DAMASK_EICMD/python/damask/orientation.py

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# -*- coding: UTF-8 no BOM -*-
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import math,os
import numpy as np
from . import Lambert
P = -1
####################################################################################################
class Quaternion2:
u"""
Quaternion with basic operations
q is the real part, p = (x, y, z) are the imaginary parts.
Defintion of multiplication depends on variable P, P {-1,1}.
"""
def __init__(self,
q = 0.0,
p = np.zeros(3,dtype=float)):
"""Initializes to identity unless specified"""
self.q = q
self.p = np.array(p)
def __copy__(self):
"""Copy"""
return self.__class__(q=self.q,
p=self.p.copy())
copy = __copy__
def __iter__(self):
"""Components"""
return iter(self.asList())
def asArray(self):
"""As numpy array"""
return np.array((self.q,self.p[0],self.p[1],self.p[2]))
def asList(self):
return [self.q]+list(self.p)
def __repr__(self):
"""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)
def __add__(self, other):
"""Addition"""
if isinstance(other, Quaternion2):
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, Quaternion2):
self.q += other.q
self.p += other.p
return self
else:
return NotImplemented
def __pos__(self):
"""Unary positive operator"""
return self
def __sub__(self, other):
"""Subtraction"""
if isinstance(other, Quaternion2):
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, Quaternion2):
self.q -= other.q
self.p -= other.p
return self
else:
return NotImplemented
def __neg__(self):
"""Unary positive operator"""
self.q *= -1.0
self.p *= -1.0
return self
def __mul__(self, other):
"""Multiplication with quaternion or scalar"""
if isinstance(other, Quaternion2):
return self.__class__(q=self.q*other.q - np.dot(self.p,other.p),
p=self.q*other.p + other.q*self.p + P * np.cross(self.p,other.p))
elif isinstance(other, (int, float)):
return self.__class__(q=self.q*other,
p=self.p*other)
else:
return NotImplemented
def __imul__(self, other):
"""In-place multiplication with quaternion or scalar"""
if isinstance(other, Quaternion2):
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)
return self
elif isinstance(other, (int, float)):
self *= other
return self
else:
return NotImplemented
def __truediv__(self, other):
"""Divsion with quaternion or scalar"""
if isinstance(other, Quaternion2):
s = other.conjugate()/abs(other)**2.
return self.__class__(q=self.q * s,
p=self.p * s)
elif isinstance(other, (int, float)):
self.q /= other
self.p /= other
return self
else:
return NotImplemented
def __itruediv__(self, other):
"""In-place divsion with quaternion or scalar"""
if isinstance(other, Quaternion2):
s = other.conjugate()/abs(other)**2.
self *= s
return self
elif isinstance(other, (int, float)):
self.q /= other
return self
else:
return NotImplemented
def __pow__(self, exponent):
"""Power"""
if isinstance(exponent, (int, float)):
omega = np.acos(self.q)
return self.__class__(q= np.cos(exponent*omega),
p=self.p * np.sin(exponent*omega)/np.sin(omega))
else:
return NotImplemented
def __ipow__(self, exponent):
"""In-place power"""
if isinstance(exponent, (int, float)):
omega = np.acos(self.q)
self.q = np.cos(exponent*omega)
self.p *= np.sin(exponent*omega)/np.sin(omega)
else:
return NotImplemented
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 normalize(self):
d = self.magnitude()
if d > 0.0:
self.q /= d
self.p /= d
return self
def normalized(self):
return self.copy().normalize()
def conjugate(self):
self.p = -self.p
return self
def conjugated(self):
return self.copy().conjugate()
def homomorph(self):
if self.q < 0.0:
self.q = -self.q
self.p = -self.p
return self
def homomorphed(self):
return self.copy().homomorph()
####################################################################################################
class Rotation:
u"""
Orientation stored as unit quaternion.
All methods and naming conventions based on Rowenhorst_etal2015
Convention 1: coordinate frames are right-handed
Convention 2: a rotation angle ω is taken to be positive for a counterclockwise rotation
when viewing from the end point of the rotation axis towards the origin
Convention 3: rotations will be interpreted in the passive sense
Convention 4: Euler angle triplets are implemented using the Bunge convention,
with the angular ranges as [0, 2π],[0, π],[0, 2π]
Convention 5: the rotation angle ω is limited to the interval [0, π]
Convention 6: P = -1 (as default)
q is the real part, p = (x, y, z) are the imaginary parts.
Vector "a" (defined in coordinate system "A") is passively rotated
resulting in new coordinates "b" when expressed in system "B".
b = Q * a
b = np.dot(Q.asMatrix(),a)
"""
__slots__ = ['quaternion']
def __init__(self,quaternion = np.array([1.0,0.0,0.0,0.0])):
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"""
Initializes to identity unless specified
If a quaternion is given, it needs to comply with the convection. Use .fromQuaternion
to check the input.
"""
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if isinstance(quaternion,Quaternion2):
self.quaternion = quaternion.copy()
else:
self.quaternion = Quaternion2(q=quaternion[0],p=quaternion[1:4])
self.quaternion.homomorph() # ToDo: Needed?
def __repr__(self):
"""Value in selected representation"""
return '\n'.join([
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'{}'.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)))) ),
])
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################################################################################################
# convert to different orientation representations (numpy arrays)
def asQuaternion(self):
return self.quaternion.asArray()
def asEulers(self,
degrees = False):
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eu = qu2eu(self.quaternion.asArray())
if degrees: eu = np.degrees(eu)
return eu
def asAngleAxis(self,
degrees = False):
ax = qu2ax(self.quaternion.asArray())
if degrees: ax[3] = np.degrees(ax[3])
return ax
def asMatrix(self):
return qu2om(self.quaternion.asArray())
def asRodrigues(self):
return qu2ro(self.quaternion.asArray())
def asHomochoric(self):
return qu2ho(self.quaternion.asArray())
def asCubochoric(self):
return qu2cu(self.quaternion.asArray())
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################################################################################################
# static constructors. The input data needs to follow the convention, options allow to
# relax these convections
@classmethod
def fromQuaternion(cls,
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quaternion,
P = -1):
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qu = quaternion if isinstance(quaternion, np.ndarray) else np.array(quaternion)
if P > 0: qu[1:4] *= -1 # convert from P=1 to P=-1
if qu[0] < 0.0:
raise ValueError('Quaternion has negative first component.\n{}'.format(qu[0]))
if not np.isclose(np.linalg.norm(qu), 1.0):
raise ValueError('Quaternion is not of unit length.\n{} {} {} {}'.format(*qu))
return cls(qu)
@classmethod
def fromEulers(cls,
eulers,
degrees = False):
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eu = eulers if isinstance(eulers, np.ndarray) else np.array(eulers)
eu = np.radians(eu) if degrees else eu
if np.any(eu < 0.0) or np.any(eu > 2.0*np.pi) or eu[1] > np.pi:
raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].\n{} {} {}.'.format(*eu))
return cls(eu2qu(eu))
@classmethod
def fromAngleAxis(cls,
angleAxis,
degrees = False,
normalise = False,
P = -1):
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ax = angleAxis if isinstance(angleAxis, np.ndarray) else np.array(angleAxis)
if P > 0: ax[0:3] *= -1 # convert from P=1 to P=-1
if degrees: ax[3] = np.radians(ax[3])
if normalise: ax[0:3] /=np.linalg.norm(ax[0:3])
if ax[3] < 0.0 or ax[3] > np.pi:
raise ValueError('Axis angle rotation angle outside of [0..π].\n'.format(ax[3]))
if not np.isclose(np.linalg.norm(ax[0:3]), 1.0):
raise ValueError('Axis angle rotation axis is not of unit length.\n{} {} {}'.format(*ax[0:3]))
return cls(ax2qu(ax))
@classmethod
def fromMatrix(cls,
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matrix,
containsStretch = False): #ToDo: better name?
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:
(U,S,Vh) = np.linalg.svd(om) # singular value decomposition
om = np.dot(U,Vh)
if not np.isclose(np.linalg.det(om),1.0):
raise ValueError('matrix is not a proper rotation.\n{}'.format(om))
if not np.isclose(np.dot(om[0],om[1]), 0.0) \
or not np.isclose(np.dot(om[1],om[2]), 0.0) \
or not np.isclose(np.dot(om[2],om[0]), 0.0):
raise ValueError('matrix is not orthogonal.\n{}'.format(om))
return cls(om2qu(om))
@classmethod
def fromRodrigues(cls,
rodrigues,
normalise = False,
P = -1):
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ro = rodrigues if isinstance(rodrigues, np.ndarray) else np.array(rodrigues)
if P > 0: ro[0:3] *= -1 # convert from P=1 to P=-1
if normalise: ro[0:3] /=np.linalg.norm(ro[0:3])
if not np.isclose(np.linalg.norm(ro[0:3]), 1.0):
raise ValueError('Rodrigues rotation axis is not of unit length.\n{} {} {}'.format(*ro[0:3]))
if ro[3] < 0.0:
raise ValueError('Rodriques rotation angle not positive.\n'.format(ro[3]))
return cls(ro2qu(ro))
def __mul__(self, other):
"""
Multiplication
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Rotation: Details needed (active/passive), rotation of (3,3,3,3)-matrix should be considered
"""
if isinstance(other, Rotation): # rotate a rotation
return self.__class__((self.quaternion * other.quaternion).asArray())
elif isinstance(other, np.ndarray):
if other.shape == (3,): # rotate a single (3)-vector
( x, y, z) = self.quaternion.p
(Vx,Vy,Vz) = other[0:3]
A = self.quaternion.q*self.quaternion.q - np.dot(self.quaternion.p,self.quaternion.p)
B = 2.0 * (x*Vx + y*Vy + z*Vz)
C = 2.0 * P*self.quaternion.q
return np.array([
A*Vx + B*x + C*(y*Vz - z*Vy),
A*Vy + B*y + C*(z*Vx - x*Vz),
A*Vz + B*z + C*(x*Vy - y*Vx),
])
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))
elif other.shape == (3,3,3,3):
raise NotImplementedError
else:
return NotImplemented
elif isinstance(other, tuple): # used to rotate a meshgrid-tuple
( x, y, z) = self.quaternion.p
(Vx,Vy,Vz) = other[0:3]
A = self.quaternion.q*self.quaternion.q - np.dot(self.quaternion.p,self.quaternion.p)
B = 2.0 * (x*Vx + y*Vy + z*Vz)
C = 2.0 * P*self.quaternion.q
return np.array([
A*Vx + B*x + C*(y*Vz - z*Vy),
A*Vy + B*y + C*(z*Vx - x*Vz),
A*Vz + B*z + C*(x*Vy - y*Vx),
])
else:
return NotImplemented
def inverse(self):
"""Inverse rotation/backward rotation"""
self.quaternion.conjugate()
return self
def inversed(self):
"""In-place inverse rotation/backward rotation"""
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return self.__class__(self.quaternion.conjugated())
def misorientation(self,other):
"""Misorientation"""
return self.__class__(other.quaternion*self.quaternion.conjugated())
# ******************************************************************************************
class Quaternion:
u"""
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Orientation represented as unit quaternion.
All methods and naming conventions based on Rowenhorst_etal2015
Convention 1: coordinate frames are right-handed
Convention 2: a rotation angle ω is taken to be positive for a counterclockwise rotation
when viewing from the end point of the rotation axis towards the origin
Convention 3: rotations will be interpreted in the passive sense
Convention 4: Euler angle triplets are implemented using the Bunge convention,
with the angular ranges as [0, 2π],[0, π],[0, 2π]
Convention 5: the rotation angle ω is limited to the interval [0, π]
Convention 6: P = -1 (as default)
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w is the real part, (x, y, z) are the imaginary parts.
Vector "a" (defined in coordinate system "A") is passively rotated
resulting in new coordinates "b" when expressed in system "B".
b = Q * a
b = np.dot(Q.asMatrix(),a)
"""
def __init__(self,
quat = None,
q = 1.0,
p = np.zeros(3,dtype=float)):
"""Initializes to identity unless specified"""
self.q = quat[0] if quat is not None else q
self.p = np.array(quat[1:4]) if quat is not None else p
self.homomorph()
def __iter__(self):
"""Components"""
return iter(self.asList())
def __copy__(self):
"""Copy"""
return self.__class__(q=self.q,p=self.p.copy())
copy = __copy__
def __repr__(self):
"""Readable string"""
return 'Quaternion(real={q:+.6f}, imag=<{p[0]:+.6f}, {p[1]:+.6f}, {p[2]:+.6f}>)'.format(q=self.q,p=self.p)
def __pow__(self, exponent):
"""Power"""
omega = math.acos(self.q)
return self.__class__(q= math.cos(exponent*omega),
p=self.p * math.sin(exponent*omega)/math.sin(omega))
def __ipow__(self, exponent):
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"""In-place power"""
omega = math.acos(self.q)
self.q = math.cos(exponent*omega)
self.p *= math.sin(exponent*omega)/math.sin(omega)
return self
def __mul__(self, other):
"""Multiplication"""
# Rowenhorst_etal2015 MSMSE: value of P is selected as -1
P = -1.0
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try: # quaternion
return self.__class__(q=self.q*other.q - np.dot(self.p,other.p),
p=self.q*other.p + other.q*self.p + P * np.cross(self.p,other.p))
except: pass
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try: # vector (perform passive rotation)
( x, y, z) = self.p
(Vx,Vy,Vz) = other[0:3]
A = self.q*self.q - np.dot(self.p,self.p)
B = 2.0 * (x*Vx + y*Vy + z*Vz)
C = 2.0 * P*self.q
return np.array([
A*Vx + B*x + C*(y*Vz - z*Vy),
A*Vy + B*y + C*(z*Vx - x*Vz),
A*Vz + B*z + C*(x*Vy - y*Vx),
])
except: pass
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try: # scalar
return self.__class__(q=self.q*other,
p=self.p*other)
except:
return self.copy()
def __imul__(self, other):
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"""In-place multiplication"""
# Rowenhorst_etal2015 MSMSE: value of P is selected as -1
P = -1.0
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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):
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"""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):
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"""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):
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"""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)
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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),
])
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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)
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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)
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# 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
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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:
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"""
Symmetry operations for lattice systems
https://en.wikipedia.org/wiki/Crystal_system
"""
lattices = [None,'orthorhombic','tetragonal','hexagonal','cubic',]
def __init__(self, symmetry = 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):
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"""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,
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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."""
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if isinstance(R, Quaternion): R = R.asRodrigues() # translate accidentally passed quaternion
# fundamental zone in Rodrigues space is point symmetric around origin
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if R.shape[0]==4: # transition old (length not stored separately) to new
Rabs = abs(R[0:3]*R[3])
else:
Rabs = abs(R)
if self.lattice == 'cubic':
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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':
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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':
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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':
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return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] and 1.0 >= Rabs[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. ] ]),
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'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.],
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[ 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
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# ******************************************************************************************
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
GTdash = {'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.GTdash,
'NW':self.NW, 'Pitsch': self.Pitsch, 'Bain':self.Bain}
relationship = models[model]
r = {'lattice':Lattice((set(relationship['mapping'])-{self.lattice}).pop()),
'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])
otherPlane = miller[otherPlane_id]/ np.linalg.norm(miller[otherPlane_id])
otherDir = miller[otherDir_id]/ np.linalg.norm(miller[otherDir_id])
myMatrix = np.array([myDir,np.cross(myPlane,myDir),myPlane]).T
otherMatrix = np.array([otherDir,np.cross(otherPlane,otherDir),otherPlane]).T
r['rotations'].append(Rotation.fromMatrix(np.dot(otherMatrix,myMatrix.T)))
return r
class Orientation2:
"""
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):
"""
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.')
mis = other.rotation*self.rotation.inversed()
mySymEqs = self.equivalentOrientations() if SST else self.equivalentOrientations()[:1] # take all or only first sym operation
otherSymEqs = other.equivalentOrientations()
for i,sA in enumerate(mySymEqs):
for j,sB in enumerate(otherSymEqs):
theQ = sB.rotation*mis*sA.rotation.inversed()
for k in range(2):
theQ.inversed()
breaker = self.lattice.symmetry.inFZ(theQ.asRodriques()) #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 theQ
def inFZ(self):
return self.lattice.symmetry.inFZ(self.rotation.asRodrigues())
def equivalentOrientations(self):
"""List of orientations which are symmetrically equivalent"""
q = self.lattice.symmetry.symmetryQuats()
q2 = [Quaternion2(q=a.asList()[0],p=a.asList()[1:4]) for a in q] # convert Quaternion to Quaternion2
x = [self.__class__(q3*self.rotation.quaternion,self.lattice) for q3 in q2]
return x
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)
# ******************************************************************************************
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.)
"""
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if self.symmetry != other.symmetry:
raise NotImplementedError('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
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if int(direction) == 0: return None
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variant = int(abs(direction))-1
(me,other) = (0,1) if direction > 0 else (1,0)
planes = {'KS': \
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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': \
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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': \
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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': \
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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': \
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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': \
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np.array([[[ 1, 0, 0],[ 1, 0, 0]],
[[ 0, 1, 0],[ 0, 1, 0]],
[[ 0, 0, 1],[ 0, 0, 1]]]),
}
normals = {'KS': \
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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': \
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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': \
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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': \
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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': \
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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]],
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[[ 1, 0, 0],[ 1, 1, 0]]]),
}
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myPlane = [float(i) for i in planes[relationModel][variant,me]] # map(float, planes[...]) does not work in python 3
myPlane /= np.linalg.norm(myPlane)
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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
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otherPlane = [float(i) for i in planes[relationModel][variant,other]] # map(float, planes[...]) does not work in python 3
otherPlane /= np.linalg.norm(otherPlane)
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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
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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 ]
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 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 om2qu(om):
"""Orientation matrix to quaternion"""
s = [+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,
-om[0,0] -om[1,1] +om[2,2] +1.0]
s = np.maximum(np.zeros(4),s)
qu = np.sqrt(s)*0.5*np.array([1.0,P,P,P])
# verify the signs (q0 always positive)
#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
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
2019-02-21 17:06:27 +05:30
if any(om2ax(om)[0:3]*qu[1:4] < 0.0): print('sign problem',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))