# -*- 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: # ****************************************************************************************** # 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 # Representation of rotation is in ACTIVE form! # (derived directly or through angleAxis, Euler angles, or active matrix) # vector "a" (defined in coordinate system "A") is actively rotated to new coordinates "b" # b = Q * a # b = np.dot(Q.asMatrix(),a) def __init__(self, quatArray = [1.0,0.0,0.0,0.0]): self.w, \ self.x, \ self.y, \ self.z = quatArray 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=%+.6f, imag=<%+.6f, %+.6f, %+.6f>)' % \ (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.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): 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): 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): 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 self.__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 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, 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, type = 'bunge', degrees = False, standardRange = False): ''' conversion of ACTIVE rotation to Euler angles 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) < 1e-4 and abs(self.y) < 1e-4: x = self.w**2 - self.z**2 y = 2.*self.w*self.z angles[0] = math.atan2(y,x) elif abs(self.w) < 1e-4 and abs(self.z) < 1e-4: 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 standardRange: angles[0] %= 2*math.pi if angles[1] < 0.0: angles[1] += math.pi angles[2] *= -1.0 angles[2] %= 2*math.pi return np.degrees(angles) if degrees else angles # # Static constructors @classmethod def fromIdentity(cls): return cls() @classmethod def fromRandom(cls,randomSeed = None): if randomSeed == 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 = c*rodrigues return cls([w,x,y,z]) @classmethod def fromAngleAxis(cls, angle, axis): if not isinstance(axis, np.ndarray): axis = np.array(axis,dtype='d') axis = axis.astype(float)/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'): eulers *= 0.5 # reduce to half angles c1 = math.cos(eulers[0]) s1 = math.sin(eulers[0]) c2 = math.cos(eulers[1]) s2 = math.sin(eulers[1]) c3 = math.cos(eulers[2]) s3 = math.sin(eulers[2]) 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 = np.trace(m) if tr > 1e-8: 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 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 map(Quaternion, np.array(symQuats)[np.atleast_1d(np.array(who)) if who != [] else xrange(len(symQuats))]) def equivalentQuaternions(self, quaternion, who = []): ''' List of symmetrically equivalent quaternions based on own symmetry. ''' return [quaternion*q 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 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 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: basis = {'improper':np.zeros((3,3),dtype=float), 'proper':np.zeros((3,3),dtype=float), } if np.all(basis == 0.0): theComponents = -np.ones(3,'d') inSST = np.all(theComponents >= 0.0) else: 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 for positive and negative values theComponents = np.dot(basis['improper'],v) 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, # put any integer to have a fixed seed or True for real random symmetry = None, ): 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,'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(str,self.asEulers('bunge',degrees=True))) ) def asQuaternion(self): return self.quaternion.asList() def asEulers(self, type = 'bunge', degrees = False, standardRange = False): return self.quaternion.asEulers(type, degrees, standardRange) 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 map(lambda q: Orientation(quaternion = q, symmetry = self.symmetry.lattice), 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 = self.quaternion.conjugated()*other.quaternion 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 = sA.conjugated()*misQ*sB for k in xrange(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 return (Orientation(quaternion = theQ,symmetry = self.symmetry.lattice), i,j,k == 1) # disorientation, own sym, other sym, self-->other: True, self<--other: False 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.conjugated()*axis # align crystal direction to axis if self.symmetry.inSST(pole,proper): break # found SST version else: pole = self.quaternion.conjugated()*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.conjugated()*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 = []): """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.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 # add (multiples) of this orientation to average 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 = 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 DOES THIS PAPER EXISTS? # 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 ??