# -*- coding: UTF-8 no BOM -*-
###################################################
# NOTE: everything here needs to be a np array #
###################################################
import math,random
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:
# ******************************************************************************************
# All methods and naming conventions based off
# http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions
# w is the real part, (x, y, z) are the imaginary parts
def __init__(self, quatArray=[1.0,0.0,0.0,0.0]):
self.w, \
self.x, \
self.y, \
self.z = quatArray
self = self.homomorph()
def __iter__(self):
return iter([self.w,self.x,self.y,self.z])
def __copy__(self):
Q = Quaternion([self.w,self.x,self.y,self.z])
return Q
copy = __copy__
def __repr__(self):
return 'Quaternion(real=%+.4f, imag=<%+.4f, %+.4f, %+.4f>)' % \
(self.w, self.x, self.y, self.z)
def __pow__(self, exponent):
omega = math.acos(self.w)
vRescale = math.sin(exponent*omega)/math.sin(omega)
Q = Quaternion()
Q.x = self.x * vRescale
Q.y = self.y * vRescale
Q.z = self.z * vRescale
Q.w = math.cos(exponent*omega)
return Q
def __ipow__(self, exponent):
omega = math.acos(self.w)
vRescale = math.sin(exponent*omega)/math.sin(omega)
self.x *= vRescale
self.y *= vRescale
self.z *= vRescale
self.w = np.cos(exponent*omega)
return self
def __mul__(self, other):
try: # quaternion
Ax = self.x
Ay = self.y
Az = self.z
Aw = self.w
Bx = other.x
By = other.y
Bz = other.z
Bw = other.w
Q = Quaternion()
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
Q.w = - Ax * Bx - Ay * By - Az * Bz + Aw * Bw
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):
try: # Quaternion
Ax = self.x
Ay = self.y
Az = self.z
Aw = self.w
Bx = other.x
By = other.y
Bz = other.z
Bw = other.w
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
self.w = -Ax * Bx - Ay * By - Az * Bz + Aw * Bw
except: pass
return self
def __div__(self, other):
if isinstance(other, (int,float,long)):
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):
if isinstance(other, (int,float,long)):
self.w /= other
self.x /= other
self.y /= other
self.z /= other
return self
def __add__(self, other):
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):
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):
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):
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):
self.w = -self.w
self.x = -self.x
self.y = -self.y
self.z = -self.z
return self
def __abs__(self):
return math.sqrt(self.w ** 2 + \
self.x ** 2 + \
self.y ** 2 + \
self.z ** 2)
magnitude = __abs__
def __eq__(self,other):
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):
return not __eq__(self,other)
def __cmp__(self,other):
return cmp(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 rotateBy_angleaxis(self, angle, axis):
self *= Quaternion.fromAngleAxis(angle, axis)
return self
def rotateBy_Eulers(self, heading, attitude, bank):
self *= Quaternion.fromEulers(eulers, type)
return self
def rotateBy_matrix(self, m):
self *= Quaternion.fromMatrix(m)
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):
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):
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 < 1e-3:
return angle, np.array([1.0, 0.0, 0.0])
else:
return angle, np.array([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'):
'''
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
'''
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)
# if angles[0] < 0.0:
# angles[0] += 2*math.pi
# if angles[1] < 0.0:
# angles[1] += math.pi
# angles[2] *= -1
# if angles[2] < 0.0:
# angles[2] += 2*math.pi
return angles
# # Static constructors
@classmethod
def fromIdentity(cls):
return cls()
@classmethod
def fromRandom(cls):
r1 = random.random()
r2 = random.random()
r3 = random.random()
w = math.cos(2.0*math.pi*r1)*math.sqrt(r3)
x = math.sin(2.0*math.pi*r2)*math.sqrt(1.0-r3)
y = math.cos(2.0*math.pi*r2)*math.sqrt(1.0-r3)
z = math.sin(2.0*math.pi*r1)*math.sqrt(r3)
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 = c*rodrigues
return cls([w,x,y,z])
@classmethod
def fromAngleAxis(cls, angle, axis):
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
return cls([w,x,y,z])
@classmethod
def fromEulers(cls, eulers, type = 'Bunge'):
c1 = math.cos(eulers[0] / 2.0)
s1 = math.sin(eulers[0] / 2.0)
c2 = math.cos(eulers[1] / 2.0)
s2 = math.sin(eulers[1] / 2.0)
c3 = math.cos(eulers[2] / 2.0)
s3 = math.sin(eulers[2] / 2.0)
if type.lower() == 'bunge' or type.lower() == 'zxz':
w = c1 * c2 * c3 - s1 * c2 * s3
x = c1 * s2 * c3 + s1 * s2 * s3
y = - c1 * s2 * s3 + s1 * s2 * c3
z = c1 * c2 * s3 + s1 * c2 * c3
else:
# print 'unknown Euler convention'
w = c1 * c2 * c3 - s1 * s2 * s3
x = s1 * s2 * c3 + c1 * c2 * s3
y = s1 * c2 * c3 + c1 * s2 * s3
z = c1 * s2 * c3 - s1 * c2 * s3
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)
tr=m[0,0]+m[1,1]+m[2,2]
if tr > 0.00000001:
s = math.sqrt(tr + 1.0)*2.0
return cls(
[ 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)
return cls(
[ (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)
return cls(
[ (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)
return cls(
[ (m[1,0] - m[0,1])/s,
(m[2,0] + m[0,2])/s,
(m[1,2] + m[2,1])/s,
s*0.25,
])
@classmethod
def new_interpolate(cls, q1, q2, t):
# 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):
if isinstance(symmetry, basestring) and symmetry.lower() in Symmetry.lattices:
self.lattice = symmetry.lower()
else:
self.lattice = None
def __copy__(self):
return self.__class__(self.lattice)
copy = __copy__
def __repr__(self):
return '%s' % (self.lattice)
def __eq__(self, other):
return self.lattice == other.lattice
def __neq__(self, other):
return not self.__eq__(other)
def __cmp__(self,other):
return cmp(Symmetry.lattices.index(self.lattice),Symmetry.lattices.index(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 ],
]
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 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 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 = {'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 = 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
# ******************************************************************************************
class Orientation:
# ******************************************************************************************
__slots__ = ['quaternion','symmetry']
def __init__(self,
quaternion = Quaternion.fromIdentity(),
Rodrigues = None,
angleAxis = None,
matrix = None,
Eulers = None,
random = False,
symmetry = None,
):
if random: # produce random orientation
self.quaternion = Quaternion.fromRandom()
elif isinstance(Eulers, np.ndarray) and Eulers.shape == (3,): # based on given Euler angles
self.quaternion = Quaternion.fromEulers(Eulers,'bunge')
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.homomorphed()
elif isinstance(quaternion, np.ndarray) and quaternion.shape == (4,): # based on given quaternion
self.quaternion = Quaternion(quaternion).homomorphed()
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 asMatrix(self):
return self.quaternion.asMatrix()
def reduced(self):
'''
Transform orientation to fall into fundamental zone according to own (or given) 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 = me * they
theQ = they * me
# if theQ.x < 0.0 or theQ.y < 0.0 or theQ.z < 0.0: theQ.conjugate() # speed up scanning since minimum angle is usually found for positive x,y,z
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)
'''
for i,q in enumerate(self.symmetry.equivalentQuaternions(self.quaternion)): # test all symmetric equivalent orientations
if SST: # pole requested to be within SST
pole = q.conjugated()*axis # align crystal direction to axis
if self.symmetry.inSST(pole): break
else:
pole = q.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 i,q in enumerate(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
@classmethod
def getAverageOrientation(cls, orientationList):
"""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
usage:
a = Orientation(Eulers=np.radians([10, 10, 0]), symmetry='hexagonal')
b = Orientation(Eulers=np.radians([20, 0, 0]), symmetry='hexagonal')
avg = Orientation.getAverageOrientation([a,b])"""
if not all(isinstance(item, Orientation) for item in orientationList):
raise TypeError("Only instances of Orientation can be averaged.")
n = len(orientationList)
tmp_m = orientationList.pop(0).quaternion.asM()
for tmp_o in orientationList:
tmp_m += tmp_o.quaternion.asM()
eig, vec = np.linalg.eig(tmp_m/n)
return Orientation( quaternion=Quaternion(quatArray=vec.T[eig.argmax()]) )
def related(self, relationModel, direction, targetSymmetry=None):
if relationModel not in ['KS','GT',"GT'",'NW','Bain']: return None
variant = int(abs(direction))
me = 0 if direction > 0 else 1
other = 1 if direction > 0 else 0
variant = variant -1
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, 1, 0]],\
[[ 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],[ -1, 0, 1]],\
[[ 1, -1, 1],[ 1, -1, 0]],\
[[ 1, -1, 1],[ 1, 0, 1]],\
[[ 1, 1, 1],[ 0, 1, 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, 1, 0]],\
[[ 1, -1, 1],[ 0, -1, 1]],\
[[ 1, -1, 1],[ 1, -1, 0]],\
[[ 1, 1, 1],[ 1, 0, 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, 1],[ 0, 1, 1]],\
[[ 1, -1, 1],[ 1, 0, 1]],\
[[ 1, -1, 1],[ 0, -1, 1]]]),
"GT'": \
np.array([[[ 17, 7, 17],[ 17, 12, 5]],\
[[-17, 7,-17],[-17, 12, -5]],\
[[-17, -7, 17],[-17,-12, 5]],\
[[ 17, -7,-17],[ 17,-12, -5]],\
[[ 17, 17, 7],[ 17, 5, 12]],\
[[-17,-17, 7],[-17, -5, 12]],\
[[ 17,-17, -7],[ 17, -5,-12]],\
[[-17, 17, -7],[-17, 5,-12]],\
[[ 17, 17, 7],[ 5, 17, 12]],\
[[-17,-17, 7],[ -5,-17, 12]],\
[[ 17,-17, -7],[ 5,-17,-12]],\
[[-17, 17, -7],[ -5, 17,-12]],\
[[ 7, 17, 17],[ 12, 17, 5]],\
[[ 7,-17,-17],[ 12,-17, -5]],\
[[ -7,-17, 17],[-12,-17, 5]],\
[[ -7, 17,-17],[-12, 17, -5]],\
[[ 7, 17, 17],[ 12, 5, 17]],\
[[ 7,-17,-17],[ 12, -5,-17]],\
[[ -7,-17, 17],[-12, -5, 17]],\
[[ -7, 17,-17],[-12, 5,-17]],\
[[ 17, 7, 17],[ 5, 12, 17]],\
[[-17, 7,-17],[ -5, 12,-17]],\
[[-17, -7, 17],[ -5,-12, 17]],\
[[ 17, -7,-17],[ 5,-12,-17]]]),
'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]]]),
'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]],\
[[ 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]],\
[[ 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]],\
[[ 0, 1, -1],[ -1, -1, 1]],\
[[ 0, 1, -1],[ -1, 1, -1]],\
[[ 1, -1, 0],[ -1, -1, 1]],\
[[ 1, -1, 0],[ -1, 1, -1]]]),
'GT': \
np.array([[[ 17, -5,-12],[ 17,-17, -7]],\
[[ 17,-12, -5],[ 17, -7,-17]],\
[[-17, 5,-12],[-17, 17, -7]],\
[[-17, 12, -5],[-17, 7,-17]],\
[[ 17, 5, 12],[ 17, 17, 7]],\
[[ 17, 12, 5],[ 17, 7, 17]],\
[[-17, -5, 12],[-17,-17, 7]],\
[[-17,-12, 5],[-17, -7, 17]],\
[[-12, 17, -5],[ -7, 17,-17]],\
[[ -5, 17,-12],[-17, 17, -7]],\
[[ 12,-17, -5],[ 7,-17,-17]],\
[[ 5,-17,-12],[ 17,-17, -7]],\
[[-12,-17, 5],[ -7,-17, 17]],\
[[ -5,-17, 12],[-17,-17, 7]],\
[[ 12, 17, 5],[ 7, 17, 17]],\
[[ 5, 17, 12],[ 17, 17, 7]],\
[[ -5,-12, 17],[-17, -7, 17]],\
[[-12, -5, 17],[ -7,-17, 17]],\
[[ 5, 12, 17],[ 17, 7, 17]],\
[[ 12, 5, 17],[ 7, 17, 17]],\
[[ -5, 12,-17],[-17, 7,-17]],\
[[-12, 5,-17],[ -7, 17,-17]],\
[[ 5,-12,-17],[ 17, -7,-17]],\
[[ 12, -5,-17],[ 7,-17,-17]]]),
"GT'": \
np.array([[[ -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, 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]],\
[[ 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]],\
[[ 1, 0, -1],[ 1, 1, -1]],\
[[ 1, 0, -1],[ 1, -1, -1]],\
[[ -1, 0, -1],[ -1, -1, -1]],\
[[ -1, 0, -1],[ -1, 1, -1]]]),
'NW': \
np.array([[[ 1, -1, 0],[ 1, 0, 0]],\
[[ 1, 0, -1],[ 1, 0, 0]],\
[[ 0, -1, 1],[ 1, 0, 0]],\
[[ 1, 1, 0],[ 1, 0, 0]],\
[[ 0, 1, -1],[ 1, 0, 0]],\
[[ 1, 0, 1],[ 1, 0, 0]],\
[[ 1, 1, 0],[ 1, 0, 0]],\
[[ 0, 1, 1],[ 1, 0, 0]],\
[[ -1, 0, 1],[ 1, 0, 0]],\
[[ 1, 0, 1],[ 1, 0, 0]],\
[[ 1, -1, 0],[ 1, 0, 0]],\
[[ 0, 1, 1],[ 1, 0, 0]]]),
'Bain': \
np.array([[[ 0, 1, 0],[ 0, 1, 1]],
[[ 0, 0, 1],[ 1, 0, 1]],
[[ 1, 0, 0],[ 1, 1, 0]]]),
}
myPlane = planes [relationModel][variant,me]
myNormal = normals[relationModel][variant,me]
myPlane = myPlane/np.linalg.norm(myPlane)
myNormal = myNormal/np.linalg.norm(myNormal)
myMatrix = np.array([myPlane,myNormal,np.cross(myNormal,myPlane)])
otherPlane = planes [relationModel][variant,other]
otherNormal = normals[relationModel][variant,other]
otherPlane = otherPlane/np.linalg.norm(otherPlane)
otherNormal = otherNormal/np.linalg.norm(otherNormal)
otherMatrix = np.array([otherPlane,otherNormal,np.cross(otherNormal,otherPlane)])
myMatrix = np.dot(self.asMatrix(),myMatrix)
return Orientation(matrix=np.dot(otherMatrix.T,myMatrix))