723965647,
|
|
+0.000003953714684212874, -0.00000036555001439719544])
|
|
hmag_squared = np.sum(ho**2.,axis=-1,keepdims=True)
|
|
hm = hmag_squared.copy()
|
|
s = tfit[0] + tfit[1] * hmag_squared
|
|
for i in range(2,16):
|
|
hm *= hmag_squared
|
|
s += tfit[i] * hm
|
|
with np.errstate(invalid='ignore'):
|
|
ax = np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-8,ho.shape[:-1]+(4,)),
|
|
[ 0.0, 0.0, 1.0, 0.0 ],
|
|
np.block([ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))]))
|
|
return ax
|
|
|
|
@staticmethod
|
|
def _ho2ro(ho):
|
|
"""Axis angle pair to Rodrigues-Frank vector."""
|
|
return Rotation._ax2ro(Rotation._ho2ax(ho))
|
|
|
|
@staticmethod
|
|
def _ho2cu(ho):
|
|
"""
|
|
Homochoric vector to cubochoric vector.
|
|
|
|
References
|
|
----------
|
|
D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
|
|
https://doi.org/10.1088/0965-0393/22/7/075013
|
|
|
|
"""
|
|
rs = np.linalg.norm(ho,axis=-1,keepdims=True)
|
|
|
|
xyz3 = np.take_along_axis(ho,Rotation._get_pyramid_order(ho,'forward'),-1)
|
|
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
# inverse M_3
|
|
xyz2 = xyz3[...,0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[...,2:3])) )
|
|
qxy = np.sum(xyz2**2,axis=-1,keepdims=True)
|
|
|
|
q2 = qxy + np.max(np.abs(xyz2),axis=-1,keepdims=True)**2
|
|
sq2 = np.sqrt(q2)
|
|
q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2))
|
|
tt = np.clip((np.min(np.abs(xyz2),axis=-1,keepdims=True)**2\
|
|
+np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
|
|
T_inv = np.where(np.abs(xyz2[...,1:2]) <= np.abs(xyz2[...,0:1]),
|
|
np.block([np.ones_like(tt),np.arccos(tt)/np.pi*12.0]),
|
|
np.block([np.arccos(tt)/np.pi*12.0,np.ones_like(tt)]))*q
|
|
T_inv[xyz2<0.0] *= -1.0
|
|
T_inv[np.broadcast_to(np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-12),T_inv.shape)] = 0.0
|
|
cu = np.block([T_inv, np.where(xyz3[...,2:3]<0.0,-np.ones_like(xyz3[...,2:3]),np.ones_like(xyz3[...,2:3])) \
|
|
* rs/np.sqrt(6.0/np.pi),
|
|
])/ _sc
|
|
|
|
cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
|
|
cu = np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1)
|
|
|
|
return cu
|
|
|
|
#---------- Cubochoric ----------
|
|
@staticmethod
|
|
def _cu2qu(cu):
|
|
"""Cubochoric vector to quaternion."""
|
|
return Rotation._ho2qu(Rotation._cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def _cu2om(cu):
|
|
"""Cubochoric vector to rotation matrix."""
|
|
return Rotation._ho2om(Rotation._cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def _cu2eu(cu):
|
|
"""Cubochoric vector to Bunge-Euler angles."""
|
|
return Rotation._ho2eu(Rotation._cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def _cu2ax(cu):
|
|
"""Cubochoric vector to axis angle pair."""
|
|
return Rotation._ho2ax(Rotation._cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def _cu2ro(cu):
|
|
"""Cubochoric vector to Rodrigues-Frank vector."""
|
|
return Rotation._ho2ro(Rotation._cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def _cu2ho(cu):
|
|
"""
|
|
Cubochoric vector to homochoric vector.
|
|
|
|
References
|
|
----------
|
|
D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
|
|
https://doi.org/10.1088/0965-0393/22/7/075013
|
|
|
|
"""
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
# get pyramide and scale by grid parameter ratio
|
|
XYZ = np.take_along_axis(cu,Rotation._get_pyramid_order(cu,'forward'),-1) * _sc
|
|
order = np.abs(XYZ[...,1:2]) <= np.abs(XYZ[...,0:1])
|
|
q = np.pi/12.0 * np.where(order,XYZ[...,1:2],XYZ[...,0:1]) \
|
|
/ np.where(order,XYZ[...,0:1],XYZ[...,1:2])
|
|
c = np.cos(q)
|
|
s = np.sin(q)
|
|
q = _R1*2.0**0.25/_beta/ np.sqrt(np.sqrt(2.0)-c) \
|
|
* np.where(order,XYZ[...,0:1],XYZ[...,1:2])
|
|
|
|
T = np.block([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
|
|
|
|
# transform to sphere grid (inverse Lambert)
|
|
c = np.sum(T**2,axis=-1,keepdims=True)
|
|
s = c * np.pi/24.0 /XYZ[...,2:3]**2
|
|
c = c * np.sqrt(np.pi/24.0)/XYZ[...,2:3]
|
|
q = np.sqrt( 1.0 - s)
|
|
|
|
ho = np.where(np.isclose(np.sum(np.abs(XYZ[...,0:2]),axis=-1,keepdims=True),0.0,rtol=0.0,atol=1.0e-16),
|
|
np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6.0/np.pi) *XYZ[...,2:3]]),
|
|
np.block([np.where(order,T[...,0:1],T[...,1:2])*q,
|
|
np.where(order,T[...,1:2],T[...,0:1])*q,
|
|
np.sqrt(6.0/np.pi) * XYZ[...,2:3] - c])
|
|
)
|
|
|
|
ho[np.isclose(np.sum(np.abs(cu),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
|
|
ho = np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1)
|
|
|
|
return ho
|
|
|
|
|
|
@staticmethod
|
|
def _get_pyramid_order(xyz,direction=None):
|
|
"""
|
|
Get order of the coordinates.
|
|
|
|
Depending on the pyramid in which the point is located, the order need to be adjusted.
|
|
|
|
Parameters
|
|
----------
|
|
xyz : numpy.ndarray
|
|
coordinates of a point on a uniform refinable grid on a ball or
|
|
in a uniform refinable cubical grid.
|
|
|
|
References
|
|
----------
|
|
D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
|
|
https://doi.org/10.1088/0965-0393/22/7/075013
|
|
|
|
"""
|
|
order = {'forward': np.array([[0,1,2],[1,2,0],[2,0,1]]),
|
|
'backward':np.array([[0,1,2],[2,0,1],[1,2,0]])}
|
|
|
|
p = np.where(np.maximum(np.abs(xyz[...,0]),np.abs(xyz[...,1])) <= np.abs(xyz[...,2]),0,
|
|
np.where(np.maximum(np.abs(xyz[...,1]),np.abs(xyz[...,2])) <= np.abs(xyz[...,0]),1,2))
|
|
|
|
return order[direction][p]
|