# -*- 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 unit vector 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,
quatArray = [1.0,0.0,0.0,0.0]):
"""Initializes to identity if not given"""
self.w, \
self.x, \
self.y, \
self.z = quatArray
self.homomorph()
def __iter__(self):
"""Components"""
return iter([self.w,self.x,self.y,self.z])
def __copy__(self):
"""Create copy"""
Q = Quaternion([self.w,self.x,self.y,self.z])
return Q
copy = __copy__
def __repr__(self):
"""Readbable string"""
return 'Quaternion(real=%+.6f, imag=<%+.6f, %+.6f, %+.6f>)' % \
(self.w, self.x, self.y, self.z)
def __pow__(self, exponent):
"""Power"""
omega = math.acos(self.w)
vRescale = math.sin(exponent*omega)/math.sin(omega)
Q = Quaternion()
Q.w = math.cos(exponent*omega)
Q.x = self.x * vRescale
Q.y = self.y * vRescale
Q.z = self.z * vRescale
return Q
def __ipow__(self, exponent):
"""In-place power"""
omega = math.acos(self.w)
vRescale = math.sin(exponent*omega)/math.sin(omega)
self.w = np.cos(exponent*omega)
self.x *= vRescale
self.y *= vRescale
self.z *= vRescale
return self
def __mul__(self, other):
"""Multiplication"""
try: # quaternion
Aw = self.w
Ax = self.x
Ay = self.y
Az = self.z
Bw = other.w
Bx = other.x
By = other.y
Bz = other.z
Q = Quaternion()
Q.w = - Ax * Bx - Ay * By - Az * Bz + Aw * Bw
Q.x = + Ax * Bw + Ay * Bz - Az * By + Aw * Bx
Q.y = - Ax * Bz + Ay * Bw + Az * Bx + Aw * By
Q.z = + Ax * By - Ay * Bx + Az * Bw + Aw * Bz
return Q
except: pass
try: # vector (perform active rotation, i.e. q*v*q.conjugated)
w = self.w
x = self.x
y = self.y
z = self.z
Vx = other[0]
Vy = other[1]
Vz = other[2]
return np.array([\
w * w * Vx + 2 * y * w * Vz - 2 * z * w * Vy + \
x * x * Vx + 2 * y * x * Vy + 2 * z * x * Vz - \
z * z * Vx - y * y * Vx,
2 * x * y * Vx + y * y * Vy + 2 * z * y * Vz + \
2 * w * z * Vx - z * z * Vy + w * w * Vy - \
2 * x * w * Vz - x * x * Vy,
2 * x * z * Vx + 2 * y * z * Vy + \
z * z * Vz - 2 * w * y * Vx - y * y * Vz + \
2 * w * x * Vy - x * x * Vz + w * w * Vz ])
except: pass
try: # scalar
Q = self.copy()
Q.w *= other
Q.x *= other
Q.y *= other
Q.z *= other
return Q
except:
return self.copy()
def __imul__(self, other):
"""In-place multiplication"""
try: # Quaternion
Aw = self.w
Ax = self.x
Ay = self.y
Az = self.z
Bw = other.w
Bx = other.x
By = other.y
Bz = other.z
self.w = - Ax * Bx - Ay * By - Az * Bz + Aw * Bw
self.x = + Ax * Bw + Ay * Bz - Az * By + Aw * Bx
self.y = - Ax * Bz + Ay * Bw + Az * Bx + Aw * By
self.z = + Ax * By - Ay * Bx + Az * Bw + Aw * Bz
except: pass
return self
def __div__(self, other):
"""Division"""
if isinstance(other, (int,float)):
w = self.w / other
x = self.x / other
y = self.y / other
z = self.z / other
return self.__class__([w,x,y,z])
else:
return NotImplemented
def __idiv__(self, other):
"""In-place division"""
if isinstance(other, (int,float)):
self.w /= other
self.x /= other
self.y /= other
self.z /= other
return self
def __add__(self, other):
"""Addition"""
if isinstance(other, Quaternion):
w = self.w + other.w
x = self.x + other.x
y = self.y + other.y
z = self.z + other.z
return self.__class__([w,x,y,z])
else:
return NotImplemented
def __iadd__(self, other):
"""In-place addition"""
if isinstance(other, Quaternion):
self.w += other.w
self.x += other.x
self.y += other.y
self.z += other.z
return self
def __sub__(self, other):
"""Subtraction"""
if isinstance(other, Quaternion):
Q = self.copy()
Q.w -= other.w
Q.x -= other.x
Q.y -= other.y
Q.z -= other.z
return Q
else:
return self.copy()
def __isub__(self, other):
"""In-place subtraction"""
if isinstance(other, Quaternion):
self.w -= other.w
self.x -= other.x
self.y -= other.y
self.z -= other.z
return self
def __neg__(self):
"""Additive inverse"""
self.w = -self.w
self.x = -self.x
self.y = -self.y
self.z = -self.z
return self
def __abs__(self):
"""Norm"""
return math.sqrt(self.w ** 2 + \
self.x ** 2 + \
self.y ** 2 + \
self.z ** 2)
magnitude = __abs__
def __eq__(self,other):
"""Equal at e-8 precision"""
return (abs(self.w-other.w) < 1e-8 and \
abs(self.x-other.x) < 1e-8 and \
abs(self.y-other.y) < 1e-8 and \
abs(self.z-other.z) < 1e-8) \
or \
(abs(-self.w-other.w) < 1e-8 and \
abs(-self.x-other.x) < 1e-8 and \
abs(-self.y-other.y) < 1e-8 and \
abs(-self.z-other.z) < 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 (self.Rodrigues()>other.Rodrigues()) - (self.Rodrigues()<other.Rodrigues())
def magnitude_squared(self):
return self.w ** 2 + \
self.x ** 2 + \
self.y ** 2 + \
self.z ** 2
def identity(self):
self.w = 1.
self.x = 0.
self.y = 0.
self.z = 0.
return self
def normalize(self):
d = self.magnitude()
if d > 0.0:
self /= d
return self
def conjugate(self):
self.x = -self.x
self.y = -self.y
self.z = -self.z
return self
def inverse(self):
d = self.magnitude()
if d > 0.0:
self.conjugate()
self /= d
return self
def homomorph(self):
if self.w < 0.0:
self.w = -self.w
self.x = -self.x
self.y = -self.y
self.z = -self.z
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 [i for i in self]
def asM(self): # to find Averaging Quaternions (see F. Landis Markley et al.)
return np.outer([i for i in self],[i for i in self])
def asMatrix(self):
qbarhalf = 0.5*(self.w**2 - self.x**2 - self.y**2 - self.z**2)
return 2.0*np.array(
[[ qbarhalf + self.x**2 , self.x*self.y - self.w*self.z, self.x*self.z + self.w*self.y],
[ self.x*self.y + self.w*self.z, qbarhalf + self.y**2 , self.y*self.z - self.w*self.x],
[ self.x*self.z - self.w*self.y, self.y*self.z + self.w*self.x, qbarhalf + self.z**2 ],
])
def asAngleAxis(self,
degrees = False):
if self.w > 1:
self.normalize()
s = math.sqrt(1. - self.w**2)
x = 2*self.w**2 - 1.
y = 2*self.w * 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.x / s, self.y / s, self.z / s]))
def asRodrigues(self):
return np.inf*np.ones(3) if self.w == 0.0 else np.array([self.x, self.y, self.z])/self.w
def asEulers(self,
degrees = False):
"""Orientation as Bunge-Euler angles."""
q03 = self.w**2+self.z**2
q12 = self.x**2+self.y**2
chi = np.sqrt(q03*q12)
if abs(chi) < 1e-10 and abs(q12) < 1e-10:
eulers = np.array([math.atan2(-2*self.w*self.z,self.w**2-self.z**2),0,0])
elif abs(chi) < 1e-10 and abs(q03) < 1e-10:
eulers = np.array([math.atan2( 2*self.x*self.y,self.x**2-self.y**2),np.pi,0])
else:
eulers = np.array([math.atan2((self.x*self.z-self.w*self.y)/chi,(-self.w*self.x-self.y*self.z)/chi),
math.atan2(2*chi,q03-q12),
math.atan2((self.w*self.y+self.x*self.z)/chi,( self.y*self.z-self.w*self.x)/chi),
])
return np.degrees(eulers) if degrees else eulers
# # Static constructors
@classmethod
def fromIdentity(cls):
return cls()
@classmethod
def fromRandom(cls,randomSeed = None):
if randomSeed is None:
randomSeed = int(os.urandom(4).encode('hex'), 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([w,x,y,z])
@classmethod
def fromRodrigues(cls, rodrigues):
if not isinstance(rodrigues, np.ndarray): rodrigues = np.array(rodrigues)
halfangle = math.atan(np.linalg.norm(rodrigues))
c = math.cos(halfangle)
w = c
x,y,z = rodrigues/c
return cls([w,x,y,z])
@classmethod
def fromAngleAxis(cls,
angle,
axis,
degrees = False):
if not isinstance(axis, np.ndarray): axis = np.array(axis,dtype='d')
axis = axis.astype(float)/np.linalg.norm(axis)
angle = np.radians(angle) if degrees else angle
s = math.sin(0.5 * angle)
w = math.cos(0.5 * angle)
x = axis[0] * s
y = axis[1] * s
z = axis[2] * s
return cls([w,x,y,z])
@classmethod
def fromEulers(cls,
eulers,
degrees = False):
if not isinstance(eulers, np.ndarray): eulers = np.array(eulers,dtype='d')
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])
w = c * np.cos(sigma)
x = -s * np.cos(delta)
y = -s * np.sin(delta)
z = -c * np.sin(sigma)
return cls([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)
w = 0.5*math.sqrt(1.+m[0,0]+m[1,1]+m[2,2])
x = 0.5*math.sqrt(1.+m[0,0]-m[1,1]-m[2,2])
y = 0.5*math.sqrt(1.-m[0,0]+m[1,1]-m[2,2])
z = 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( 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.w * q2.w + q1.x * q2.x + q1.y * q2.y + q1.z * q2.z
if costheta < 0.:
costheta = -costheta
q1 = q1.conjugated()
elif costheta > 1:
costheta = 1
theta = math.acos(costheta)
if abs(theta) < 0.01:
Q.w = q2.w
Q.x = q2.x
Q.y = q2.y
Q.z = q2.z
return Q
sintheta = math.sqrt(1.0 - costheta * costheta)
if abs(sintheta) < 0.01:
Q.w = (q1.w + q2.w) * 0.5
Q.x = (q1.x + q2.x) * 0.5
Q.y = (q1.y + q2.y) * 0.5
Q.z = (q1.z + q2.z) * 0.5
return Q
ratio1 = math.sin((1 - t) * theta) / sintheta
ratio2 = math.sin(t * theta) / sintheta
Q.w = q1.w * ratio1 + q2.w * ratio2
Q.x = q1.x * ratio1 + q2.x * ratio2
Q.y = q1.y * ratio1 + q2.y * ratio2
Q.z = q1.z * ratio1 + q2.z * 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 '%s' % (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,'d'),rgb) # limit to maximum intensity
rgb /= max(rgb) # normalize to (HS)V = 1
else:
rgb = np.zeros(3,'d')
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,): # based on given quaternion-like array
self.quaternion = Quaternion(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 'Symmetry: %s\n' % (self.symmetry) + \
'Quaternion: %s\n' % (self.quaternion) + \
'Matrix:\n%s\n' % ( '\n'.join(['\t'.join(map(str,self.asMatrix()[i,:])) for i in range(3)]) ) + \
'Bunge Eulers / deg: %s' % ('\t'.join(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(quatArray = 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)