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DAMASK_EICMD
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DAMASK_EICMD/lib/damask/corientation.pyx

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add Cython version of orientation module to provide faster orientation calculation.
2015-07-30 00:53:06 +05:30
#!/usr/bin/env python
# encoding: utf-8
# filename: corientation.pyx
# __ __ __________ ____ __ ____ ______ ____
# / //_// ____/ __ \/ __ \/ //_/ / / / __ \/ __ \
# / ,< / __/ / / / / / / / ,< / / / / / / / / / /
# / /| |/ /___/ /_/ / /_/ / /| / /_/ / /_/ / /_/ /
# /_/ |_/_____/_____/\____/_/ |_\____/_____/\____/
######################################################
# This is a Cython implementation of original DAMASK #
# orientation class, mainly for speed improvement. #
######################################################
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"""
NOTE
----
The static method in Cython is different from Python, need more
time to figure out details.
"""
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import math, random, os
import numpy as np
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cimport numpy as np
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##
# This Rodrigues class is odd, not sure if it will function
# properly or not
cdef class Rodrigues:
"""Rodrigues representation of orientation """
cdef public double[3] r
def __init__(self, vector):
if isinstance(vector, Rodrigues):
self.r[0] = vector.r[0]
self.r[1] = vector.r[1]
self.r[2] = vector.r[2]
else:
self.r[0] = vector[0]
self.r[1] = vector[1]
self.r[2] = vector[2]
def asQuaternion(self):
cdef double norm, halfAngle
cdef double[4] q
norm = np.linalg.norm(self.vector)
halfAngle = np.arctan(norm)
q[0] = np.cos(halfAngle)
tmp = np.sin(halfAngle)*self.vector/norm
q[1],q[2],q[3] = tmp[0],tmp[1],tmp[2]
return Quaternion(q)
def asAngleAxis(self):
cdef double norm, halfAngle
norm = np.linalg.norm(self.vector)
halfAngle = np.arctan(norm)
return (2.0*halfAngle,self.vector/norm)
##
# The Quaternion class do the heavy lifting of orientation
# calculation
cdef class Quaternion:
""" Quaternion representation of orientation """
# All methods and naming conventions based off
# http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions
cdef public double w,x,y,z
def __init__(self, data):
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"""
@description
------------
copy constructor friendly
@parameters
-----------
data: array
"""
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cdef double[4] q
if isinstance(data, Quaternion):
q[0] = data.w
q[1] = data.x
q[2] = data.y
q[3] = data.z
else:
q[0] = data[0]
q[1] = data[1]
q[2] = data[2]
q[3] = data[3]
self.Quaternion(q)
cdef Quaternion(self, double* quatArray):
"""
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@description
------------
internal constructor for Quaternion
@parameters
-----------
quatArray: double[4] // <w, x, y, z>
w is the real part, (x, y, z) are the imaginary parts
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"""
if quatArray[0] < 0:
self.w = -quatArray[0]
self.x = -quatArray[1]
self.y = -quatArray[2]
self.z = -quatArray[3]
else:
self.w = quatArray[0]
self.x = quatArray[1]
self.y = quatArray[2]
self.z = quatArray[3]
def __copy__(self):
cdef double[4] q = [self.w,self.x,self.y,self.z]
return Quaternion(q)
copy = __copy__
def __iter__(self):
return iter([self.w,self.x,self.y,self.z])
def __repr__(self):
return 'Quaternion(real={:.4f},imag=<{:.4f},{:.4f}, {:.4f}>)'.format(self.w,
self.x,
self.y,
self.z)
def __pow__(self, exponent, modulo):
# declare local var for speed gain
cdef double omega, vRescale
cdef double[4] q
omega = math.acos(self.w)
vRescale = math.sin(exponent*omega)/math.sin(omega)
q[0] = math.cos(exponent*omega)
q[1] = self.x*vRescale
q[2] = self.y*vRescale
q[3] = self.z*vRescale
return Quaternion(q)
def __ipow__(self, exponent):
self = self.__pow__(self, exponent, 1.0)
return self
def __mul__(self, other):
# declare local var for speed gain
cdef double Aw,Ax,Ay,Az,Bw,Bx,By,Bz
cdef double w,x,y,z,Vx,Vy,Vz
cdef double[4] q
# quaternion * quaternion
try:
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[0] = - Ax * Bx - Ay * By - Az * Bz + Aw * Bw
q[1] = + Ax * Bw + Ay * Bz - Az * By + Aw * Bx
q[2] = - Ax * Bz + Ay * Bw + Az * Bx + Aw * By
q[3] = + Ax * By - Ay * Bx + Az * Bw + Aw * Bz
return Quaternion(q)
except:
pass
# vector (perform active rotation, i.e. q*v*q.conjugated)
try:
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
# quaternion * scalar
try:
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):
if isinstance(other, Quaternion):
self = self.__mul__(other)
return self
else:
return NotImplemented
def __div__(self, other):
cdef double[4] q
if isinstance(other, (int,float,long)):
q[0] = self.w / other
q[1] = self.x / other
q[2] = self.y / other
q[3] = self.z / other
return Quaternion(q)
else:
NotImplemented
def __idiv__(self, other):
self = self.__div__(other)
return self
def __add__(self, other):
cdef double[4] q
if isinstance(other, Quaternion):
q[0] = self.w + other.w
q[1] = self.x + other.x
q[2] = self.y + other.y
q[3] = self.z + other.z
return self.__class__(q)
else:
return NotImplemented
def __iadd__(self, other):
self = self.__add__(other)
return self
def __sub__(self, other):
cdef double[4] q
if isinstance(other, Quaternion):
q[0] = self.w - other.w
q[1] = self.x - other.x
q[2] = self.y - other.y
q[3] = self.z - other.z
return self.__class__(q)
else:
return NotImplemented
def __isub__(self, other):
self = self.__sub__(other)
return self
def __neg__(self):
cdef double[4] q
q[0] = -self.w
q[1] = -self.x
q[2] = -self.y
q[3] = -self.z
return self.__class__(q)
def __abs__(self):
cdef double tmp
tmp = self.w**2 + self.x**2 + self.y**2 + self.z**2
tmp = math.sqrt(tmp)
return tmp
magnitude = __abs__
def __richcmp__(Quaternion self, Quaternion other, int op):
cdef bint tmp
tmp = (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)
if op == 2: #__eq__
return tmp
elif op ==3: #__ne__
return not tmp
else:
return NotImplemented
def __cmp__(self,other):
# not sure if this actually works or not
return cmp(self.Rodrigues(),other.Rodrigues())
def magnitude_squared(self):
cdef double tmp
tmp = self.w**2 + self.x**2 + self.y**2 + self.z**2
return tmp
def identity(self):
self.w = 1.0
self.x = 0.0
self.y = 0.0
self.z = 0.0
return self
def rotateBy_angleaxis(self, angle, axis):
self *= Quaternion.fromAngleAxis(angle, axis)
return self
def rotateBy_Eulers(self, eulers):
self *= Quaternion.fromEulers(eulers, type)
return self
def rotateBy_matrix(self, m):
self *= Quaternion.fromMatrix(m)
return self
def normalize(self):
cdef double d
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):
cdef double d
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
# return a copy of me
def normalized(self):
cdef Quaternion q
q = Quaternion(self.normalize())
return q
def conjugated(self):
cdef Quaternion q
q = Quaternion(self.conjugate())
return q
def asList(self):
cdef double[4] q = [self.w, self.x, self.y, self.z]
return list(q)
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):
return np.array([[1.0-2.0*(self.y*self.y+self.z*self.z), 2.0*(self.x*self.y-self.z*self.w), 2.0*(self.x*self.z+self.y*self.w)],
[ 2.0*(self.x*self.y+self.z*self.w), 1.0-2.0*(self.x*self.x+self.z*self.z), 2.0*(self.y*self.z-self.x*self.w)],
[ 2.0*(self.x*self.z-self.y*self.w), 2.0*(self.x*self.w+self.y*self.z), 1.0-2.0*(self.x*self.x+self.y*self.y)]])
def asAngleAxis(self):
cdef double s,x,y
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)
return angle, np.array([1.0, 0.0, 0.0] if angle < 1e-3 else [self.x/s, self.y/s, self.z/s])
def asRodrigues(self):
if self.w != 0.0:
return np.array([self.x, self.y, self.z])/self.w
else:
return np.array([float('inf')]*3)
def asEulers(self,type='bunge',degrees=False):
"""conversion taken from:
Melcher, A.; Unser, A.; Reichhardt, M.; Nestler, B.; Pötschke, M.; Selzer, M.
Conversion of EBSD data by a quaternion based algorithm to be used for grain structure simulations
Technische Mechanik 30 (2010) pp 401--413
"""
cdef double x,y
angles = [0.0,0.0,0.0]
if type.lower() == 'bunge' or type.lower() == 'zxz':
if abs(self.x - self.y) < 1e-8:
x = self.w**2 - self.z**2
y = 2.*self.w*self.z
angles[0] = math.atan2(y,x)
elif abs(self.w - self.z) < 1e-8:
x = self.x**2 - self.y**2
y = 2.*self.x*self.y
angles[0] = math.atan2(y,x)
angles[1] = math.pi
else:
chi = math.sqrt((self.w**2 + self.z**2)*(self.x**2 + self.y**2))
x = (self.w * self.x - self.y * self.z)/2./chi
y = (self.w * self.y + self.x * self.z)/2./chi
angles[0] = math.atan2(y,x)
x = self.w**2 + self.z**2 - (self.x**2 + self.y**2)
y = 2.*chi
angles[1] = math.atan2(y,x)
x = (self.w * self.x + self.y * self.z)/2./chi
y = (self.z * self.x - self.y * self.w)/2./chi
angles[2] = math.atan2(y,x)
return np.degrees(angles) if degrees else angles
@staticmethod
def fromIdentity():
cdef double[4] q = [1.0, 0.0, 0.0, 0.0]
return Quaternion(q)
@staticmethod
def fromRandom(randomSeed=None):
cdef double r1,r2,r3
cdef double[4] q
if randomSeed == None:
randomSeed = int(os.urandom(4).encode('hex'), 16)
random.seed(randomSeed)
r1 = random.random()
r2 = random.random()
r3 = random.random()
q[0] = math.cos(2.0*math.pi*r1)*math.sqrt(r3)
q[1] = math.sin(2.0*math.pi*r2)*math.sqrt(1.0-r3)
q[2] = math.cos(2.0*math.pi*r2)*math.sqrt(1.0-r3)
q[3] = math.sin(2.0*math.pi*r1)*math.sqrt(r3)
return Quaternion(q)
@staticmethod
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def fromRodrigues(rodrigues):
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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 = c*rodrigues
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return Quaternion([w,x,y,z])
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@staticmethod
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def fromAngleAxis(angle, axis):
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if not isinstance(axis, np.ndarray): axis = np.array(axis)
axis /= np.linalg.norm(axis)
s = math.sin(angle / 2.0)
w = math.cos(angle / 2.0)
x = axis[0] * s
y = axis[1] * s
z = axis[2] * s
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return Quaternion([w,x,y,z])
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@staticmethod
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def fromEulers(eulers, type = 'Bunge'):
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cdef double c1,s1,c2,s2,c3,s3
cdef double[4] q
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cdef double[3] halfEulers
cdef int i
for i in range(3):
halfEulers[i] = eulers[i] * 0.5 # reduce to half angles
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c1 = math.cos(halfEulers[0])
s1 = math.sin(halfEulers[0])
c2 = math.cos(halfEulers[1])
s2 = math.sin(halfEulers[1])
c3 = math.cos(halfEulers[2])
s3 = math.sin(halfEulers[2])
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if type.lower() == 'bunge' or type.lower() == 'zxz':
q[0] = c1 * c2 * c3 - s1 * c2 * s3
q[1] = c1 * s2 * c3 + s1 * s2 * s3
q[2] = - c1 * s2 * s3 + s1 * s2 * c3
q[3] = c1 * c2 * s3 + s1 * c2 * c3
else:
q[0] = c1 * c2 * c3 - s1 * s2 * s3
q[1] = s1 * s2 * c3 + c1 * c2 * s3
q[2] = s1 * c2 * c3 + c1 * s2 * s3
q[3] = c1 * s2 * c3 - s1 * c2 * s3
return Quaternion(q)
## Modified Method to calculate Quaternion from Orientation Matrix, Source: http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/
@staticmethod
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def fromMatrix(m):
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# This is a slow implementation
if m.shape != (3,3) and np.prod(m.shape) == 9:
m = m.reshape(3,3)
tr=m[0,0]+m[1,1]+m[2,2]
if tr > 0.00000001:
s = math.sqrt(tr + 1.0)*2.0
return Quaternion(
[ s*0.25,
(m[2,1] - m[1,2])/s,
(m[0,2] - m[2,0])/s,
(m[1,0] - m[0,1])/s
])
elif m[0,0] > m[1,1] and m[0,0] > m[2,2]:
t = m[0,0] - m[1,1] - m[2,2] + 1.0
s = 2.0*math.sqrt(t)
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return Quaternion(
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[ (m[2,1] - m[1,2])/s,
s*0.25,
(m[0,1] + m[1,0])/s,
(m[2,0] + m[0,2])/s,
])
elif m[1,1] > m[2,2]:
t = -m[0,0] + m[1,1] - m[2,2] + 1.0
s = 2.0*math.sqrt(t)
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return Quaternion(
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[ (m[0,2] - m[2,0])/s,
(m[0,1] + m[1,0])/s,
s*0.25,
(m[1,2] + m[2,1])/s,
])
else:
t = -m[0,0] - m[1,1] + m[2,2] + 1.0
s = 2.0*math.sqrt(t)
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return Quaternion(
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[ (m[1,0] - m[0,1])/s,
(m[2,0] + m[0,2])/s,
(m[1,2] + m[2,1])/s,
s*0.25,
])
@staticmethod
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def new_interpolate(q1, q2, t):
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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 = Quaternion.fromIdentity()
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
##
# Define lattice_type to make it easier for future
# development
cdef enum lattice_type:
NONE = 0
ORTHORHOMBIC= 1
TETRAGONAL = 2
HEXAGONAL = 3
CUBIC = 4
##
# Symmetry class
cdef class Symmetry:
cdef public lattice_type lattice
# cdef enum LATTICES:
# NONE = 0
# ORTHORHOMBIC= 1
# TETRAGONAL = 2
# HEXAGONAL = 3
# CUBIC = 4
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def __init__(self, symmetry):
if symmetry == 0 or symmetry == None:
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self.lattice = NONE
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elif symmetry == 1 or symmetry == 'orthorhombic':
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self.lattice = ORTHORHOMBIC
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elif symmetry == 2 or symmetry == 'tetragonal':
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self.lattice = TETRAGONAL
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elif symmetry == 3 or symmetry == 'hexagonal':
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self.lattice = HEXAGONAL
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elif symmetry == 4 or symmetry == 'cubic':
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self.lattice = CUBIC
else:
self.lattice = NONE
def __copy__(self):
return self.__class__(self.lattice)
copy = __copy__
def __repr__(self):
return '{}'.format(self.lattice)
def __richcmp__(Symmetry self, Symmetry other, int op):
cdef bint tmp
tmp = self.lattice == other.lattice
if op == 2: #__eq__
return tmp
elif op ==3: #__ne__
return not tmp
else:
return NotImplemented
def __cmp__(self,other):
return cmp(self.lattice,other.lattice)
def equivalentQuaternions(self,quaternion):
'''
List of symmetrically equivalent quaternions based on own symmetry.
'''
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 ],
[ 0.0,1.0,0.0,0.0 ],
[ 0.0,0.0,1.0,0.0 ],
[ 0.0,0.0,0.0,1.0 ],
[-0.5*math.sqrt(3), 0.0, 0.0, 0.5 ],
[-0.5*math.sqrt(3), 0.0, 0.0,-0.5 ],
[ 0.0, 0.5*math.sqrt(3), 0.5, 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,-0.5,-0.5*math.sqrt(3), 0.0 ],
[ 0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
[-0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
]
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 ],
]
# due to the use of list comprehension, the speed grain is quite
# limited
return [quaternion*Quaternion(q) for q in symQuats]
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
R = abs(R) # fundamental zone in Rodrigues space is point symmetric around origin
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 accidentally passed quaternion
cdef double epsilon = 0.0
if self.lattice == CUBIC:
return R[0] >= R[1]+epsilon and R[1] >= R[2]+epsilon and R[2] >= epsilon and self.inFZ(R)
elif self.lattice == HEXAGONAL:
return R[0] >= math.sqrt(3)*(R[1]+epsilon) and R[1] >= epsilon and R[2] >= epsilon and self.inFZ(R)
elif self.lattice == TETRAGONAL:
return R[0] >= R[1]+epsilon and R[1] >= epsilon and R[2] >= epsilon and self.inFZ(R)
elif self.lattice == ORTHORHOMBIC:
return R[0] >= epsilon and R[1] >= epsilon and R[2] >= epsilon and self.inFZ(R)
else:
return True
def inSST(self,vector,color = False):
'''
Check whether given vector falls into standard stereographic triangle of own symmetry.
Return inverse pole figure color if requested.
'''
# basis = {4 : 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
# 3 : 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
# 2 : 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
# 1 : 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 = np.array([ [-1. , 0. , 1. ],
[ np.sqrt(2.), -np.sqrt(2.), 0. ],
[ 0. , np.sqrt(3.), 0. ] ])
elif self.lattice == HEXAGONAL:
basis = np.array([ [ 0. , 0. , 1. ],
[ 1. , -np.sqrt(3.), 0. ],
[ 0. , 2. , 0. ] ])
elif self.lattice == TETRAGONAL:
basis = np.array([ [ 0. , 0. , 1. ],
[ 1. , -1. , 0. ],
[ 0. , np.sqrt(2.), 0. ] ])
elif self.lattice == ORTHORHOMBIC:
basis = np.array([ [ 0., 0., 1.],
[ 1., 0., 0.],
[ 0., 1., 0.] ])
else:
basis = np.zeros((3,3),dtype=float)
if np.all(basis == 0.0):
theComponents = -np.ones(3,'d')
else:
theComponents = np.dot(basis,np.array([vector[0],vector[1],abs(vector[2])]))
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 http://pyeuclid.googlecode.com/svn/trunk/euclid.py
# suggested reading: http://web.mit.edu/2.998/www/QuaternionReport1.pdf
##
# Orientation class is a composite class of Symmetry and Quaternion
cdef class Orientation:
cdef public Quaternion quaternion
cdef public Symmetry symmetry
def __init__(self,
quaternion = Quaternion.fromIdentity(),
Rodrigues = None,
angleAxis = None,
matrix = None,
Eulers = None,
fix syntax error regarding several static method
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random = False, # put any integer to have a fixed seed or True for real random
symmetry = None
add Cython version of orientation module to provide faster orientation calculation.
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):
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
fix syntax error regarding several static method
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self.quaternion = Quaternion.fromEulers(Eulers, type='bunge')
add Cython version of orientation module to provide faster orientation calculation.
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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])
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.homomorph()
elif isinstance(quaternion, np.ndarray) and quaternion.shape == (4,): # based on given quaternion
self.quaternion = Quaternion(quaternion).homomorph()
self.symmetry = Symmetry(symmetry)
def __copy__(self):
return self.__class__(quaternion=self.quaternion,symmetry=self.symmetry.lattice)
copy = __copy__
def __repr__(self):
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(lambda x:str(np.degrees(x)),self.asEulers('bunge'))) )
def asQuaternion(self):
return self.quaternion.asList()
def asEulers(self,type='bunge'):
return self.quaternion.asEulers(type)
def asRodrigues(self):
return self.quaternion.asRodrigues()
def asAngleAxis(self):
return self.quaternion.asAngleAxis()
def asMatrix(self):
return self.quaternion.asMatrix()
def inFZ(self):
return self.symmetry.inFZ(self.quaternion.asRodrigues())
def equivalentQuaternions(self):
return self.symmetry.equivalentQuaternions(self.quaternion)
def equivalentOrientations(self):
return map(lambda q: Orientation(quaternion=q,symmetry=self.symmetry.lattice),
self.equivalentQuaternions())
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):
'''
Disorientation between myself and given other orientation
(either reduced according to my own symmetry or given one)
'''
lowerSymmetry = min(self.symmetry,other.symmetry)
breaker = False
for me in self.symmetry.equivalentQuaternions(self.quaternion):
me.conjugate()
for they in other.symmetry.equivalentQuaternions(other.quaternion):
theQ = they * me
breaker = lowerSymmetry.inDisorientationSST(theQ.asRodrigues()) #\
# or lowerSymmetry.inDisorientationSST(theQ.conjugated().asRodrigues())
if breaker: break
if breaker: break
return Orientation(quaternion=theQ,symmetry=self.symmetry.lattice) #, me.conjugated(), they
def inversePole(self,axis,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.conjugated()*axis # align crystal direction to axis
if self.symmetry.inSST(pole): break # found SST version
else:
pole = self.quaternion.conjugated()*axis # align crystal direction to axis
return pole
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.conjugated()*axis # align crystal direction to axis
inSST,color = self.symmetry.inSST(pole,color=True)
if inSST: break
return color
@staticmethod
fix syntax error regarding several static method
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def getAverageOrientation(orientationList):
add Cython version of orientation module to provide faster orientation calculation.
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"""
RETURN THE 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
sample usage:
a = Orientation(Eulers=np.radians([10, 10, 0]), symmetry=3)
b = Orientation(Eulers=np.radians([20, 0, 0]), symmetry=3)
avg = Orientation.getAverageOrientation([a,b])
fix syntax error regarding several static method
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NOTE
----
No symmetry information is available for the average orientation.
add Cython version of orientation module to provide faster orientation calculation.
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"""
if not all(isinstance(item, Orientation) for item in orientationList):
raise TypeError("Only instances of Orientation can be averaged.")
N = len(orientationList)
M = orientationList.pop(0).quaternion.asM()
for o in orientationList:
M += o.quaternion.asM()
eig, vec = np.linalg.eig(M/N)
fix syntax error regarding several static method
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return Orientation(quaternion = Quaternion(vec.T[eig.argmax()]))
add Cython version of orientation module to provide faster orientation calculation.
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def related(self, relationModel, direction, targetSymmetry = None):
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
# 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([myPlane,myNormal,np.cross(myPlane,myNormal)])
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([otherPlane,otherNormal,np.cross(otherPlane,otherNormal)])
rot=np.dot(otherMatrix.T,myMatrix)
return Orientation(matrix=np.dot(rot,self.asMatrix())) # no symmetry information ??