1478 lines
55 KiB
Python
1478 lines
55 KiB
Python
# -*- coding: UTF-8 no BOM -*-
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import math
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import numpy as np
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from . import Lambert
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from .quaternion import Quaternion
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from .quaternion import P
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####################################################################################################
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class Rotation:
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u"""
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Orientation stored as unit quaternion.
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Following: D Rowenhorst et al. Consistent representations of and conversions between 3D rotations
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10.1088/0965-0393/23/8/083501
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Convention 1: coordinate frames are right-handed
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Convention 2: 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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Convention 3: rotations will be interpreted in the passive sense
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Convention 4: 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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Convention 5: the rotation angle ω is limited to the interval [0, π]
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Convention 6: P = -1 (as default)
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q is the real part, p = (x, y, z) are the imaginary parts.
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Vector "a" (defined in coordinate system "A") is passively rotated
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resulting in new coordinates "b" when 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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"""
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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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If a quaternion is given, it needs to comply with the convection. Use .fromQuaternion
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to check the input.
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"""
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if isinstance(quaternion,Quaternion):
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self.quaternion = quaternion.copy()
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else:
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self.quaternion = Quaternion(q=quaternion[0],p=quaternion[1:4])
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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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"""Value in selected representation"""
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return '\n'.join([
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'{}'.format(self.quaternion),
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'Matrix:\n{}'.format( '\n'.join(['\t'.join(list(map(str,self.asMatrix()[i,:]))) for i in range(3)]) ),
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'Bunge Eulers / deg: {}'.format('\t'.join(list(map(str,self.asEulers(degrees=True)))) ),
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])
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def __mul__(self, other):
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"""
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Multiplication
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Rotation: Details needed (active/passive), rotation of (3,3,3,3)-matrix should be considered
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"""
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if isinstance(other, Rotation): # rotate a rotation
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return self.__class__(self.quaternion * other.quaternion).standardize()
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elif isinstance(other, np.ndarray):
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if other.shape == (3,): # rotate a single (3)-vector
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( x, y, z) = self.quaternion.p
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(Vx,Vy,Vz) = other[0:3]
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A = self.quaternion.q*self.quaternion.q - np.dot(self.quaternion.p,self.quaternion.p)
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B = 2.0 * (x*Vx + y*Vy + z*Vz)
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C = 2.0 * P*self.quaternion.q
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return np.array([
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A*Vx + B*x + C*(y*Vz - z*Vy),
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A*Vy + B*y + C*(z*Vx - x*Vz),
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A*Vz + B*z + C*(x*Vy - y*Vx),
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])
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elif other.shape == (3,3,): # rotate a single (3x3)-matrix
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return np.dot(self.asMatrix(),np.dot(other,self.asMatrix().T))
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elif other.shape == (3,3,3,3):
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raise NotImplementedError
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else:
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return NotImplemented
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elif isinstance(other, tuple): # used to rotate a meshgrid-tuple
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( x, y, z) = self.quaternion.p
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(Vx,Vy,Vz) = other[0:3]
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A = self.quaternion.q*self.quaternion.q - np.dot(self.quaternion.p,self.quaternion.p)
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B = 2.0 * (x*Vx + y*Vy + z*Vz)
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C = 2.0 * P*self.quaternion.q
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return np.array([
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A*Vx + B*x + C*(y*Vz - z*Vy),
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A*Vy + B*y + C*(z*Vx - x*Vz),
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A*Vz + B*z + C*(x*Vy - y*Vx),
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])
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else:
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return NotImplemented
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def inverse(self):
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"""In-place inverse rotation/backward rotation"""
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self.quaternion.conjugate()
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return self
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def inversed(self):
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"""Inverse rotation/backward rotation"""
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return self.copy().inverse()
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def standardize(self):
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"""In-place quaternion representation with positive q"""
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if self.quaternion.q < 0.0: self.quaternion.homomorph()
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return self
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def standardized(self):
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"""Quaternion representation with positive q"""
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return self.copy().standardize()
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def misorientation(self,other):
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"""Misorientation"""
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return other*self.inversed()
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def average(self,other):
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"""Calculate the average rotation"""
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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 asQuaternion(self):
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"""Unit quaternion: (q, [p_1, p_2, p_3])"""
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return self.quaternion.asArray()
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def asEulers(self,
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degrees = False):
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"""Bunge-Euler angles: (φ_1, ϕ, φ_2)"""
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eu = qu2eu(self.quaternion.asArray())
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if degrees: eu = np.degrees(eu)
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return eu
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def asAxisAngle(self,
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degrees = False):
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"""Axis-angle pair: ([n_1, n_2, n_3], ω)"""
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ax = qu2ax(self.quaternion.asArray())
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if degrees: ax[3] = np.degrees(ax[3])
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return ax
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def asMatrix(self):
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"""Rotation matrix"""
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return qu2om(self.quaternion.asArray())
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def asRodrigues(self,vector=False):
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"""Rodrigues-Frank vector: ([n_1, n_2, n_3], tan(ω/2))"""
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ro = qu2ro(self.quaternion.asArray())
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return ro[:3]*ro[3] if vector else ro
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def asHomochoric(self):
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"""Homochoric vector: (h_1, h_2, h_3)"""
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return qu2ho(self.quaternion.asArray())
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def asCubochoric(self):
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return qu2cu(self.quaternion.asArray())
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def asM(self):
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"""Intermediate representation supporting quaternion averaging (see F. Landis Markley et al.)"""
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return self.quaternion.asM()
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################################################################################################
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# static constructors. The input data needs to follow the convention, options allow to
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# relax these convections
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@classmethod
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def fromQuaternion(cls,
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quaternion,
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acceptHomomorph = False,
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P = -1):
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qu = quaternion if isinstance(quaternion, np.ndarray) else np.array(quaternion)
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if P > 0: qu[1:4] *= -1 # convert from P=1 to P=-1
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if qu[0] < 0.0:
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if acceptHomomorph:
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qu *= -1.
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else:
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raise ValueError('Quaternion has negative first component.\n{}'.format(qu[0]))
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if not np.isclose(np.linalg.norm(qu), 1.0):
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raise ValueError('Quaternion is not of unit length.\n{} {} {} {}'.format(*qu))
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return cls(qu)
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@classmethod
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def fromEulers(cls,
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eulers,
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degrees = False):
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eu = eulers if isinstance(eulers, np.ndarray) else np.array(eulers)
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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 eu[1] > np.pi:
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raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].\n{} {} {}.'.format(*eu))
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return cls(eu2qu(eu))
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@classmethod
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def fromAxisAngle(cls,
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angleAxis,
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degrees = False,
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normalise = False,
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P = -1):
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ax = angleAxis if isinstance(angleAxis, np.ndarray) else np.array(angleAxis)
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if P > 0: ax[0:3] *= -1 # convert from P=1 to P=-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])
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if ax[3] < 0.0 or ax[3] > np.pi:
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raise ValueError('Axis angle rotation angle outside of [0..π].\n'.format(ax[3]))
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if not np.isclose(np.linalg.norm(ax[0:3]), 1.0):
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raise ValueError('Axis angle rotation axis is not of unit length.\n{} {} {}'.format(*ax[0:3]))
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return cls(ax2qu(ax))
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@classmethod
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def fromMatrix(cls,
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matrix,
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containsStretch = False): #ToDo: better name?
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om = matrix if isinstance(matrix, np.ndarray) else np.array(matrix).reshape((3,3)) # ToDo: Reshape here or require explicit?
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if containsStretch:
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(U,S,Vh) = np.linalg.svd(om) # singular value decomposition
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om = np.dot(U,Vh)
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if not np.isclose(np.linalg.det(om),1.0):
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raise ValueError('matrix is not a proper rotation.\n{}'.format(om))
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if not np.isclose(np.dot(om[0],om[1]), 0.0) \
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or not np.isclose(np.dot(om[1],om[2]), 0.0) \
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or not np.isclose(np.dot(om[2],om[0]), 0.0):
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raise ValueError('matrix is not orthogonal.\n{}'.format(om))
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return cls(om2qu(om))
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@classmethod
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def fromRodrigues(cls,
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rodrigues,
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normalise = False,
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P = -1):
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ro = rodrigues if isinstance(rodrigues, np.ndarray) else np.array(rodrigues)
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if P > 0: ro[0:3] *= -1 # convert from P=1 to P=-1
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if normalise: ro[0:3] /=np.linalg.norm(ro[0:3])
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if not np.isclose(np.linalg.norm(ro[0:3]), 1.0):
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raise ValueError('Rodrigues rotation axis is not of unit length.\n{} {} {}'.format(*ro[0:3]))
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if ro[3] < 0.0:
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raise ValueError('Rodriques rotation angle not positive.\n'.format(ro[3]))
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return cls(ro2qu(ro))
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@classmethod
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def fromHomochoric(cls,
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homochoric,
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P = -1):
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ho = homochoric if isinstance(homochoric, np.ndarray) else np.array(homochoric)
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if P > 0: ho *= -1 # convert from P=1 to P=-1
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return cls(ho2qu(ho))
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@classmethod
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def fromCubochoric(cls,
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cubochoric,
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P = -1):
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cu = cubochoric if isinstance(cubochoric, np.ndarray) else np.array(cubochoric)
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ho = cu2ho(cu)
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if P > 0: ho *= -1 # convert from P=1 to P=-1
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return cls(ho2qu(ho))
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@classmethod
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def fromAverage(cls,
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rotations,
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weights = []):
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"""
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Average rotation
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ref: F. Landis Markley, Yang Cheng, John Lucas Crassidis, and Yaakov Oshman.
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Averaging Quaternions,
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Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4 (2007), pp. 1193-1197.
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doi: 10.2514/1.28949
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usage: input a list of rotations and optional weights
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"""
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if not all(isinstance(item, Rotation) for item in rotations):
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raise TypeError("Only instances of Rotation can be averaged.")
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N = len(rotations)
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if weights == [] or not weights:
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weights = np.ones(N,dtype='i')
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for i,(r,n) in enumerate(zip(rotations,weights)):
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M = r.asM() * n if i == 0 \
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else M + r.asM() * n # noqa add (multiples) of this rotation to average noqa
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eig, vec = np.linalg.eig(M/N)
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return cls.fromQuaternion(np.real(vec.T[eig.argmax()]),acceptHomomorph = True)
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@classmethod
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def fromRandom(cls):
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r = np.random.random(3)
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A = np.sqrt(r[2])
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B = np.sqrt(1.0-r[2])
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w = np.cos(2.0*np.pi*r[0])*A
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x = np.sin(2.0*np.pi*r[1])*B
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y = np.cos(2.0*np.pi*r[1])*B
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z = np.sin(2.0*np.pi*r[0])*A
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return cls.fromQuaternion([w,x,y,z],acceptHomomorph=True)
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# ******************************************************************************************
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class Symmetry:
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"""
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Symmetry operations for lattice systems
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https://en.wikipedia.org/wiki/Crystal_system
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"""
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lattices = [None,'orthorhombic','tetragonal','hexagonal','cubic',]
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def __init__(self, symmetry = None):
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if isinstance(symmetry, str) and symmetry.lower() in Symmetry.lattices:
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self.lattice = symmetry.lower()
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else:
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self.lattice = None
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def __copy__(self):
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"""Copy"""
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return self.__class__(self.lattice)
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copy = __copy__
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def __repr__(self):
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"""Readable string"""
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return '{}'.format(self.lattice)
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def __eq__(self, other):
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"""Equal to other"""
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return self.lattice == other.lattice
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def __neq__(self, other):
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"""Not equal to other"""
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return not self.__eq__(other)
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def __cmp__(self,other):
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"""Linear ordering"""
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myOrder = Symmetry.lattices.index(self.lattice)
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otherOrder = Symmetry.lattices.index(other.lattice)
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return (myOrder > otherOrder) - (myOrder < otherOrder)
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def symmetryOperations(self,members=[]):
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"""List (or single element) of symmetry operations as rotations."""
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if self.lattice == 'cubic':
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symQuats = [
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[ 1.0, 0.0, 0.0, 0.0 ],
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[ 0.0, 1.0, 0.0, 0.0 ],
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[ 0.0, 0.0, 1.0, 0.0 ],
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[ 0.0, 0.0, 0.0, 1.0 ],
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[ 0.0, 0.0, 0.5*np.sqrt(2), 0.5*np.sqrt(2) ],
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[ 0.0, 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2) ],
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[ 0.0, 0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2) ],
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[ 0.0, 0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2) ],
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[ 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ],
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[ 0.0, -0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ],
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[ 0.5, 0.5, 0.5, 0.5 ],
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[-0.5, 0.5, 0.5, 0.5 ],
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[-0.5, 0.5, 0.5, -0.5 ],
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[-0.5, 0.5, -0.5, 0.5 ],
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[-0.5, -0.5, 0.5, 0.5 ],
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[-0.5, -0.5, 0.5, -0.5 ],
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[-0.5, -0.5, -0.5, 0.5 ],
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[-0.5, 0.5, -0.5, -0.5 ],
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[-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
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[ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
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[-0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2), 0.0 ],
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[-0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2), 0.0 ],
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[-0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0, 0.0 ],
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[-0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0, 0.0 ],
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]
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elif self.lattice == 'hexagonal':
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symQuats = [
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[ 1.0, 0.0, 0.0, 0.0 ],
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[-0.5*np.sqrt(3), 0.0, 0.0, -0.5 ],
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[ 0.5, 0.0, 0.0, 0.5*np.sqrt(3) ],
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[ 0.0, 0.0, 0.0, 1.0 ],
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[-0.5, 0.0, 0.0, 0.5*np.sqrt(3) ],
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[-0.5*np.sqrt(3), 0.0, 0.0, 0.5 ],
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[ 0.0, 1.0, 0.0, 0.0 ],
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[ 0.0, -0.5*np.sqrt(3), 0.5, 0.0 ],
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[ 0.0, 0.5, -0.5*np.sqrt(3), 0.0 ],
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[ 0.0, 0.0, 1.0, 0.0 ],
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[ 0.0, -0.5, -0.5*np.sqrt(3), 0.0 ],
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[ 0.0, 0.5*np.sqrt(3), 0.5, 0.0 ],
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]
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elif self.lattice == 'tetragonal':
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symQuats = [
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[ 1.0, 0.0, 0.0, 0.0 ],
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[ 0.0, 1.0, 0.0, 0.0 ],
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[ 0.0, 0.0, 1.0, 0.0 ],
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[ 0.0, 0.0, 0.0, 1.0 ],
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[ 0.0, 0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ],
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[ 0.0, -0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ],
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[ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
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[-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ],
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]
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elif self.lattice == 'orthorhombic':
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symQuats = [
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[ 1.0,0.0,0.0,0.0 ],
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[ 0.0,1.0,0.0,0.0 ],
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[ 0.0,0.0,1.0,0.0 ],
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[ 0.0,0.0,0.0,1.0 ],
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]
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else:
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symQuats = [
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[ 1.0,0.0,0.0,0.0 ],
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]
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symOps = list(map(Rotation,
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np.array(symQuats)[np.atleast_1d(members) if members != [] else range(len(symQuats))]))
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try:
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iter(members) # asking for (even empty) list of members?
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except TypeError:
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return symOps[0] # no, return rotation object
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else:
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return symOps # yes, return list of rotations
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def inFZ(self,rodrigues):
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"""
|
|
Check whether given Rodrigues vector falls into fundamental zone of own symmetry.
|
|
|
|
Fundamental zone in Rodrigues space is point symmetric around origin.
|
|
"""
|
|
if (len(rodrigues) != 3):
|
|
raise ValueError('Input is not a Rodriques-Frank vector.\n')
|
|
|
|
if np.any(rodrigues == np.inf): return False
|
|
|
|
Rabs = abs(rodrigues)
|
|
|
|
if self.lattice == 'cubic':
|
|
return math.sqrt(2.0)-1.0 >= Rabs[0] \
|
|
and math.sqrt(2.0)-1.0 >= Rabs[1] \
|
|
and math.sqrt(2.0)-1.0 >= Rabs[2] \
|
|
and 1.0 >= Rabs[0] + Rabs[1] + Rabs[2]
|
|
elif self.lattice == 'hexagonal':
|
|
return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] and 1.0 >= Rabs[2] \
|
|
and 2.0 >= math.sqrt(3)*Rabs[0] + Rabs[1] \
|
|
and 2.0 >= math.sqrt(3)*Rabs[1] + Rabs[0] \
|
|
and 2.0 >= math.sqrt(3) + Rabs[2]
|
|
elif self.lattice == 'tetragonal':
|
|
return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] \
|
|
and math.sqrt(2.0) >= Rabs[0] + Rabs[1] \
|
|
and math.sqrt(2.0) >= Rabs[2] + 1.0
|
|
elif self.lattice == 'orthorhombic':
|
|
return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] and 1.0 >= Rabs[2]
|
|
else:
|
|
return True
|
|
|
|
|
|
def inDisorientationSST(self,rodrigues):
|
|
"""
|
|
Check whether given Rodrigues vector (of misorientation) falls into standard stereographic triangle of own symmetry.
|
|
|
|
Determination of disorientations follow the work of A. Heinz and P. Neumann:
|
|
Representation of Orientation and Disorientation Data for Cubic, Hexagonal, Tetragonal and Orthorhombic Crystals
|
|
Acta Cryst. (1991). A47, 780-789
|
|
"""
|
|
if (len(rodrigues) != 3):
|
|
raise ValueError('Input is not a Rodriques-Frank vector.\n')
|
|
R = rodrigues
|
|
|
|
epsilon = 0.0
|
|
if self.lattice == 'cubic':
|
|
return R[0] >= R[1]+epsilon and R[1] >= R[2]+epsilon and R[2] >= epsilon
|
|
|
|
elif self.lattice == 'hexagonal':
|
|
return R[0] >= math.sqrt(3)*(R[1]-epsilon) and R[1] >= epsilon and R[2] >= epsilon
|
|
|
|
elif self.lattice == 'tetragonal':
|
|
return R[0] >= R[1]-epsilon and R[1] >= epsilon and R[2] >= epsilon
|
|
|
|
elif self.lattice == 'orthorhombic':
|
|
return R[0] >= epsilon and R[1] >= epsilon and R[2] >= epsilon
|
|
|
|
else:
|
|
return True
|
|
|
|
|
|
def inSST(self,
|
|
vector,
|
|
proper = False,
|
|
color = False):
|
|
"""
|
|
Check whether given vector falls into standard stereographic triangle of own symmetry.
|
|
|
|
proper considers only vectors with z >= 0, hence uses two neighboring SSTs.
|
|
Return inverse pole figure color if requested.
|
|
Bases are computed from
|
|
|
|
basis = {'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
|
|
[1.,0.,1.]/np.sqrt(2.), # direction of green
|
|
[1.,1.,1.]/np.sqrt(3.)]).T), # direction of blue
|
|
'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
|
|
[1.,0.,0.], # direction of green
|
|
[np.sqrt(3.),1.,0.]/np.sqrt(4.)]).T), # direction of blue
|
|
'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
|
|
[1.,0.,0.], # direction of green
|
|
[1.,1.,0.]/np.sqrt(2.)]).T), # direction of blue
|
|
'orthorhombic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
|
|
[1.,0.,0.], # direction of green
|
|
[0.,1.,0.]]).T), # direction of blue
|
|
}
|
|
"""
|
|
if self.lattice == 'cubic':
|
|
basis = {'improper':np.array([ [-1. , 0. , 1. ],
|
|
[ np.sqrt(2.) , -np.sqrt(2.) , 0. ],
|
|
[ 0. , np.sqrt(3.) , 0. ] ]),
|
|
'proper':np.array([ [ 0. , -1. , 1. ],
|
|
[-np.sqrt(2.) , np.sqrt(2.) , 0. ],
|
|
[ np.sqrt(3.) , 0. , 0. ] ]),
|
|
}
|
|
elif self.lattice == 'hexagonal':
|
|
basis = {'improper':np.array([ [ 0. , 0. , 1. ],
|
|
[ 1. , -np.sqrt(3.) , 0. ],
|
|
[ 0. , 2. , 0. ] ]),
|
|
'proper':np.array([ [ 0. , 0. , 1. ],
|
|
[-1. , np.sqrt(3.) , 0. ],
|
|
[ np.sqrt(3.) , -1. , 0. ] ]),
|
|
}
|
|
elif self.lattice == 'tetragonal':
|
|
basis = {'improper':np.array([ [ 0. , 0. , 1. ],
|
|
[ 1. , -1. , 0. ],
|
|
[ 0. , np.sqrt(2.) , 0. ] ]),
|
|
'proper':np.array([ [ 0. , 0. , 1. ],
|
|
[-1. , 1. , 0. ],
|
|
[ np.sqrt(2.) , 0. , 0. ] ]),
|
|
}
|
|
elif self.lattice == 'orthorhombic':
|
|
basis = {'improper':np.array([ [ 0., 0., 1.],
|
|
[ 1., 0., 0.],
|
|
[ 0., 1., 0.] ]),
|
|
'proper':np.array([ [ 0., 0., 1.],
|
|
[-1., 0., 0.],
|
|
[ 0., 1., 0.] ]),
|
|
}
|
|
else: # direct exit for unspecified symmetry
|
|
if color:
|
|
return (True,np.zeros(3,'d'))
|
|
else:
|
|
return True
|
|
|
|
v = np.array(vector,dtype=float)
|
|
if proper: # check both improper ...
|
|
theComponents = np.dot(basis['improper'],v)
|
|
inSST = np.all(theComponents >= 0.0)
|
|
if not inSST: # ... and proper SST
|
|
theComponents = np.dot(basis['proper'],v)
|
|
inSST = np.all(theComponents >= 0.0)
|
|
else:
|
|
v[2] = abs(v[2]) # z component projects identical
|
|
theComponents = np.dot(basis['improper'],v) # for positive and negative values
|
|
inSST = np.all(theComponents >= 0.0)
|
|
|
|
if color: # have to return color array
|
|
if inSST:
|
|
rgb = np.power(theComponents/np.linalg.norm(theComponents),0.5) # smoothen color ramps
|
|
rgb = np.minimum(np.ones(3,dtype=float),rgb) # limit to maximum intensity
|
|
rgb /= max(rgb) # normalize to (HS)V = 1
|
|
else:
|
|
rgb = np.zeros(3,dtype=float)
|
|
return (inSST,rgb)
|
|
else:
|
|
return inSST
|
|
|
|
# code derived from https://github.com/ezag/pyeuclid
|
|
# suggested reading: http://web.mit.edu/2.998/www/QuaternionReport1.pdf
|
|
|
|
|
|
# ******************************************************************************************
|
|
class Lattice:
|
|
"""
|
|
Lattice system
|
|
|
|
Currently, this contains only a mapping from Bravais lattice to symmetry
|
|
and orientation relationships. It could include twin and slip systems.
|
|
https://en.wikipedia.org/wiki/Bravais_lattice
|
|
"""
|
|
|
|
lattices = {
|
|
'triclinic':{'symmetry':None},
|
|
'bct':{'symmetry':'tetragonal'},
|
|
'hex':{'symmetry':'hexagonal'},
|
|
'fcc':{'symmetry':'cubic','c/a':1.0},
|
|
'bcc':{'symmetry':'cubic','c/a':1.0},
|
|
}
|
|
|
|
|
|
def __init__(self, lattice):
|
|
self.lattice = lattice
|
|
self.symmetry = Symmetry(self.lattices[lattice]['symmetry'])
|
|
|
|
|
|
def __repr__(self):
|
|
"""Report basic lattice information"""
|
|
return 'Bravais lattice {} ({} symmetry)'.format(self.lattice,self.symmetry)
|
|
|
|
|
|
# Kurdjomov--Sachs orientation relationship for fcc <-> bcc transformation
|
|
# from S. Morito et al./Journal of Alloys and Compounds 5775 (2013) S587-S592
|
|
# also see K. Kitahara et al./Acta Materialia 54 (2006) 1279-1288
|
|
KS = {'mapping':{'fcc':0,'bcc':1},
|
|
'planes': np.array([
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]]],dtype='float'),
|
|
'directions': np.array([
|
|
[[ -1, 0, 1],[ -1, -1, 1]],
|
|
[[ -1, 0, 1],[ -1, 1, -1]],
|
|
[[ 0, 1, -1],[ -1, -1, 1]],
|
|
[[ 0, 1, -1],[ -1, 1, -1]],
|
|
[[ 1, -1, 0],[ -1, -1, 1]],
|
|
[[ 1, -1, 0],[ -1, 1, -1]],
|
|
[[ 1, 0, -1],[ -1, -1, 1]],
|
|
[[ 1, 0, -1],[ -1, 1, -1]],
|
|
[[ -1, -1, 0],[ -1, -1, 1]],
|
|
[[ -1, -1, 0],[ -1, 1, -1]],
|
|
[[ 0, 1, 1],[ -1, -1, 1]],
|
|
[[ 0, 1, 1],[ -1, 1, -1]],
|
|
[[ 0, -1, 1],[ -1, -1, 1]],
|
|
[[ 0, -1, 1],[ -1, 1, -1]],
|
|
[[ -1, 0, -1],[ -1, -1, 1]],
|
|
[[ -1, 0, -1],[ -1, 1, -1]],
|
|
[[ 1, 1, 0],[ -1, -1, 1]],
|
|
[[ 1, 1, 0],[ -1, 1, -1]],
|
|
[[ -1, 1, 0],[ -1, -1, 1]],
|
|
[[ -1, 1, 0],[ -1, 1, -1]],
|
|
[[ 0, -1, -1],[ -1, -1, 1]],
|
|
[[ 0, -1, -1],[ -1, 1, -1]],
|
|
[[ 1, 0, 1],[ -1, -1, 1]],
|
|
[[ 1, 0, 1],[ -1, 1, -1]]],dtype='float')}
|
|
|
|
# Greninger--Troiano orientation relationship for fcc <-> bcc transformation
|
|
# from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
GT = {'mapping':{'fcc':0,'bcc':1},
|
|
'planes': np.array([
|
|
[[ 1, 1, 1],[ 1, 0, 1]],
|
|
[[ 1, 1, 1],[ 1, 1, 0]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, -1, 1],[ -1, 0, 1]],
|
|
[[ -1, -1, 1],[ -1, -1, 0]],
|
|
[[ -1, -1, 1],[ 0, -1, 1]],
|
|
[[ -1, 1, 1],[ -1, 0, 1]],
|
|
[[ -1, 1, 1],[ -1, 1, 0]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 1, 0, 1]],
|
|
[[ 1, -1, 1],[ 1, -1, 0]],
|
|
[[ 1, -1, 1],[ 0, -1, 1]],
|
|
[[ 1, 1, 1],[ 1, 1, 0]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 1, 0, 1]],
|
|
[[ -1, -1, 1],[ -1, -1, 0]],
|
|
[[ -1, -1, 1],[ 0, -1, 1]],
|
|
[[ -1, -1, 1],[ -1, 0, 1]],
|
|
[[ -1, 1, 1],[ -1, 1, 0]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ -1, 0, 1]],
|
|
[[ 1, -1, 1],[ 1, -1, 0]],
|
|
[[ 1, -1, 1],[ 0, -1, 1]],
|
|
[[ 1, -1, 1],[ 1, 0, 1]]],dtype='float'),
|
|
'directions': np.array([
|
|
[[ -5,-12, 17],[-17, -7, 17]],
|
|
[[ 17, -5,-12],[ 17,-17, -7]],
|
|
[[-12, 17, -5],[ -7, 17,-17]],
|
|
[[ 5, 12, 17],[ 17, 7, 17]],
|
|
[[-17, 5,-12],[-17, 17, -7]],
|
|
[[ 12,-17, -5],[ 7,-17,-17]],
|
|
[[ -5, 12,-17],[-17, 7,-17]],
|
|
[[ 17, 5, 12],[ 17, 17, 7]],
|
|
[[-12,-17, 5],[ -7,-17, 17]],
|
|
[[ 5,-12,-17],[ 17, -7,-17]],
|
|
[[-17, -5, 12],[-17,-17, 7]],
|
|
[[ 12, 17, 5],[ 7, 17, 17]],
|
|
[[ -5, 17,-12],[-17, 17, -7]],
|
|
[[-12, -5, 17],[ -7,-17, 17]],
|
|
[[ 17,-12, -5],[ 17, -7,-17]],
|
|
[[ 5,-17,-12],[ 17,-17, -7]],
|
|
[[ 12, 5, 17],[ 7, 17, 17]],
|
|
[[-17, 12, -5],[-17, 7,-17]],
|
|
[[ -5,-17, 12],[-17,-17, 7]],
|
|
[[-12, 5,-17],[ -7, 17,-17]],
|
|
[[ 17, 12, 5],[ 17, 7, 17]],
|
|
[[ 5, 17, 12],[ 17, 17, 7]],
|
|
[[ 12, -5,-17],[ 7,-17,-17]],
|
|
[[-17,-12, 5],[-17, 7, 17]]],dtype='float')}
|
|
|
|
# Greninger--Troiano' orientation relationship for fcc <-> bcc transformation
|
|
# from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
GTprime = {'mapping':{'fcc':0,'bcc':1},
|
|
'planes': np.array([
|
|
[[ 7, 17, 17],[ 12, 5, 17]],
|
|
[[ 17, 7, 17],[ 17, 12, 5]],
|
|
[[ 17, 17, 7],[ 5, 17, 12]],
|
|
[[ -7,-17, 17],[-12, -5, 17]],
|
|
[[-17, -7, 17],[-17,-12, 5]],
|
|
[[-17,-17, 7],[ -5,-17, 12]],
|
|
[[ 7,-17,-17],[ 12, -5,-17]],
|
|
[[ 17, -7,-17],[ 17,-12, -5]],
|
|
[[ 17,-17, -7],[ 5,-17,-12]],
|
|
[[ -7, 17,-17],[-12, 5,-17]],
|
|
[[-17, 7,-17],[-17, 12, -5]],
|
|
[[-17, 17, -7],[ -5, 17,-12]],
|
|
[[ 7, 17, 17],[ 12, 17, 5]],
|
|
[[ 17, 7, 17],[ 5, 12, 17]],
|
|
[[ 17, 17, 7],[ 17, 5, 12]],
|
|
[[ -7,-17, 17],[-12,-17, 5]],
|
|
[[-17, -7, 17],[ -5,-12, 17]],
|
|
[[-17,-17, 7],[-17, -5, 12]],
|
|
[[ 7,-17,-17],[ 12,-17, -5]],
|
|
[[ 17, -7,-17],[ 5, -12,-17]],
|
|
[[ 17,-17, 7],[ 17, -5,-12]],
|
|
[[ -7, 17,-17],[-12, 17, -5]],
|
|
[[-17, 7,-17],[ -5, 12,-17]],
|
|
[[-17, 17, -7],[-17, 5,-12]]],dtype='float'),
|
|
'directions': np.array([
|
|
[[ 0, 1, -1],[ 1, 1, -1]],
|
|
[[ -1, 0, 1],[ -1, 1, 1]],
|
|
[[ 1, -1, 0],[ 1, -1, 1]],
|
|
[[ 0, -1, -1],[ -1, -1, -1]],
|
|
[[ 1, 0, 1],[ 1, -1, 1]],
|
|
[[ 1, -1, 0],[ 1, -1, -1]],
|
|
[[ 0, 1, -1],[ -1, 1, -1]],
|
|
[[ 1, 0, 1],[ 1, 1, 1]],
|
|
[[ -1, -1, 0],[ -1, -1, 1]],
|
|
[[ 0, -1, -1],[ 1, -1, -1]],
|
|
[[ -1, 0, 1],[ -1, -1, 1]],
|
|
[[ -1, -1, 0],[ -1, -1, -1]],
|
|
[[ 0, -1, 1],[ 1, -1, 1]],
|
|
[[ 1, 0, -1],[ 1, 1, -1]],
|
|
[[ -1, 1, 0],[ -1, 1, 1]],
|
|
[[ 0, 1, 1],[ -1, 1, 1]],
|
|
[[ -1, 0, -1],[ -1, -1, -1]],
|
|
[[ -1, 1, 0],[ -1, 1, -1]],
|
|
[[ 0, -1, 1],[ -1, -1, 1]],
|
|
[[ -1, 0, -1],[ -1, 1, -1]],
|
|
[[ 1, 1, 0],[ 1, 1, 1]],
|
|
[[ 0, 1, 1],[ 1, 1, 1]],
|
|
[[ 1, 0, -1],[ 1, -1, -1]],
|
|
[[ 1, 1, 0],[ 1, 1, -1]]],dtype='float')}
|
|
|
|
# Nishiyama--Wassermann orientation relationship for fcc <-> bcc transformation
|
|
# from H. Kitahara et al./Materials Characterization 54 (2005) 378-386
|
|
NW = {'mapping':{'fcc':0,'bcc':1},
|
|
'planes': np.array([
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ -1, -1, 1],[ 0, 1, 1]],
|
|
[[ -1, -1, 1],[ 0, 1, 1]],
|
|
[[ -1, -1, 1],[ 0, 1, 1]]],dtype='float'),
|
|
'directions': np.array([
|
|
[[ 2, -1, -1],[ 0, -1, 1]],
|
|
[[ -1, 2, -1],[ 0, -1, 1]],
|
|
[[ -1, -1, 2],[ 0, -1, 1]],
|
|
[[ -2, -1, -1],[ 0, -1, 1]],
|
|
[[ 1, 2, -1],[ 0, -1, 1]],
|
|
[[ 1, -1, 2],[ 0, -1, 1]],
|
|
[[ 2, 1, -1],[ 0, -1, 1]],
|
|
[[ -1, -2, -1],[ 0, -1, 1]],
|
|
[[ -1, 1, 2],[ 0, -1, 1]],
|
|
[[ -1, 2, 1],[ 0, -1, 1]],
|
|
[[ -1, 2, 1],[ 0, -1, 1]],
|
|
[[ -1, -1, -2],[ 0, -1, 1]]],dtype='float')}
|
|
|
|
# Pitsch orientation relationship for fcc <-> bcc transformation
|
|
# from Y. He et al./Acta Materialia 53 (2005) 1179-1190
|
|
Pitsch = {'mapping':{'fcc':0,'bcc':1},
|
|
'planes': np.array([
|
|
[[ 0, 1, 0],[ -1, 0, 1]],
|
|
[[ 0, 0, 1],[ 1, -1, 0]],
|
|
[[ 1, 0, 0],[ 0, 1, -1]],
|
|
[[ 1, 0, 0],[ 0, -1, -1]],
|
|
[[ 0, 1, 0],[ -1, 0, -1]],
|
|
[[ 0, 0, 1],[ -1, -1, 0]],
|
|
[[ 0, 1, 0],[ -1, 0, -1]],
|
|
[[ 0, 0, 1],[ -1, -1, 0]],
|
|
[[ 1, 0, 0],[ 0, -1, -1]],
|
|
[[ 1, 0, 0],[ 0, -1, 1]],
|
|
[[ 0, 1, 0],[ 1, 0, -1]],
|
|
[[ 0, 0, 1],[ -1, 1, 0]]],dtype='float'),
|
|
'directions': np.array([
|
|
[[ 1, 0, 1],[ 1, -1, 1]],
|
|
[[ 1, 1, 0],[ 1, 1, -1]],
|
|
[[ 0, 1, 1],[ -1, 1, 1]],
|
|
[[ 0, 1, -1],[ -1, 1, -1]],
|
|
[[ -1, 0, 1],[ -1, -1, 1]],
|
|
[[ 1, -1, 0],[ 1, -1, -1]],
|
|
[[ 1, 0, -1],[ 1, -1, -1]],
|
|
[[ -1, 1, 0],[ -1, 1, -1]],
|
|
[[ 0, -1, 1],[ -1, -1, 1]],
|
|
[[ 0, 1, 1],[ -1, 1, 1]],
|
|
[[ 1, 0, 1],[ 1, -1, 1]],
|
|
[[ 1, 1, 0],[ 1, 1, -1]]],dtype='float')}
|
|
|
|
# Bain orientation relationship for fcc <-> bcc transformation
|
|
# from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
Bain = {'mapping':{'fcc':0,'bcc':1},
|
|
'planes': np.array([
|
|
[[ 1, 0, 0],[ 1, 0, 0]],
|
|
[[ 0, 1, 0],[ 0, 1, 0]],
|
|
[[ 0, 0, 1],[ 0, 0, 1]]],dtype='float'),
|
|
'directions': np.array([
|
|
[[ 0, 1, 0],[ 0, 1, 1]],
|
|
[[ 0, 0, 1],[ 1, 0, 1]],
|
|
[[ 1, 0, 0],[ 1, 1, 0]]],dtype='float')}
|
|
|
|
def relationOperations(self,model):
|
|
|
|
models={'KS':self.KS, 'GT':self.GT, "GT'":self.GTprime,
|
|
'NW':self.NW, 'Pitsch': self.Pitsch, 'Bain':self.Bain}
|
|
try:
|
|
relationship = models[model]
|
|
except:
|
|
raise KeyError('Orientation relationship "{}" is unknown'.format(model))
|
|
|
|
if self.lattice not in relationship['mapping']:
|
|
raise ValueError('Relationship "{}" not supported for lattice "{}"'.format(model,self.lattice))
|
|
|
|
r = {'lattice':Lattice((set(relationship['mapping'])-{self.lattice}).pop()), # target lattice
|
|
'rotations':[] }
|
|
|
|
myPlane_id = relationship['mapping'][self.lattice]
|
|
otherPlane_id = (myPlane_id+1)%2
|
|
myDir_id = myPlane_id +2
|
|
otherDir_id = otherPlane_id +2
|
|
|
|
for miller in np.hstack((relationship['planes'],relationship['directions'])):
|
|
myPlane = miller[myPlane_id]/ np.linalg.norm(miller[myPlane_id])
|
|
myDir = miller[myDir_id]/ np.linalg.norm(miller[myDir_id])
|
|
myMatrix = np.array([myDir,np.cross(myPlane,myDir),myPlane]).T
|
|
|
|
otherPlane = miller[otherPlane_id]/ np.linalg.norm(miller[otherPlane_id])
|
|
otherDir = miller[otherDir_id]/ np.linalg.norm(miller[otherDir_id])
|
|
otherMatrix = np.array([otherDir,np.cross(otherPlane,otherDir),otherPlane]).T
|
|
|
|
r['rotations'].append(Rotation.fromMatrix(np.dot(otherMatrix,myMatrix.T)))
|
|
|
|
return r
|
|
|
|
|
|
|
|
class Orientation:
|
|
"""
|
|
Crystallographic orientation
|
|
|
|
A crystallographic orientation contains a rotation and a lattice
|
|
"""
|
|
|
|
__slots__ = ['rotation','lattice']
|
|
|
|
def __repr__(self):
|
|
"""Report lattice type and orientation"""
|
|
return self.lattice.__repr__()+'\n'+self.rotation.__repr__()
|
|
|
|
def __init__(self, rotation, lattice):
|
|
|
|
if isinstance(lattice, Lattice):
|
|
self.lattice = lattice
|
|
else:
|
|
self.lattice = Lattice(lattice) # assume string
|
|
|
|
if isinstance(rotation, Rotation):
|
|
self.rotation = rotation
|
|
else:
|
|
self.rotation = Rotation(rotation) # assume quaternion
|
|
|
|
def disorientation(self,
|
|
other,
|
|
SST = True,
|
|
symmetries = False):
|
|
"""
|
|
Disorientation between myself and given other orientation.
|
|
|
|
Rotation axis falls into SST if SST == True.
|
|
(Currently requires same symmetry for both orientations.
|
|
Look into A. Heinz and P. Neumann 1991 for cases with differing sym.)
|
|
"""
|
|
if self.lattice.symmetry != other.lattice.symmetry:
|
|
raise NotImplementedError('disorientation between different symmetry classes not supported yet.')
|
|
|
|
mySymEqs = self.equivalentOrientations() if SST else self.equivalentOrientations([0]) # take all or only first sym operation
|
|
otherSymEqs = other.equivalentOrientations()
|
|
|
|
for i,sA in enumerate(mySymEqs):
|
|
aInv = sA.rotation.inversed()
|
|
for j,sB in enumerate(otherSymEqs):
|
|
b = sB.rotation
|
|
r = b*aInv
|
|
for k in range(2):
|
|
r.inverse()
|
|
breaker = self.lattice.symmetry.inFZ(r.asRodrigues(vector=True)) \
|
|
and (not SST or other.lattice.symmetry.inDisorientationSST(r.asRodrigues(vector=True)))
|
|
if breaker: break
|
|
if breaker: break
|
|
if breaker: break
|
|
|
|
return (Orientation(r,self.lattice), i,j, k == 1) if symmetries else r # disorientation ...
|
|
# ... own sym, other sym,
|
|
# self-->other: True, self<--other: False
|
|
|
|
|
|
def inFZ(self):
|
|
return self.lattice.symmetry.inFZ(self.rotation.asRodrigues(vector=True))
|
|
|
|
def equivalentOrientations(self,members=[]):
|
|
"""List of orientations which are symmetrically equivalent"""
|
|
try:
|
|
iter(members) # asking for (even empty) list of members?
|
|
except TypeError:
|
|
return self.__class__(self.lattice.symmetry.symmetryOperations(members)*self.rotation,self.lattice) # no, return rotation object
|
|
else:
|
|
return [self.__class__(q*self.rotation,self.lattice) \
|
|
for q in self.lattice.symmetry.symmetryOperations(members)] # yes, return list of rotations
|
|
|
|
def relatedOrientations(self,model):
|
|
"""List of orientations related by the given orientation relationship"""
|
|
r = self.lattice.relationOperations(model)
|
|
return [self.__class__(self.rotation*o,r['lattice']) for o in r['rotations']]
|
|
|
|
def reduced(self):
|
|
"""Transform orientation to fall into fundamental zone according to symmetry"""
|
|
for me in self.equivalentOrientations():
|
|
if self.lattice.symmetry.inFZ(me.rotation.asRodrigues()): break
|
|
|
|
return self.__class__(me.rotation,self.lattice)
|
|
|
|
def inversePole(self,
|
|
axis,
|
|
proper = False,
|
|
SST = True):
|
|
"""Axis rotated according to orientation (using crystal symmetry to ensure location falls into SST)"""
|
|
if SST: # pole requested to be within SST
|
|
for i,o in enumerate(self.equivalentOrientations()): # test all symmetric equivalent quaternions
|
|
pole = o.rotation*axis # align crystal direction to axis
|
|
if self.lattice.symmetry.inSST(pole,proper): break # found SST version
|
|
else:
|
|
pole = self.rotation*axis # align crystal direction to axis
|
|
|
|
return (pole,i if SST else 0)
|
|
|
|
|
|
def IPFcolor(self,axis):
|
|
"""TSL color of inverse pole figure for given axis"""
|
|
color = np.zeros(3,'d')
|
|
|
|
for o in self.equivalentOrientations():
|
|
pole = o.rotation*axis # align crystal direction to axis
|
|
inSST,color = self.lattice.symmetry.inSST(pole,color=True)
|
|
if inSST: break
|
|
|
|
return color
|
|
|
|
|
|
@classmethod
|
|
def fromAverage(cls,
|
|
orientations,
|
|
weights = []):
|
|
"""Create orientation from average of list of orientations"""
|
|
if not all(isinstance(item, Orientation) for item in orientations):
|
|
raise TypeError("Only instances of Orientation can be averaged.")
|
|
|
|
closest = []
|
|
ref = orientations[0]
|
|
for o in orientations:
|
|
closest.append(o.equivalentOrientations(
|
|
ref.disorientation(o,
|
|
SST = False, # select (o[ther]'s) sym orientation
|
|
symmetries = True)[2]).rotation) # with lowest misorientation
|
|
|
|
return Orientation(Rotation.fromAverage(closest,weights),ref.lattice)
|
|
|
|
|
|
def average(self,other):
|
|
"""Calculate the average rotation"""
|
|
return Orientation.fromAverage([self,other])
|
|
|
|
|
|
####################################################################################################
|
|
# 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.
|
|
####################################################################################################
|
|
|
|
def isone(a):
|
|
return np.isclose(a,1.0,atol=1.0e-7,rtol=0.0)
|
|
|
|
def iszero(a):
|
|
return np.isclose(a,0.0,atol=1.0e-12,rtol=0.0)
|
|
|
|
#---------- quaternion ----------
|
|
|
|
def qu2om(qu):
|
|
"""Quaternion to orientation matrix"""
|
|
qq = qu[0]**2-(qu[1]**2 + qu[2]**2 + qu[3]**2)
|
|
om = np.diag(qq + 2.0*np.array([qu[1],qu[2],qu[3]])**2)
|
|
|
|
om[1,0] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3])
|
|
om[0,1] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3])
|
|
om[2,1] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1])
|
|
om[1,2] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1])
|
|
om[0,2] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2])
|
|
om[2,0] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2])
|
|
return om if P > 0.0 else om.T
|
|
|
|
|
|
def qu2eu(qu):
|
|
"""Quaternion to Euler angles"""
|
|
q03 = qu[0]**2+qu[3]**2
|
|
q12 = qu[1]**2+qu[2]**2
|
|
chi = np.sqrt(q03*q12)
|
|
|
|
if iszero(chi):
|
|
eu = np.array([np.arctan2(-P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0, 0.0]) if iszero(q12) else \
|
|
np.array([np.arctan2(2.0*qu[1]*qu[2],qu[1]**2-qu[2]**2), np.pi, 0.0])
|
|
else:
|
|
eu = np.array([np.arctan2((-P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-P*qu[0]*qu[1]-qu[2]*qu[3])*chi ),
|
|
np.arctan2( 2.0*chi, q03-q12 ),
|
|
np.arctan2(( P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-P*qu[0]*qu[1]+qu[2]*qu[3])*chi )])
|
|
|
|
# reduce Euler angles to definition range, i.e a lower limit of 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
|
|
|
|
|
|
def qu2ax(qu):
|
|
"""
|
|
Quaternion to axis angle
|
|
|
|
Modified version of the original formulation, should be numerically more stable
|
|
"""
|
|
if iszero(qu[1]**2+qu[2]**2+qu[3]**2): # set axis to [001] if the angle is 0/360
|
|
ax = [ 0.0, 0.0, 1.0, 0.0 ]
|
|
elif not iszero(qu[0]):
|
|
s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2)
|
|
omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
|
|
ax = [ qu[1]*s, qu[2]*s, qu[3]*s, omega ]
|
|
else:
|
|
ax = [ qu[1], qu[2], qu[3], np.pi]
|
|
|
|
return np.array(ax)
|
|
|
|
|
|
def qu2ro(qu):
|
|
"""Quaternion to Rodrigues vector"""
|
|
if iszero(qu[0]):
|
|
ro = [qu[1], qu[2], qu[3], np.inf]
|
|
else:
|
|
s = np.linalg.norm([qu[1],qu[2],qu[3]])
|
|
ro = [0.0,0.0,P,0.0] if iszero(s) else \
|
|
[ qu[1]/s, qu[2]/s, qu[3]/s, np.tan(np.arccos(np.clip(qu[0],-1.0,1.0)))] # avoid numerical difficulties
|
|
|
|
return np.array(ro)
|
|
|
|
|
|
def qu2ho(qu):
|
|
"""Quaternion to homochoric"""
|
|
omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0)) # avoid numerical difficulties
|
|
|
|
if iszero(omega):
|
|
ho = np.array([ 0.0, 0.0, 0.0 ])
|
|
else:
|
|
ho = np.array([qu[1], qu[2], qu[3]])
|
|
f = 0.75 * ( omega - np.sin(omega) )
|
|
ho = ho/np.linalg.norm(ho) * f**(1./3.)
|
|
|
|
return ho
|
|
|
|
|
|
def qu2cu(qu):
|
|
"""Quaternion to cubochoric"""
|
|
return ho2cu(qu2ho(qu))
|
|
|
|
|
|
#---------- orientation matrix ----------
|
|
|
|
def om2qu(om):
|
|
"""
|
|
Orientation matrix to quaternion
|
|
|
|
The original formulation (direct conversion) had (numerical?) issues
|
|
"""
|
|
return eu2qu(om2eu(om))
|
|
|
|
|
|
def om2eu(om):
|
|
"""Orientation matrix to Euler angles"""
|
|
if abs(om[2,2]) < 1.0:
|
|
zeta = 1.0/np.sqrt(1.0-om[2,2]**2)
|
|
eu = np.array([np.arctan2(om[2,0]*zeta,-om[2,1]*zeta),
|
|
np.arccos(om[2,2]),
|
|
np.arctan2(om[0,2]*zeta, om[1,2]*zeta)])
|
|
else:
|
|
eu = np.array([np.arctan2( om[0,1],om[0,0]), np.pi*0.5*(1-om[2,2]),0.0]) # following the paper, not the reference implementation
|
|
|
|
# reduce Euler angles to definition range, i.e a lower limit of 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
|
|
|
|
|
|
def om2ax(om):
|
|
"""Orientation matrix to axis angle"""
|
|
ax=np.empty(4)
|
|
|
|
# first get the rotation angle
|
|
t = 0.5*(om.trace() -1.0)
|
|
ax[3] = np.arccos(np.clip(t,-1.0,1.0))
|
|
|
|
if iszero(ax[3]):
|
|
ax = [ 0.0, 0.0, 1.0, 0.0]
|
|
else:
|
|
w,vr = np.linalg.eig(om)
|
|
# next, find the eigenvalue (1,0j)
|
|
i = np.where(np.isclose(w,1.0+0.0j))[0][0]
|
|
ax[0:3] = np.real(vr[0:3,i])
|
|
diagDelta = np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]])
|
|
ax[0:3] = np.where(iszero(diagDelta), ax[0:3],np.abs(ax[0:3])*np.sign(-P*diagDelta))
|
|
|
|
return np.array(ax)
|
|
|
|
|
|
def om2ro(om):
|
|
"""Orientation matrix to Rodriques vector"""
|
|
return eu2ro(om2eu(om))
|
|
|
|
|
|
def om2ho(om):
|
|
"""Orientation matrix to homochoric"""
|
|
return ax2ho(om2ax(om))
|
|
|
|
|
|
def om2cu(om):
|
|
"""Orientation matrix to cubochoric"""
|
|
return ho2cu(om2ho(om))
|
|
|
|
|
|
#---------- Euler angles ----------
|
|
|
|
def eu2qu(eu):
|
|
"""Euler angles to quaternion"""
|
|
ee = 0.5*eu
|
|
cPhi = np.cos(ee[1])
|
|
sPhi = np.sin(ee[1])
|
|
qu = np.array([ cPhi*np.cos(ee[0]+ee[2]),
|
|
-P*sPhi*np.cos(ee[0]-ee[2]),
|
|
-P*sPhi*np.sin(ee[0]-ee[2]),
|
|
-P*cPhi*np.sin(ee[0]+ee[2]) ])
|
|
if qu[0] < 0.0: qu*=-1
|
|
return qu
|
|
|
|
|
|
def eu2om(eu):
|
|
"""Euler angles to orientation matrix"""
|
|
c = np.cos(eu)
|
|
s = np.sin(eu)
|
|
|
|
om = np.array([[+c[0]*c[2]-s[0]*s[2]*c[1], +s[0]*c[2]+c[0]*s[2]*c[1], +s[2]*s[1]],
|
|
[-c[0]*s[2]-s[0]*c[2]*c[1], -s[0]*s[2]+c[0]*c[2]*c[1], +c[2]*s[1]],
|
|
[+s[0]*s[1], -c[0]*s[1], +c[1] ]])
|
|
|
|
om[np.where(iszero(om))] = 0.0
|
|
return om
|
|
|
|
|
|
def eu2ax(eu):
|
|
"""Euler angles to axis angle"""
|
|
t = np.tan(eu[1]*0.5)
|
|
sigma = 0.5*(eu[0]+eu[2])
|
|
delta = 0.5*(eu[0]-eu[2])
|
|
tau = np.linalg.norm([t,np.sin(sigma)])
|
|
alpha = np.pi if iszero(np.cos(sigma)) else \
|
|
2.0*np.arctan(tau/np.cos(sigma))
|
|
|
|
if iszero(alpha):
|
|
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
|
else:
|
|
ax = -P/tau * np.array([ t*np.cos(delta), t*np.sin(delta), np.sin(sigma) ]) # passive axis-angle pair so a minus sign in front
|
|
ax = np.append(ax,alpha)
|
|
if alpha < 0.0: ax *= -1.0 # ensure alpha is positive
|
|
|
|
return ax
|
|
|
|
|
|
def eu2ro(eu):
|
|
"""Euler angles to Rodrigues vector"""
|
|
ro = eu2ax(eu) # convert to axis angle representation
|
|
if ro[3] >= np.pi: # Differs from original implementation. check convention 5
|
|
ro[3] = np.inf
|
|
elif iszero(ro[3]):
|
|
ro = np.array([ 0.0, 0.0, P, 0.0 ])
|
|
else:
|
|
ro[3] = np.tan(ro[3]*0.5)
|
|
|
|
return ro
|
|
|
|
|
|
def eu2ho(eu):
|
|
"""Euler angles to homochoric"""
|
|
return ax2ho(eu2ax(eu))
|
|
|
|
|
|
def eu2cu(eu):
|
|
"""Euler angles to cubochoric"""
|
|
return ho2cu(eu2ho(eu))
|
|
|
|
|
|
#---------- axis angle ----------
|
|
|
|
def ax2qu(ax):
|
|
"""Axis angle to quaternion"""
|
|
if iszero(ax[3]):
|
|
qu = np.array([ 1.0, 0.0, 0.0, 0.0 ])
|
|
else:
|
|
c = np.cos(ax[3]*0.5)
|
|
s = np.sin(ax[3]*0.5)
|
|
qu = np.array([ c, ax[0]*s, ax[1]*s, ax[2]*s ])
|
|
|
|
return qu
|
|
|
|
|
|
def ax2om(ax):
|
|
"""Axis angle to orientation matrix"""
|
|
c = np.cos(ax[3])
|
|
s = np.sin(ax[3])
|
|
omc = 1.0-c
|
|
om=np.diag(ax[0:3]**2*omc + c)
|
|
|
|
for idx in [[0,1,2],[1,2,0],[2,0,1]]:
|
|
q = omc*ax[idx[0]] * ax[idx[1]]
|
|
om[idx[0],idx[1]] = q + s*ax[idx[2]]
|
|
om[idx[1],idx[0]] = q - s*ax[idx[2]]
|
|
|
|
return om if P < 0.0 else om.T
|
|
|
|
|
|
def ax2eu(ax):
|
|
"""Orientation matrix to Euler angles"""
|
|
return om2eu(ax2om(ax))
|
|
|
|
|
|
def ax2ro(ax):
|
|
"""Axis angle to Rodrigues vector"""
|
|
if iszero(ax[3]):
|
|
ro = [ 0.0, 0.0, P, 0.0 ]
|
|
else:
|
|
ro = [ax[0], ax[1], ax[2]]
|
|
# 180 degree case
|
|
ro += [np.inf] if np.isclose(ax[3],np.pi,atol=1.0e-15,rtol=0.0) else \
|
|
[np.tan(ax[3]*0.5)]
|
|
|
|
return np.array(ro)
|
|
|
|
|
|
def ax2ho(ax):
|
|
"""Axis angle to homochoric"""
|
|
f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0)
|
|
ho = ax[0:3] * f
|
|
return ho
|
|
|
|
|
|
def ax2cu(ax):
|
|
"""Axis angle to cubochoric"""
|
|
return ho2cu(ax2ho(ax))
|
|
|
|
|
|
#---------- Rodrigues--Frank ----------
|
|
|
|
def ro2qu(ro):
|
|
"""Rodrigues vector to quaternion"""
|
|
return ax2qu(ro2ax(ro))
|
|
|
|
|
|
def ro2om(ro):
|
|
"""Rodgrigues vector to orientation matrix"""
|
|
return ax2om(ro2ax(ro))
|
|
|
|
|
|
def ro2eu(ro):
|
|
"""Rodrigues vector to orientation matrix"""
|
|
return om2eu(ro2om(ro))
|
|
|
|
|
|
def ro2ax(ro):
|
|
"""Rodrigues vector to axis angle"""
|
|
ta = ro[3]
|
|
|
|
if iszero(ta):
|
|
ax = [ 0.0, 0.0, 1.0, 0.0 ]
|
|
elif not np.isfinite(ta):
|
|
ax = [ ro[0], ro[1], ro[2], np.pi ]
|
|
else:
|
|
angle = 2.0*np.arctan(ta)
|
|
ta = 1.0/np.linalg.norm(ro[0:3])
|
|
ax = [ ro[0]/ta, ro[1]/ta, ro[2]/ta, angle ]
|
|
|
|
return np.array(ax)
|
|
|
|
|
|
def ro2ho(ro):
|
|
"""Rodrigues vector to homochoric"""
|
|
if iszero(np.sum(ro[0:3]**2.0)):
|
|
ho = [ 0.0, 0.0, 0.0 ]
|
|
else:
|
|
f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
|
|
ho = ro[0:3] * (0.75*f)**(1.0/3.0)
|
|
|
|
return np.array(ho)
|
|
|
|
|
|
def ro2cu(ro):
|
|
"""Rodrigues vector to cubochoric"""
|
|
return ho2cu(ro2ho(ro))
|
|
|
|
|
|
#---------- homochoric ----------
|
|
|
|
def ho2qu(ho):
|
|
"""Homochoric to quaternion"""
|
|
return ax2qu(ho2ax(ho))
|
|
|
|
|
|
def ho2om(ho):
|
|
"""Homochoric to orientation matrix"""
|
|
return ax2om(ho2ax(ho))
|
|
|
|
|
|
def ho2eu(ho):
|
|
"""Homochoric to Euler angles"""
|
|
return ax2eu(ho2ax(ho))
|
|
|
|
|
|
def ho2ax(ho):
|
|
"""Homochoric to axis angle"""
|
|
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])
|
|
# normalize h and store the magnitude
|
|
hmag_squared = np.sum(ho**2.)
|
|
if iszero(hmag_squared):
|
|
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
|
else:
|
|
hm = hmag_squared
|
|
|
|
# convert the magnitude to the rotation angle
|
|
s = tfit[0] + tfit[1] * hmag_squared
|
|
for i in range(2,16):
|
|
hm *= hmag_squared
|
|
s += tfit[i] * hm
|
|
ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0)))
|
|
return ax
|
|
|
|
|
|
def ho2ro(ho):
|
|
"""Axis angle to Rodriques vector"""
|
|
return ax2ro(ho2ax(ho))
|
|
|
|
|
|
def ho2cu(ho):
|
|
"""Homochoric to cubochoric"""
|
|
return Lambert.BallToCube(ho)
|
|
|
|
|
|
#---------- cubochoric ----------
|
|
|
|
def cu2qu(cu):
|
|
"""Cubochoric to quaternion"""
|
|
return ho2qu(cu2ho(cu))
|
|
|
|
|
|
def cu2om(cu):
|
|
"""Cubochoric to orientation matrix"""
|
|
return ho2om(cu2ho(cu))
|
|
|
|
|
|
def cu2eu(cu):
|
|
"""Cubochoric to Euler angles"""
|
|
return ho2eu(cu2ho(cu))
|
|
|
|
|
|
def cu2ax(cu):
|
|
"""Cubochoric to axis angle"""
|
|
return ho2ax(cu2ho(cu))
|
|
|
|
|
|
def cu2ro(cu):
|
|
"""Cubochoric to Rodrigues vector"""
|
|
return ho2ro(cu2ho(cu))
|
|
|
|
|
|
def cu2ho(cu):
|
|
"""Cubochoric to homochoric"""
|
|
return Lambert.CubeToBall(cu)
|