1949 lines
75 KiB
Python
1949 lines
75 KiB
Python
# -*- coding: UTF-8 no BOM -*-
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import math,os
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import numpy as np
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from . import Lambert
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P = -1
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####################################################################################################
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class Quaternion2:
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u"""
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Quaternion with basic operations
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q is the real part, p = (x, y, z) are the imaginary parts.
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Defintion of multiplication depends on variable P, P ∉ {-1,1}.
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"""
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def __init__(self,
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q = 0.0,
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p = np.zeros(3,dtype=float)):
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"""Initializes to identity unless specified"""
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self.q = q
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self.p = np.array(p)
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def __copy__(self):
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"""Copy"""
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return self.__class__(q=self.q,
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p=self.p.copy())
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copy = __copy__
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def __iter__(self):
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"""Components"""
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return iter(self.asList())
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def asArray(self):
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"""As numpy array"""
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return np.array((self.q,self.p[0],self.p[1],self.p[2]))
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def asList(self):
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return [self.q]+list(self.p)
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def __repr__(self):
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"""Readable string"""
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return 'Quaternion: (real={q:+.6f}, imag=<{p[0]:+.6f}, {p[1]:+.6f}, {p[2]:+.6f}>)'.format(q=self.q,p=self.p)
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def __add__(self, other):
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"""Addition"""
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if isinstance(other, Quaternion2):
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return self.__class__(q=self.q + other.q,
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p=self.p + other.p)
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else:
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return NotImplemented
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def __iadd__(self, other):
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"""In-place addition"""
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if isinstance(other, Quaternion2):
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self.q += other.q
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self.p += other.p
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return self
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else:
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return NotImplemented
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def __pos__(self):
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"""Unary positive operator"""
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return self
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def __sub__(self, other):
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"""Subtraction"""
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if isinstance(other, Quaternion2):
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return self.__class__(q=self.q - other.q,
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p=self.p - other.p)
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else:
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return NotImplemented
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def __isub__(self, other):
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"""In-place subtraction"""
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if isinstance(other, Quaternion2):
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self.q -= other.q
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self.p -= other.p
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return self
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else:
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return NotImplemented
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def __neg__(self):
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"""Unary positive operator"""
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self.q *= -1.0
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self.p *= -1.0
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return self
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def __mul__(self, other):
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"""Multiplication with quaternion or scalar"""
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if isinstance(other, Quaternion2):
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return self.__class__(q=self.q*other.q - np.dot(self.p,other.p),
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p=self.q*other.p + other.q*self.p + P * np.cross(self.p,other.p))
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elif isinstance(other, (int, float)):
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return self.__class__(q=self.q*other,
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p=self.p*other)
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else:
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return NotImplemented
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def __imul__(self, other):
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"""In-place multiplication with quaternion or scalar"""
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if isinstance(other, Quaternion2):
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self.q = self.q*other.q - np.dot(self.p,other.p)
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self.p = self.q*other.p + other.q*self.p + P * np.cross(self.p,other.p)
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return self
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elif isinstance(other, (int, float)):
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self *= other
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return self
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else:
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return NotImplemented
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def __truediv__(self, other):
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"""Divsion with quaternion or scalar"""
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if isinstance(other, Quaternion2):
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s = other.conjugate()/abs(other)**2.
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return self.__class__(q=self.q * s,
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p=self.p * s)
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elif isinstance(other, (int, float)):
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self.q /= other
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self.p /= other
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return self
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else:
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return NotImplemented
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def __itruediv__(self, other):
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"""In-place divsion with quaternion or scalar"""
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if isinstance(other, Quaternion2):
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s = other.conjugate()/abs(other)**2.
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self *= s
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return self
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elif isinstance(other, (int, float)):
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self.q /= other
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return self
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else:
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return NotImplemented
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def __pow__(self, exponent):
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"""Power"""
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if isinstance(exponent, (int, float)):
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omega = np.acos(self.q)
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return self.__class__(q= np.cos(exponent*omega),
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p=self.p * np.sin(exponent*omega)/np.sin(omega))
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else:
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return NotImplemented
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def __ipow__(self, exponent):
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"""In-place power"""
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if isinstance(exponent, (int, float)):
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omega = np.acos(self.q)
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self.q = np.cos(exponent*omega)
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self.p *= np.sin(exponent*omega)/np.sin(omega)
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else:
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return NotImplemented
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def __abs__(self):
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"""Norm"""
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return math.sqrt(self.q ** 2 + np.dot(self.p,self.p))
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magnitude = __abs__
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def __eq__(self,other):
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"""Equal (sufficiently close) to each other"""
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return np.isclose(( self-other).magnitude(),0.0) \
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or np.isclose((-self-other).magnitude(),0.0)
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def __ne__(self,other):
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"""Not equal (sufficiently close) to each other"""
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return not self.__eq__(other)
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def normalize(self):
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d = self.magnitude()
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if d > 0.0:
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self.q /= d
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self.p /= d
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return self
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def normalized(self):
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return self.copy().normalize()
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def conjugate(self):
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self.p = -self.p
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return self
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def conjugated(self):
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return self.copy().conjugate()
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def homomorph(self):
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if self.q < 0.0:
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self.q = -self.q
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self.p = -self.p
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return self
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def homomorphed(self):
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return self.copy().homomorph()
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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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All methods and naming conventions based on Rowenhorst_etal2015
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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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self.quaternion = Quaternion2(q=quaternion[0],p=quaternion[1:4])
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self.quaternion.homomorph() # ToDo: Needed?
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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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################################################################################################
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# convert to different orientation representations (numpy arrays)
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def asQuaternion(self):
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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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return np.degrees(qu2eu(self.quaternion.asArray())) if degrees else qu2eu(self.quaternion.asArray())
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def asAngleAxis(self,
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degrees = False):
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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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return qu2om(self.quaternion.asArray())
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def asRodrigues(self):
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return qu2ro(self.quaternion.asArray())
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def asHomochoric(self):
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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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################################################################################################
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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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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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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 fromAngleAxis(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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def __mul__(self, other):
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"""
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Multiplication
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Rotation: Details needed (active/passive), more rotation of (3,3,3,3) 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).asArray())
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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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"""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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"""In-place inverse rotation/backward rotation"""
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return self.__class__(self.quaternion.conjugated().asArray())
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# ******************************************************************************************
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class Quaternion:
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u"""
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Orientation represented as unit quaternion.
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All methods and naming conventions based on Rowenhorst_etal2015
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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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w is the real part, (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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def __init__(self,
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quat = None,
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q = 1.0,
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p = np.zeros(3,dtype=float)):
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"""Initializes to identity unless specified"""
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self.q = quat[0] if quat is not None else q
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self.p = np.array(quat[1:4]) if quat is not None else p
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self.homomorph()
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def __iter__(self):
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"""Components"""
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return iter(self.asList())
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def __copy__(self):
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"""Copy"""
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return self.__class__(q=self.q,p=self.p.copy())
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copy = __copy__
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def __repr__(self):
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"""Readable string"""
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return 'Quaternion(real={q:+.6f}, imag=<{p[0]:+.6f}, {p[1]:+.6f}, {p[2]:+.6f}>)'.format(q=self.q,p=self.p)
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def __pow__(self, exponent):
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"""Power"""
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omega = math.acos(self.q)
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return self.__class__(q= math.cos(exponent*omega),
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p=self.p * math.sin(exponent*omega)/math.sin(omega))
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def __ipow__(self, exponent):
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"""In-place power"""
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omega = math.acos(self.q)
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self.q = math.cos(exponent*omega)
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self.p *= math.sin(exponent*omega)/math.sin(omega)
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return self
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def __mul__(self, other):
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"""Multiplication"""
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# Rowenhorst_etal2015 MSMSE: value of P is selected as -1
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P = -1.0
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try: # quaternion
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return self.__class__(q=self.q*other.q - np.dot(self.p,other.p),
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p=self.q*other.p + other.q*self.p + P * np.cross(self.p,other.p))
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except: pass
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try: # vector (perform passive rotation)
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( x, y, z) = self.p
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(Vx,Vy,Vz) = other[0:3]
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A = self.q*self.q - np.dot(self.p,self.p)
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B = 2.0 * (x*Vx + y*Vy + z*Vz)
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C = 2.0 * P*self.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),
|
|
])
|
|
except: pass
|
|
try: # scalar
|
|
return self.__class__(q=self.q*other,
|
|
p=self.p*other)
|
|
except:
|
|
return self.copy()
|
|
|
|
def __imul__(self, other):
|
|
"""In-place multiplication"""
|
|
# Rowenhorst_etal2015 MSMSE: value of P is selected as -1
|
|
P = -1.0
|
|
try: # Quaternion
|
|
self.q = self.q*other.q - np.dot(self.p,other.p)
|
|
self.p = self.q*other.p + other.q*self.p + P * np.cross(self.p,other.p)
|
|
except: pass
|
|
return self
|
|
|
|
def __div__(self, other):
|
|
"""Division"""
|
|
if isinstance(other, (int,float)):
|
|
return self.__class__(q=self.q / other,
|
|
p=self.p / other)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __idiv__(self, other):
|
|
"""In-place division"""
|
|
if isinstance(other, (int,float)):
|
|
self.q /= other
|
|
self.p /= other
|
|
return self
|
|
|
|
def __add__(self, other):
|
|
"""Addition"""
|
|
if isinstance(other, Quaternion):
|
|
return self.__class__(q=self.q + other.q,
|
|
p=self.p + other.p)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __iadd__(self, other):
|
|
"""In-place addition"""
|
|
if isinstance(other, Quaternion):
|
|
self.q += other.q
|
|
self.p += other.p
|
|
return self
|
|
|
|
def __sub__(self, other):
|
|
"""Subtraction"""
|
|
if isinstance(other, Quaternion):
|
|
return self.__class__(q=self.q - other.q,
|
|
p=self.p - other.p)
|
|
else:
|
|
return NotImplemented
|
|
|
|
def __isub__(self, other):
|
|
"""In-place subtraction"""
|
|
if isinstance(other, Quaternion):
|
|
self.q -= other.q
|
|
self.p -= other.p
|
|
return self
|
|
|
|
def __neg__(self):
|
|
"""Additive inverse"""
|
|
self.q = -self.q
|
|
self.p = -self.p
|
|
return self
|
|
|
|
def __abs__(self):
|
|
"""Norm"""
|
|
return math.sqrt(self.q ** 2 + np.dot(self.p,self.p))
|
|
|
|
magnitude = __abs__
|
|
|
|
def __eq__(self,other):
|
|
"""Equal (sufficiently close) to each other"""
|
|
return np.isclose(( self-other).magnitude(),0.0) \
|
|
or np.isclose((-self-other).magnitude(),0.0)
|
|
|
|
def __ne__(self,other):
|
|
"""Not equal (sufficiently close) to each other"""
|
|
return not self.__eq__(other)
|
|
|
|
def __cmp__(self,other):
|
|
"""Linear ordering"""
|
|
return (1 if np.linalg.norm(self.asRodrigues()) > np.linalg.norm(other.asRodrigues()) else 0) \
|
|
- (1 if np.linalg.norm(self.asRodrigues()) < np.linalg.norm(other.asRodrigues()) else 0)
|
|
|
|
def magnitude_squared(self):
|
|
return self.q ** 2 + np.dot(self.p,self.p)
|
|
|
|
def normalize(self):
|
|
d = self.magnitude()
|
|
if d > 0.0:
|
|
self.q /= d
|
|
self.p /= d
|
|
return self
|
|
|
|
def conjugate(self):
|
|
self.p = -self.p
|
|
return self
|
|
|
|
def homomorph(self):
|
|
if self.q < 0.0:
|
|
self.q = -self.q
|
|
self.p = -self.p
|
|
return self
|
|
|
|
def normalized(self):
|
|
return self.copy().normalize()
|
|
|
|
def conjugated(self):
|
|
return self.copy().conjugate()
|
|
|
|
def homomorphed(self):
|
|
return self.copy().homomorph()
|
|
|
|
def asList(self):
|
|
return [self.q]+list(self.p)
|
|
|
|
def asM(self): # to find Averaging Quaternions (see F. Landis Markley et al.)
|
|
return np.outer(self.asList(),self.asList())
|
|
|
|
def asMatrix(self):
|
|
# Rowenhorst_etal2015 MSMSE: value of P is selected as -1
|
|
P = -1.0
|
|
qbarhalf = 0.5*(self.q**2 - np.dot(self.p,self.p))
|
|
return 2.0*np.array(
|
|
[[ qbarhalf + self.p[0]**2 ,
|
|
self.p[0]*self.p[1] -P* self.q*self.p[2],
|
|
self.p[0]*self.p[2] +P* self.q*self.p[1] ],
|
|
[ self.p[0]*self.p[1] +P* self.q*self.p[2],
|
|
qbarhalf + self.p[1]**2 ,
|
|
self.p[1]*self.p[2] -P* self.q*self.p[0] ],
|
|
[ self.p[0]*self.p[2] -P* self.q*self.p[1],
|
|
self.p[1]*self.p[2] +P* self.q*self.p[0],
|
|
qbarhalf + self.p[2]**2 ],
|
|
])
|
|
|
|
def asAngleAxis(self,
|
|
degrees = False,
|
|
flat = False):
|
|
|
|
angle = 2.0*math.acos(self.q)
|
|
|
|
if np.isclose(angle,0.0):
|
|
angle = 0.0
|
|
axis = np.array([0.0,0.0,1.0])
|
|
elif np.isclose(self.q,0.0):
|
|
angle = math.pi
|
|
axis = self.p
|
|
else:
|
|
axis = np.sign(self.q)*self.p/np.linalg.norm(self.p)
|
|
|
|
angle = np.degrees(angle) if degrees else angle
|
|
|
|
return np.hstack((angle,axis)) if flat else (angle,axis)
|
|
|
|
def asRodrigues(self):
|
|
return np.inf*np.ones(3) if np.isclose(self.q,0.0) else self.p/self.q
|
|
|
|
def asEulers(self,
|
|
degrees = False):
|
|
"""Orientation as Bunge-Euler angles."""
|
|
# Rowenhorst_etal2015 MSMSE: value of P is selected as -1
|
|
P = -1.0
|
|
q03 = self.q**2 + self.p[2]**2
|
|
q12 = self.p[0]**2 + self.p[1]**2
|
|
chi = np.sqrt(q03*q12)
|
|
|
|
if np.isclose(chi,0.0) and np.isclose(q12,0.0):
|
|
eulers = np.array([math.atan2(-2*P*self.q*self.p[2],self.q**2-self.p[2]**2),0,0])
|
|
elif np.isclose(chi,0.0) and np.isclose(q03,0.0):
|
|
eulers = np.array([math.atan2( 2 *self.p[0]*self.p[1],self.p[0]**2-self.p[1]**2),np.pi,0])
|
|
else:
|
|
eulers = np.array([math.atan2((self.p[0]*self.p[2]-P*self.q*self.p[1])/chi,(-P*self.q*self.p[0]-self.p[1]*self.p[2])/chi),
|
|
math.atan2(2*chi,q03-q12),
|
|
math.atan2((P*self.q*self.p[1]+self.p[0]*self.p[2])/chi,( self.p[1]*self.p[2]-P*self.q*self.p[0])/chi),
|
|
])
|
|
|
|
eulers %= 2.0*math.pi # enforce positive angles
|
|
return np.degrees(eulers) if degrees else eulers
|
|
|
|
|
|
# # Static constructors
|
|
@classmethod
|
|
def fromIdentity(cls):
|
|
return cls()
|
|
|
|
|
|
@classmethod
|
|
def fromRandom(cls,randomSeed = None):
|
|
import binascii
|
|
if randomSeed is None:
|
|
randomSeed = int(binascii.hexlify(os.urandom(4)),16)
|
|
np.random.seed(randomSeed)
|
|
r = np.random.random(3)
|
|
A = math.sqrt(max(0.0,r[2]))
|
|
B = math.sqrt(max(0.0,1.0-r[2]))
|
|
w = math.cos(2.0*math.pi*r[0])*A
|
|
x = math.sin(2.0*math.pi*r[1])*B
|
|
y = math.cos(2.0*math.pi*r[1])*B
|
|
z = math.sin(2.0*math.pi*r[0])*A
|
|
return cls(quat=[w,x,y,z])
|
|
|
|
|
|
@classmethod
|
|
def fromRodrigues(cls, rodrigues):
|
|
if not isinstance(rodrigues, np.ndarray): rodrigues = np.array(rodrigues)
|
|
norm = np.linalg.norm(rodrigues)
|
|
halfangle = math.atan(norm)
|
|
s = math.sin(halfangle)
|
|
c = math.cos(halfangle)
|
|
return cls(q=c,p=s*rodrigues/norm)
|
|
|
|
|
|
@classmethod
|
|
def fromAngleAxis(cls,
|
|
angle,
|
|
axis,
|
|
degrees = False):
|
|
if not isinstance(axis, np.ndarray): axis = np.array(axis,dtype=float)
|
|
axis = axis.astype(float)/np.linalg.norm(axis)
|
|
angle = np.radians(angle) if degrees else angle
|
|
s = math.sin(0.5 * angle)
|
|
c = math.cos(0.5 * angle)
|
|
return cls(q=c,p=axis*s)
|
|
|
|
|
|
@classmethod
|
|
def fromEulers(cls,
|
|
eulers,
|
|
degrees = False):
|
|
if not isinstance(eulers, np.ndarray): eulers = np.array(eulers,dtype=float)
|
|
eulers = np.radians(eulers) if degrees else eulers
|
|
|
|
sigma = 0.5*(eulers[0]+eulers[2])
|
|
delta = 0.5*(eulers[0]-eulers[2])
|
|
c = np.cos(0.5*eulers[1])
|
|
s = np.sin(0.5*eulers[1])
|
|
|
|
# Rowenhorst_etal2015 MSMSE: value of P is selected as -1
|
|
P = -1.0
|
|
w = c * np.cos(sigma)
|
|
x = -P * s * np.cos(delta)
|
|
y = -P * s * np.sin(delta)
|
|
z = -P * c * np.sin(sigma)
|
|
return cls(quat=[w,x,y,z])
|
|
|
|
|
|
# Modified Method to calculate Quaternion from Orientation Matrix,
|
|
# Source: http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/
|
|
|
|
@classmethod
|
|
def fromMatrix(cls, m):
|
|
if m.shape != (3,3) and np.prod(m.shape) == 9:
|
|
m = m.reshape(3,3)
|
|
|
|
# Rowenhorst_etal2015 MSMSE: value of P is selected as -1
|
|
P = -1.0
|
|
w = 0.5*math.sqrt(max(0.0,1.0+m[0,0]+m[1,1]+m[2,2]))
|
|
x = P*0.5*math.sqrt(max(0.0,1.0+m[0,0]-m[1,1]-m[2,2]))
|
|
y = P*0.5*math.sqrt(max(0.0,1.0-m[0,0]+m[1,1]-m[2,2]))
|
|
z = P*0.5*math.sqrt(max(0.0,1.0-m[0,0]-m[1,1]+m[2,2]))
|
|
|
|
x *= -1 if m[2,1] < m[1,2] else 1
|
|
y *= -1 if m[0,2] < m[2,0] else 1
|
|
z *= -1 if m[1,0] < m[0,1] else 1
|
|
|
|
return cls(quat=np.array([w,x,y,z])/math.sqrt(w**2 + x**2 + y**2 + z**2))
|
|
|
|
|
|
@classmethod
|
|
def new_interpolate(cls, q1, q2, t):
|
|
"""
|
|
Interpolation
|
|
|
|
See http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20070017872_2007014421.pdf
|
|
for (another?) way to interpolate quaternions.
|
|
"""
|
|
assert isinstance(q1, Quaternion) and isinstance(q2, Quaternion)
|
|
Q = cls()
|
|
|
|
costheta = q1.q*q2.q + np.dot(q1.p,q2.p)
|
|
if costheta < 0.:
|
|
costheta = -costheta
|
|
q1 = q1.conjugated()
|
|
elif costheta > 1.:
|
|
costheta = 1.
|
|
|
|
theta = math.acos(costheta)
|
|
if abs(theta) < 0.01:
|
|
Q.q = q2.q
|
|
Q.p = q2.p
|
|
return Q
|
|
|
|
sintheta = math.sqrt(1.0 - costheta * costheta)
|
|
if abs(sintheta) < 0.01:
|
|
Q.q = (q1.q + q2.q) * 0.5
|
|
Q.p = (q1.p + q2.p) * 0.5
|
|
return Q
|
|
|
|
ratio1 = math.sin((1.0 - t) * theta) / sintheta
|
|
ratio2 = math.sin( t * theta) / sintheta
|
|
|
|
Q.q = q1.q * ratio1 + q2.q * ratio2
|
|
Q.p = q1.p * ratio1 + q2.p * ratio2
|
|
return Q
|
|
|
|
|
|
# ******************************************************************************************
|
|
class Symmetry:
|
|
|
|
lattices = [None,'orthorhombic','tetragonal','hexagonal','cubic',]
|
|
|
|
def __init__(self, symmetry = None):
|
|
"""Lattice with given symmetry, defaults to None"""
|
|
if isinstance(symmetry, str) and symmetry.lower() in Symmetry.lattices:
|
|
self.lattice = symmetry.lower()
|
|
else:
|
|
self.lattice = None
|
|
|
|
|
|
def __copy__(self):
|
|
"""Copy"""
|
|
return self.__class__(self.lattice)
|
|
|
|
copy = __copy__
|
|
|
|
|
|
def __repr__(self):
|
|
"""Readable string"""
|
|
return '{}'.format(self.lattice)
|
|
|
|
|
|
def __eq__(self, other):
|
|
"""Equal to other"""
|
|
return self.lattice == other.lattice
|
|
|
|
def __neq__(self, other):
|
|
"""Not equal to other"""
|
|
return not self.__eq__(other)
|
|
|
|
def __cmp__(self,other):
|
|
"""Linear ordering"""
|
|
myOrder = Symmetry.lattices.index(self.lattice)
|
|
otherOrder = Symmetry.lattices.index(other.lattice)
|
|
return (myOrder > otherOrder) - (myOrder < otherOrder)
|
|
|
|
def symmetryQuats(self,who = []):
|
|
"""List of symmetry operations as quaternions."""
|
|
if self.lattice == 'cubic':
|
|
symQuats = [
|
|
[ 1.0, 0.0, 0.0, 0.0 ],
|
|
[ 0.0, 1.0, 0.0, 0.0 ],
|
|
[ 0.0, 0.0, 1.0, 0.0 ],
|
|
[ 0.0, 0.0, 0.0, 1.0 ],
|
|
[ 0.0, 0.0, 0.5*math.sqrt(2), 0.5*math.sqrt(2) ],
|
|
[ 0.0, 0.0, 0.5*math.sqrt(2),-0.5*math.sqrt(2) ],
|
|
[ 0.0, 0.5*math.sqrt(2), 0.0, 0.5*math.sqrt(2) ],
|
|
[ 0.0, 0.5*math.sqrt(2), 0.0, -0.5*math.sqrt(2) ],
|
|
[ 0.0, 0.5*math.sqrt(2),-0.5*math.sqrt(2), 0.0 ],
|
|
[ 0.0, -0.5*math.sqrt(2),-0.5*math.sqrt(2), 0.0 ],
|
|
[ 0.5, 0.5, 0.5, 0.5 ],
|
|
[-0.5, 0.5, 0.5, 0.5 ],
|
|
[-0.5, 0.5, 0.5, -0.5 ],
|
|
[-0.5, 0.5, -0.5, 0.5 ],
|
|
[-0.5, -0.5, 0.5, 0.5 ],
|
|
[-0.5, -0.5, 0.5, -0.5 ],
|
|
[-0.5, -0.5, -0.5, 0.5 ],
|
|
[-0.5, 0.5, -0.5, -0.5 ],
|
|
[-0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
|
|
[ 0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
|
|
[-0.5*math.sqrt(2), 0.0, 0.5*math.sqrt(2), 0.0 ],
|
|
[-0.5*math.sqrt(2), 0.0, -0.5*math.sqrt(2), 0.0 ],
|
|
[-0.5*math.sqrt(2), 0.5*math.sqrt(2), 0.0, 0.0 ],
|
|
[-0.5*math.sqrt(2),-0.5*math.sqrt(2), 0.0, 0.0 ],
|
|
]
|
|
elif self.lattice == 'hexagonal':
|
|
symQuats = [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
[-0.5*math.sqrt(3), 0.0, 0.0,-0.5 ],
|
|
[ 0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
|
|
[ 0.0,0.0,0.0,1.0 ],
|
|
[-0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
|
|
[-0.5*math.sqrt(3), 0.0, 0.0, 0.5 ],
|
|
[ 0.0,1.0,0.0,0.0 ],
|
|
[ 0.0,-0.5*math.sqrt(3), 0.5, 0.0 ],
|
|
[ 0.0, 0.5,-0.5*math.sqrt(3), 0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
[ 0.0,-0.5,-0.5*math.sqrt(3), 0.0 ],
|
|
[ 0.0, 0.5*math.sqrt(3), 0.5, 0.0 ],
|
|
]
|
|
elif self.lattice == 'tetragonal':
|
|
symQuats = [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
[ 0.0,1.0,0.0,0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
[ 0.0,0.0,0.0,1.0 ],
|
|
[ 0.0, 0.5*math.sqrt(2), 0.5*math.sqrt(2), 0.0 ],
|
|
[ 0.0,-0.5*math.sqrt(2), 0.5*math.sqrt(2), 0.0 ],
|
|
[ 0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
|
|
[-0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
|
|
]
|
|
elif self.lattice == 'orthorhombic':
|
|
symQuats = [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
[ 0.0,1.0,0.0,0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
[ 0.0,0.0,0.0,1.0 ],
|
|
]
|
|
else:
|
|
symQuats = [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
]
|
|
|
|
return list(map(Quaternion,
|
|
np.array(symQuats)[np.atleast_1d(np.array(who)) if who != [] else range(len(symQuats))]))
|
|
|
|
|
|
def equivalentQuaternions(self,
|
|
quaternion,
|
|
who = []):
|
|
"""List of symmetrically equivalent quaternions based on own symmetry."""
|
|
return [q*quaternion for q in self.symmetryQuats(who)]
|
|
|
|
|
|
def inFZ(self,R):
|
|
"""Check whether given Rodrigues vector falls into fundamental zone of own symmetry."""
|
|
if isinstance(R, Quaternion): R = R.asRodrigues() # translate accidentally passed quaternion
|
|
# fundamental zone in Rodrigues space is point symmetric around origin
|
|
R = abs(R)
|
|
if self.lattice == 'cubic':
|
|
return math.sqrt(2.0)-1.0 >= R[0] \
|
|
and math.sqrt(2.0)-1.0 >= R[1] \
|
|
and math.sqrt(2.0)-1.0 >= R[2] \
|
|
and 1.0 >= R[0] + R[1] + R[2]
|
|
elif self.lattice == 'hexagonal':
|
|
return 1.0 >= R[0] and 1.0 >= R[1] and 1.0 >= R[2] \
|
|
and 2.0 >= math.sqrt(3)*R[0] + R[1] \
|
|
and 2.0 >= math.sqrt(3)*R[1] + R[0] \
|
|
and 2.0 >= math.sqrt(3) + R[2]
|
|
elif self.lattice == 'tetragonal':
|
|
return 1.0 >= R[0] and 1.0 >= R[1] \
|
|
and math.sqrt(2.0) >= R[0] + R[1] \
|
|
and math.sqrt(2.0) >= R[2] + 1.0
|
|
elif self.lattice == 'orthorhombic':
|
|
return 1.0 >= R[0] and 1.0 >= R[1] and 1.0 >= R[2]
|
|
else:
|
|
return True
|
|
|
|
|
|
def inDisorientationSST(self,R):
|
|
"""
|
|
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 isinstance(R, Quaternion): R = R.asRodrigues() # translate accidentially passed quaternion
|
|
|
|
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 Orientation:
|
|
|
|
__slots__ = ['quaternion','symmetry']
|
|
|
|
def __init__(self,
|
|
quaternion = Quaternion.fromIdentity(),
|
|
Rodrigues = None,
|
|
angleAxis = None,
|
|
matrix = None,
|
|
Eulers = None,
|
|
random = False, # integer to have a fixed seed or True for real random
|
|
symmetry = None,
|
|
degrees = False,
|
|
):
|
|
if random: # produce random orientation
|
|
if isinstance(random, bool ):
|
|
self.quaternion = Quaternion.fromRandom()
|
|
else:
|
|
self.quaternion = Quaternion.fromRandom(randomSeed=random)
|
|
elif isinstance(Eulers, np.ndarray) and Eulers.shape == (3,): # based on given Euler angles
|
|
self.quaternion = Quaternion.fromEulers(Eulers,degrees=degrees)
|
|
elif isinstance(matrix, np.ndarray) : # based on given rotation matrix
|
|
self.quaternion = Quaternion.fromMatrix(matrix)
|
|
elif isinstance(angleAxis, np.ndarray) and angleAxis.shape == (4,): # based on given angle and rotation axis
|
|
self.quaternion = Quaternion.fromAngleAxis(angleAxis[0],angleAxis[1:4],degrees=degrees)
|
|
elif isinstance(Rodrigues, np.ndarray) and Rodrigues.shape == (3,): # based on given Rodrigues vector
|
|
self.quaternion = Quaternion.fromRodrigues(Rodrigues)
|
|
elif isinstance(quaternion, Quaternion): # based on given quaternion
|
|
self.quaternion = quaternion.homomorphed()
|
|
elif (isinstance(quaternion, np.ndarray) and quaternion.shape == (4,)) or \
|
|
(isinstance(quaternion, list) and len(quaternion) == 4 ): # based on given quaternion-like array
|
|
self.quaternion = Quaternion(quat=quaternion).homomorphed()
|
|
|
|
self.symmetry = Symmetry(symmetry)
|
|
|
|
def __copy__(self):
|
|
"""Copy"""
|
|
return self.__class__(quaternion=self.quaternion,symmetry=self.symmetry.lattice)
|
|
|
|
copy = __copy__
|
|
|
|
|
|
def __repr__(self):
|
|
"""Value as all implemented representations"""
|
|
return '\n'.join([
|
|
'Symmetry: {}'.format(self.symmetry),
|
|
'Quaternion: {}'.format(self.quaternion),
|
|
'Matrix:\n{}'.format( '\n'.join(['\t'.join(list(map(str,self.asMatrix()[i,:]))) for i in range(3)]) ),
|
|
'Bunge Eulers / deg: {}'.format('\t'.join(list(map(str,self.asEulers(degrees=True)))) ),
|
|
])
|
|
|
|
def asQuaternion(self):
|
|
return self.quaternion.asList()
|
|
|
|
def asEulers(self,
|
|
degrees = False,
|
|
):
|
|
return self.quaternion.asEulers(degrees)
|
|
eulers = property(asEulers)
|
|
|
|
def asRodrigues(self):
|
|
return self.quaternion.asRodrigues()
|
|
rodrigues = property(asRodrigues)
|
|
|
|
def asAngleAxis(self,
|
|
degrees = False,
|
|
flat = False):
|
|
return self.quaternion.asAngleAxis(degrees,flat)
|
|
angleAxis = property(asAngleAxis)
|
|
|
|
def asMatrix(self):
|
|
return self.quaternion.asMatrix()
|
|
matrix = property(asMatrix)
|
|
|
|
def inFZ(self):
|
|
return self.symmetry.inFZ(self.quaternion.asRodrigues())
|
|
infz = property(inFZ)
|
|
|
|
def equivalentQuaternions(self,
|
|
who = []):
|
|
return self.symmetry.equivalentQuaternions(self.quaternion,who)
|
|
|
|
def equivalentOrientations(self,
|
|
who = []):
|
|
return [Orientation(quaternion = q, symmetry = self.symmetry.lattice) for q in self.equivalentQuaternions(who)]
|
|
|
|
def reduced(self):
|
|
"""Transform orientation to fall into fundamental zone according to symmetry"""
|
|
for me in self.symmetry.equivalentQuaternions(self.quaternion):
|
|
if self.symmetry.inFZ(me.asRodrigues()): break
|
|
|
|
return Orientation(quaternion=me,symmetry=self.symmetry.lattice)
|
|
|
|
|
|
def disorientation(self,
|
|
other,
|
|
SST = True):
|
|
"""
|
|
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.symmetry != other.symmetry: raise TypeError('disorientation between different symmetry classes not supported yet.')
|
|
|
|
misQ = other.quaternion*self.quaternion.conjugated()
|
|
mySymQs = self.symmetry.symmetryQuats() if SST else self.symmetry.symmetryQuats()[:1] # take all or only first sym operation
|
|
otherSymQs = other.symmetry.symmetryQuats()
|
|
|
|
for i,sA in enumerate(mySymQs):
|
|
for j,sB in enumerate(otherSymQs):
|
|
theQ = sB*misQ*sA.conjugated()
|
|
for k in range(2):
|
|
theQ.conjugate()
|
|
breaker = self.symmetry.inFZ(theQ) \
|
|
and (not SST or other.symmetry.inDisorientationSST(theQ))
|
|
if breaker: break
|
|
if breaker: break
|
|
if breaker: break
|
|
|
|
# disorientation, own sym, other sym, self-->other: True, self<--other: False
|
|
return (Orientation(quaternion = theQ,symmetry = self.symmetry.lattice),
|
|
i,j, k == 1)
|
|
|
|
|
|
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,q in enumerate(self.symmetry.equivalentQuaternions(self.quaternion)): # test all symmetric equivalent quaternions
|
|
pole = q*axis # align crystal direction to axis
|
|
if self.symmetry.inSST(pole,proper): break # found SST version
|
|
else:
|
|
pole = self.quaternion*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 q in self.symmetry.equivalentQuaternions(self.quaternion):
|
|
pole = q*axis # align crystal direction to axis
|
|
inSST,color = self.symmetry.inSST(pole,color=True)
|
|
if inSST: break
|
|
|
|
return color
|
|
|
|
@classmethod
|
|
def average(cls,
|
|
orientations,
|
|
multiplicity = []):
|
|
"""
|
|
Average orientation
|
|
|
|
ref: F. Landis Markley, Yang Cheng, John Lucas Crassidis, and Yaakov Oshman.
|
|
Averaging Quaternions,
|
|
Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4 (2007), pp. 1193-1197.
|
|
doi: 10.2514/1.28949
|
|
usage:
|
|
a = Orientation(Eulers=np.radians([10, 10, 0]), symmetry='hexagonal')
|
|
b = Orientation(Eulers=np.radians([20, 0, 0]), symmetry='hexagonal')
|
|
avg = Orientation.average([a,b])
|
|
"""
|
|
if not all(isinstance(item, Orientation) for item in orientations):
|
|
raise TypeError("Only instances of Orientation can be averaged.")
|
|
|
|
N = len(orientations)
|
|
if multiplicity == [] or not multiplicity:
|
|
multiplicity = np.ones(N,dtype='i')
|
|
|
|
reference = orientations[0] # take first as reference
|
|
for i,(o,n) in enumerate(zip(orientations,multiplicity)):
|
|
closest = o.equivalentOrientations(reference.disorientation(o,SST = False)[2])[0] # select sym orientation with lowest misorientation
|
|
M = closest.quaternion.asM() * n if i == 0 else M + closest.quaternion.asM() * n # noqa add (multiples) of this orientation to average noqa
|
|
eig, vec = np.linalg.eig(M/N)
|
|
|
|
return Orientation(quaternion = Quaternion(quat = np.real(vec.T[eig.argmax()])),
|
|
symmetry = reference.symmetry.lattice)
|
|
|
|
|
|
def related(self,
|
|
relationModel,
|
|
direction,
|
|
targetSymmetry = 'cubic'):
|
|
"""
|
|
Orientation relationship
|
|
|
|
positive number: fcc --> bcc
|
|
negative number: bcc --> fcc
|
|
"""
|
|
if relationModel not in ['KS','GT','GTdash','NW','Pitsch','Bain']: return None
|
|
if int(direction) == 0: return None
|
|
|
|
# KS from S. Morito et al./Journal of Alloys and Compounds 5775 (2013) S587-S592
|
|
# for KS rotation matrices also check K. Kitahara et al./Acta Materialia 54 (2006) 1279-1288
|
|
# GT from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
# GT' from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
# NW from H. Kitahara et al./Materials Characterization 54 (2005) 378-386
|
|
# Pitsch from Y. He et al./Acta Materialia 53 (2005) 1179-1190
|
|
# Bain from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
|
|
variant = int(abs(direction))-1
|
|
(me,other) = (0,1) if direction > 0 else (1,0)
|
|
|
|
planes = {'KS': \
|
|
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]]]),
|
|
'GT': \
|
|
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]]]),
|
|
'GTdash': \
|
|
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]]]),
|
|
'NW': \
|
|
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]]]),
|
|
'Pitsch': \
|
|
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]]]),
|
|
'Bain': \
|
|
np.array([[[ 1, 0, 0],[ 1, 0, 0]],
|
|
[[ 0, 1, 0],[ 0, 1, 0]],
|
|
[[ 0, 0, 1],[ 0, 0, 1]]]),
|
|
}
|
|
|
|
normals = {'KS': \
|
|
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]]]),
|
|
'GT': \
|
|
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]]]),
|
|
'GTdash': \
|
|
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]]]),
|
|
'NW': \
|
|
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]]]),
|
|
'Pitsch': \
|
|
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]]]),
|
|
'Bain': \
|
|
np.array([[[ 0, 1, 0],[ 0, 1, 1]],
|
|
[[ 0, 0, 1],[ 1, 0, 1]],
|
|
[[ 1, 0, 0],[ 1, 1, 0]]]),
|
|
}
|
|
myPlane = [float(i) for i in planes[relationModel][variant,me]] # map(float, planes[...]) does not work in python 3
|
|
myPlane /= np.linalg.norm(myPlane)
|
|
myNormal = [float(i) for i in normals[relationModel][variant,me]] # map(float, planes[...]) does not work in python 3
|
|
myNormal /= np.linalg.norm(myNormal)
|
|
myMatrix = np.array([myNormal,np.cross(myPlane,myNormal),myPlane]).T
|
|
|
|
otherPlane = [float(i) for i in planes[relationModel][variant,other]] # map(float, planes[...]) does not work in python 3
|
|
otherPlane /= np.linalg.norm(otherPlane)
|
|
otherNormal = [float(i) for i in normals[relationModel][variant,other]] # map(float, planes[...]) does not work in python 3
|
|
otherNormal /= np.linalg.norm(otherNormal)
|
|
otherMatrix = np.array([otherNormal,np.cross(otherPlane,otherNormal),otherPlane]).T
|
|
|
|
rot=np.dot(otherMatrix,myMatrix.T)
|
|
|
|
return Orientation(matrix=np.dot(rot,self.asMatrix()),symmetry=targetSymmetry)
|
|
|
|
####################################################################################################
|
|
# Code below available according to the followin 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-15,rtol=0.0)
|
|
|
|
def iszero(a):
|
|
return np.isclose(a,0.0,atol=1.0e-300,rtol=0.0)
|
|
|
|
|
|
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 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.homomorph() !ToDo: Check with original
|
|
return qu
|
|
|
|
|
|
def om2eu(om):
|
|
"""Euler angles to orientation matrix"""
|
|
if isone(om[2,2]**2):
|
|
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
|
|
else:
|
|
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)])
|
|
|
|
# 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 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 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:
|
|
#chiInv = 1.0/chi ToDo: needed for what?
|
|
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 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 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(s)) # ToDo: Check sanity check in reference implementation
|
|
|
|
return ax
|
|
|
|
|
|
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 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 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 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 ro2ho(ro):
|
|
"""Rodrigues vector to homochoric"""
|
|
if iszero(np.sum(ro[0:3]**2.0)):
|
|
ho = [ 0.0, 0.0, 0.0 ]
|
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else:
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f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
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ho = ro[0:3] * (0.75*f)**(1.0/3.0)
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return np.array(ho)
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|
|
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def qu2om(qu):
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|
"""Quaternion to orientation matrix"""
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|
qq = qu[0]**2-(qu[1]**2 + qu[2]**2 + qu[3]**2)
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om = np.diag(qq + 2.0*np.array([qu[1],qu[2],qu[3]])**2)
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|
|
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om[1,0] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3])
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om[0,1] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3])
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om[2,1] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1])
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om[1,2] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1])
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om[0,2] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2])
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om[2,0] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2])
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return om if P > 0.0 else om.T
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|
|
|
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def om2qu(om):
|
|
"""Orientation matrix to quaternion"""
|
|
s = [+om[0,0] +om[1,1] +om[2,2] +1.0,
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+om[0,0] -om[1,1] -om[2,2] +1.0,
|
|
-om[0,0] +om[1,1] -om[2,2] +1.0,
|
|
-om[0,0] -om[1,1] +om[2,2] +1.0]
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s = np.maximum(np.zeros(4),s)
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qu = np.sqrt(s)*0.5*np.array([1.0,P,P,P])
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# verify the signs (q0 always positive)
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#ToDo: Here I donot understand the original shortcut from paper to implementation
|
|
|
|
qu /= np.linalg.norm(qu)
|
|
if any(isone(abs(qu))): qu[np.where(np.logical_not(isone(qu)))] = 0.0
|
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if om[2,1] < om[1,2]: qu[1] *= -1.0
|
|
if om[0,2] < om[2,0]: qu[2] *= -1.0
|
|
if om[1,0] < om[0,1]: qu[3] *= -1.0
|
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if any(om2ax(om)[0:3]*qu[1:4] < 0.0): print(om2ax(om),qu) # something is wrong here
|
|
return qu
|
|
|
|
|
|
def qu2ax(qu):
|
|
"""Quaternion to axis angle"""
|
|
omega = 2.0 * np.arccos(qu[0])
|
|
if iszero(omega): # return axis as [001] if the angle is zero
|
|
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)
|
|
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(qu[0]))]
|
|
|
|
return np.array(ro)
|
|
|
|
|
|
def qu2ho(qu):
|
|
"""Quaternion to homochoric"""
|
|
omega = 2.0 * np.arccos(qu[0])
|
|
|
|
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 ho2cu(ho):
|
|
"""Homochoric to cubochoric"""
|
|
return Lambert.BallToCube(ho)
|
|
|
|
|
|
def cu2ho(cu):
|
|
"""Cubochoric to homochoric"""
|
|
return Lambert.CubeToBall(cu)
|
|
|
|
|
|
def ro2eu(ro):
|
|
"""Rodrigues vector to orientation matrix"""
|
|
return om2eu(ro2om(ro))
|
|
|
|
|
|
def eu2ho(eu):
|
|
"""Euler angles to homochoric"""
|
|
return ax2ho(eu2ax(eu))
|
|
|
|
|
|
def om2ro(om):
|
|
"""Orientation matrix to Rodriques vector"""
|
|
return eu2ro(om2eu(om))
|
|
|
|
|
|
def om2ho(om):
|
|
"""Orientation matrix to homochoric"""
|
|
return ax2ho(om2ax(om))
|
|
|
|
|
|
def ax2eu(ax):
|
|
"""Orientation matrix to Euler angles"""
|
|
return om2eu(ax2om(ax))
|
|
|
|
|
|
def ro2om(ro):
|
|
"""Rodgrigues vector to orientation matrix"""
|
|
return ax2om(ro2ax(ro))
|
|
|
|
|
|
def ro2qu(ro):
|
|
"""Rodrigues vector to quaternion"""
|
|
return ax2qu(ro2ax(ro))
|
|
|
|
|
|
def ho2eu(ho):
|
|
"""Homochoric to Euler angles"""
|
|
return ax2eu(ho2ax(ho))
|
|
|
|
|
|
def ho2om(ho):
|
|
"""Homochoric to orientation matrix"""
|
|
return ax2om(ho2ax(ho))
|
|
|
|
|
|
def ho2ro(ho):
|
|
"""Axis angle to Rodriques vector"""
|
|
return ax2ro(ho2ax(ho))
|
|
|
|
|
|
def ho2qu(ho):
|
|
"""Homochoric to quaternion"""
|
|
return ax2qu(ho2ax(ho))
|
|
|
|
|
|
def eu2cu(eu):
|
|
"""Euler angles to cubochoric"""
|
|
return ho2cu(eu2ho(eu))
|
|
|
|
|
|
def om2cu(om):
|
|
"""Orientation matrix to cubochoric"""
|
|
return ho2cu(om2ho(om))
|
|
|
|
|
|
def ax2cu(ax):
|
|
"""Axis angle to cubochoric"""
|
|
return ho2cu(ax2ho(ax))
|
|
|
|
|
|
def ro2cu(ro):
|
|
"""Rodrigues vector to cubochoric"""
|
|
return ho2cu(ro2ho(ro))
|
|
|
|
|
|
def qu2cu(qu):
|
|
"""Quaternion to cubochoric"""
|
|
return ho2cu(qu2ho(qu))
|
|
|
|
|
|
def cu2eu(cu):
|
|
"""Cubochoric to Euler angles"""
|
|
return ho2eu(cu2ho(cu))
|
|
|
|
|
|
def cu2om(cu):
|
|
"""Cubochoric to orientation matrix"""
|
|
return ho2om(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 cu2qu(cu):
|
|
"""Cubochoric to quaternion"""
|
|
return ho2qu(cu2ho(cu))
|