# -*- coding: UTF-8 no BOM -*-
###################################################
# NOTE: everything here needs to be a np array #
###################################################
import math,os
import numpy as np
# ******************************************************************************************
class Rodrigues:
def __init__(self, vector = np.zeros(3)):
self.vector = vector
def asQuaternion(self):
norm = np.linalg.norm(self.vector)
halfAngle = np.arctan(norm)
return Quaternion(np.cos(halfAngle),np.sin(halfAngle)*self.vector/norm)
def asAngleAxis(self):
norm = np.linalg.norm(self.vector)
halfAngle = np.arctan(norm)
return (2.0*halfAngle,self.vector/norm)
# ******************************************************************************************
class Quaternion:
u"""
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]
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):
"""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
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
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
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 at e-8 precision"""
return (self-other).magnitude() < 1e-8 or (-self-other).magnitude() < 1e-8
def __ne__(self,other):
"""Not equal at e-8 precision"""
return not self.__eq__(self,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 identity(self):
self.q = 1.
self.p = np.zeros(3,dtype=float)
return self
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 inverse(self):
d = self.magnitude()
if d > 0.0:
self.conjugate()
self /= d
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 inversed(self):
return self.copy().inverse()
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):
if self.q > 1.:
self.normalize()
s = math.sqrt(1. - self.q**2)
x = 2*self.q**2 - 1.
y = 2*self.q * s
angle = math.atan2(y,x)
if angle < 0.0:
angle *= -1.
s *= -1.
return (np.degrees(angle) if degrees else angle,
np.array([1.0, 0.0, 0.0] if np.abs(angle) < 1e-6 else self.p / s))
def asRodrigues(self):
return np.inf*np.ones(3) if 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 abs(chi) < 1e-10 and abs(q12) < 1e-10:
eulers = np.array([math.atan2(-2*P*self.q*self.p[2],self.q**2-self.p[2]**2),0,0])
elif abs(chi) < 1e-10 and abs(q03) < 1e-10:
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),
])
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)
w = math.cos(2.0*math.pi*r[0])*math.sqrt(r[2])
x = math.sin(2.0*math.pi*r[1])*math.sqrt(1.0-r[2])
y = math.cos(2.0*math.pi*r[1])*math.sqrt(1.0-r[2])
z = math.sin(2.0*math.pi*r[0])*math.sqrt(r[2])
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(1.+m[0,0]+m[1,1]+m[2,2])
x = P*0.5*math.sqrt(1.+m[0,0]-m[1,1]-m[2,2])
y = P*0.5*math.sqrt(1.-m[0,0]+m[1,1]-m[2,2])
z = P*0.5*math.sqrt(1.-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):
"""Readbable string"""
return '{}'.format(self.lattice)
def __eq__(self, other):
"""Equal"""
return self.lattice == other.lattice
def __neq__(self, other):
"""Not equal"""
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 accidentially 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.
"""
# 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.)]).transpose()), # 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.)]).transpose()), # 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.)]).transpose()), # direction of blue
# 'orthorhombic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
# [1.,0.,0.], # direction of green
# [0.,1.,0.]]).transpose()), # 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):
return self.quaternion.asAngleAxis(degrees)
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)