# -*- coding: UTF-8 no BOM -*- import math import numpy as np from . import Lambert P = -1 #################################################################################################### class Quaternion: u""" Quaternion with basic operations q is the real part, p = (x, y, z) are the imaginary parts. Defintion of multiplication depends on variable P, P ∉ {-1,1}. """ def __init__(self, q = 0.0, p = np.zeros(3,dtype=float)): """Initializes to identity unless specified""" self.q = q self.p = np.array(p) def __copy__(self): """Copy""" return self.__class__(q=self.q, p=self.p.copy()) copy = __copy__ def __iter__(self): """Components""" return iter(self.asList()) def asArray(self): """As numpy array""" return np.array((self.q,self.p[0],self.p[1],self.p[2])) def asList(self): return [self.q]+list(self.p) def __repr__(self): """Readable string""" return 'Quaternion: (real={q:+.6f}, imag=<{p[0]:+.6f}, {p[1]:+.6f}, {p[2]:+.6f}>)'.format(q=self.q,p=self.p) def __add__(self, other): """Addition""" if isinstance(other, Quaternion): return self.__class__(q=self.q + other.q, p=self.p + other.p) else: return NotImplemented def __iadd__(self, other): """In-place addition""" if isinstance(other, Quaternion): self.q += other.q self.p += other.p return self else: return NotImplemented def __pos__(self): """Unary positive operator""" return self def __sub__(self, other): """Subtraction""" if isinstance(other, Quaternion): return self.__class__(q=self.q - other.q, p=self.p - other.p) else: return NotImplemented def __isub__(self, other): """In-place subtraction""" if isinstance(other, Quaternion): self.q -= other.q self.p -= other.p return self else: return NotImplemented def __neg__(self): """Unary positive operator""" self.q *= -1.0 self.p *= -1.0 return self def __mul__(self, other): """Multiplication with quaternion or scalar""" if isinstance(other, Quaternion): return self.__class__(q=self.q*other.q - np.dot(self.p,other.p), p=self.q*other.p + other.q*self.p + P * np.cross(self.p,other.p)) elif isinstance(other, (int, float)): return self.__class__(q=self.q*other, p=self.p*other) else: return NotImplemented def __imul__(self, other): """In-place multiplication with quaternion or scalar""" if isinstance(other, Quaternion): self.q = self.q*other.q - np.dot(self.p,other.p) self.p = self.q*other.p + other.q*self.p + P * np.cross(self.p,other.p) return self elif isinstance(other, (int, float)): self *= other return self else: return NotImplemented def __truediv__(self, other): """Divsion with quaternion or scalar""" if isinstance(other, Quaternion): s = other.conjugate()/abs(other)**2. return self.__class__(q=self.q * s, p=self.p * s) elif isinstance(other, (int, float)): self.q /= other self.p /= other return self else: return NotImplemented def __itruediv__(self, other): """In-place divsion with quaternion or scalar""" if isinstance(other, Quaternion): s = other.conjugate()/abs(other)**2. self *= s return self elif isinstance(other, (int, float)): self.q /= other return self else: return NotImplemented def __pow__(self, exponent): """Power""" if isinstance(exponent, (int, float)): omega = np.acos(self.q) return self.__class__(q= np.cos(exponent*omega), p=self.p * np.sin(exponent*omega)/np.sin(omega)) else: return NotImplemented def __ipow__(self, exponent): """In-place power""" if isinstance(exponent, (int, float)): omega = np.acos(self.q) self.q = np.cos(exponent*omega) self.p *= np.sin(exponent*omega)/np.sin(omega) else: return NotImplemented def __abs__(self): """Norm""" return math.sqrt(self.q ** 2 + np.dot(self.p,self.p)) magnitude = __abs__ def __eq__(self,other): """Equal (sufficiently close) to each other""" return np.isclose(( self-other).magnitude(),0.0) \ or np.isclose((-self-other).magnitude(),0.0) def __ne__(self,other): """Not equal (sufficiently close) to each other""" return not self.__eq__(other) def normalize(self): d = self.magnitude() if d > 0.0: self.q /= d self.p /= d return self def normalized(self): return self.copy().normalize() def conjugate(self): self.p = -self.p return self def conjugated(self): return self.copy().conjugate() def homomorph(self): if self.q < 0.0: self.q = -self.q self.p = -self.p return self def homomorphed(self): return self.copy().homomorph() #################################################################################################### class Rotation: u""" Orientation stored as unit quaternion. Following: D Rowenhorst et al. Consistent representations of and conversions between 3D rotations 10.1088/0965-0393/23/8/083501 Convention 1: coordinate frames are right-handed Convention 2: a rotation angle ω is taken to be positive for a counterclockwise rotation when viewing from the end point of the rotation axis towards the origin Convention 3: rotations will be interpreted in the passive sense Convention 4: Euler angle triplets are implemented using the Bunge convention, with the angular ranges as [0, 2π],[0, π],[0, 2π] Convention 5: the rotation angle ω is limited to the interval [0, π] Convention 6: P = -1 (as default) q is the real part, p = (x, y, z) are the imaginary parts. Vector "a" (defined in coordinate system "A") is passively rotated resulting in new coordinates "b" when expressed in system "B". b = Q * a b = np.dot(Q.asMatrix(),a) """ __slots__ = ['quaternion'] def __init__(self,quaternion = np.array([1.0,0.0,0.0,0.0])): """ Initializes to identity unless specified If a quaternion is given, it needs to comply with the convection. Use .fromQuaternion to check the input. """ if isinstance(quaternion,Quaternion): self.quaternion = quaternion.copy() else: self.quaternion = Quaternion(q=quaternion[0],p=quaternion[1:4]) self.quaternion.homomorph() # ToDo: Needed? def __repr__(self): """Value in selected representation""" return '\n'.join([ '{}'.format(self.quaternion), 'Matrix:\n{}'.format( '\n'.join(['\t'.join(list(map(str,self.asMatrix()[i,:]))) for i in range(3)]) ), 'Bunge Eulers / deg: {}'.format('\t'.join(list(map(str,self.asEulers(degrees=True)))) ), ]) ################################################################################################ # convert to different orientation representations (numpy arrays) def asQuaternion(self): """Unit quaternion: (q, [p_1, p_2, p_3])""" return self.quaternion.asArray() def asEulers(self, degrees = False): """Bunge-Euler angles: (φ_1, ϕ, φ_2)""" eu = qu2eu(self.quaternion.asArray()) if degrees: eu = np.degrees(eu) return eu def asAxisAngle(self, degrees = False): """Axis-angle pair: ([n_1, n_2, n_3], ω)""" ax = qu2ax(self.quaternion.asArray()) if degrees: ax[3] = np.degrees(ax[3]) return ax def asMatrix(self): """Rotation matrix""" return qu2om(self.quaternion.asArray()) def asRodrigues(self): """Rodrigues-Frank vector: ([n_1, n_2, n_3], tan(ω/2))""" return qu2ro(self.quaternion.asArray()) def asHomochoric(self): """Homochoric vector: (h_1, h_2, h_3)""" return qu2ho(self.quaternion.asArray()) def asCubochoric(self): return qu2cu(self.quaternion.asArray()) ################################################################################################ # static constructors. The input data needs to follow the convention, options allow to # relax these convections @classmethod def fromQuaternion(cls, quaternion, acceptHomomorph = False, P = -1): qu = quaternion if isinstance(quaternion, np.ndarray) else np.array(quaternion) if P > 0: qu[1:4] *= -1 # convert from P=1 to P=-1 if qu[0] < 0.0: if acceptHomomorph: qu *= -1. else: raise ValueError('Quaternion has negative first component.\n{}'.format(qu[0])) if not np.isclose(np.linalg.norm(qu), 1.0): raise ValueError('Quaternion is not of unit length.\n{} {} {} {}'.format(*qu)) return cls(qu) @classmethod def fromEulers(cls, eulers, degrees = False): eu = eulers if isinstance(eulers, np.ndarray) else np.array(eulers) eu = np.radians(eu) if degrees else eu if np.any(eu < 0.0) or np.any(eu > 2.0*np.pi) or eu[1] > np.pi: raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].\n{} {} {}.'.format(*eu)) return cls(eu2qu(eu)) @classmethod def fromAxisAngle(cls, angleAxis, degrees = False, normalise = False, P = -1): ax = angleAxis if isinstance(angleAxis, np.ndarray) else np.array(angleAxis) if P > 0: ax[0:3] *= -1 # convert from P=1 to P=-1 if degrees: ax[3] = np.radians(ax[3]) if normalise: ax[0:3] /=np.linalg.norm(ax[0:3]) if ax[3] < 0.0 or ax[3] > np.pi: raise ValueError('Axis angle rotation angle outside of [0..π].\n'.format(ax[3])) if not np.isclose(np.linalg.norm(ax[0:3]), 1.0): raise ValueError('Axis angle rotation axis is not of unit length.\n{} {} {}'.format(*ax[0:3])) return cls(ax2qu(ax)) @classmethod def fromMatrix(cls, matrix, containsStretch = False): #ToDo: better name? om = matrix if isinstance(matrix, np.ndarray) else np.array(matrix).reshape((3,3)) # ToDo: Reshape here or require explicit? if containsStretch: (U,S,Vh) = np.linalg.svd(om) # singular value decomposition om = np.dot(U,Vh) if not np.isclose(np.linalg.det(om),1.0): raise ValueError('matrix is not a proper rotation.\n{}'.format(om)) if not np.isclose(np.dot(om[0],om[1]), 0.0) \ or not np.isclose(np.dot(om[1],om[2]), 0.0) \ or not np.isclose(np.dot(om[2],om[0]), 0.0): raise ValueError('matrix is not orthogonal.\n{}'.format(om)) return cls(om2qu(om)) @classmethod def fromRodrigues(cls, rodrigues, normalise = False, P = -1): ro = rodrigues if isinstance(rodrigues, np.ndarray) else np.array(rodrigues) if P > 0: ro[0:3] *= -1 # convert from P=1 to P=-1 if normalise: ro[0:3] /=np.linalg.norm(ro[0:3]) if not np.isclose(np.linalg.norm(ro[0:3]), 1.0): raise ValueError('Rodrigues rotation axis is not of unit length.\n{} {} {}'.format(*ro[0:3])) if ro[3] < 0.0: raise ValueError('Rodriques rotation angle not positive.\n'.format(ro[3])) return cls(ro2qu(ro)) @classmethod def fromHomochoric(cls, homochoric, P = -1): ho = homochoric if isinstance(homochoric, np.ndarray) else np.array(homochoric) if P > 0: ho *= -1 # convert from P=1 to P=-1 return cls(ho2qu(ho)) @classmethod def fromCubochoric(cls, cubochoric, P = -1): cu = cubochoric if isinstance(cubochoric, np.ndarray) else np.array(cubochoric) ho = cu2ho(cu) if P > 0: ho *= -1 # convert from P=1 to P=-1 return cls(ho2qu(ho)) def __mul__(self, other): """ Multiplication Rotation: Details needed (active/passive), rotation of (3,3,3,3)-matrix should be considered """ if isinstance(other, Rotation): # rotate a rotation return self.__class__((self.quaternion * other.quaternion).asArray()) elif isinstance(other, np.ndarray): if other.shape == (3,): # rotate a single (3)-vector ( x, y, z) = self.quaternion.p (Vx,Vy,Vz) = other[0:3] A = self.quaternion.q*self.quaternion.q - np.dot(self.quaternion.p,self.quaternion.p) B = 2.0 * (x*Vx + y*Vy + z*Vz) C = 2.0 * P*self.quaternion.q return np.array([ A*Vx + B*x + C*(y*Vz - z*Vy), A*Vy + B*y + C*(z*Vx - x*Vz), A*Vz + B*z + C*(x*Vy - y*Vx), ]) elif other.shape == (3,3,): # rotate a single (3x3)-matrix return np.dot(self.asMatrix(),np.dot(other,self.asMatrix().T)) elif other.shape == (3,3,3,3): raise NotImplementedError else: return NotImplemented elif isinstance(other, tuple): # used to rotate a meshgrid-tuple ( x, y, z) = self.quaternion.p (Vx,Vy,Vz) = other[0:3] A = self.quaternion.q*self.quaternion.q - np.dot(self.quaternion.p,self.quaternion.p) B = 2.0 * (x*Vx + y*Vy + z*Vz) C = 2.0 * P*self.quaternion.q return np.array([ A*Vx + B*x + C*(y*Vz - z*Vy), A*Vy + B*y + C*(z*Vx - x*Vz), A*Vz + B*z + C*(x*Vy - y*Vx), ]) else: return NotImplemented def inverse(self): """Inverse rotation/backward rotation""" self.quaternion.conjugate() return self def inversed(self): """In-place inverse rotation/backward rotation""" return self.__class__(self.quaternion.conjugated()) def misorientation(self,other): """Misorientation""" return self.__class__(other.quaternion*self.quaternion.conjugated()) # ****************************************************************************************** class Symmetry: """ Symmetry operations for lattice systems https://en.wikipedia.org/wiki/Crystal_system """ lattices = [None,'orthorhombic','tetragonal','hexagonal','cubic',] def __init__(self, symmetry = None): if isinstance(symmetry, str) and symmetry.lower() in Symmetry.lattices: self.lattice = symmetry.lower() else: self.lattice = None def __copy__(self): """Copy""" return self.__class__(self.lattice) copy = __copy__ def __repr__(self): """Readable string""" return '{}'.format(self.lattice) def __eq__(self, other): """Equal to other""" return self.lattice == other.lattice def __neq__(self, other): """Not equal to other""" return not self.__eq__(other) def __cmp__(self,other): """Linear ordering""" myOrder = Symmetry.lattices.index(self.lattice) otherOrder = Symmetry.lattices.index(other.lattice) return (myOrder > otherOrder) - (myOrder < otherOrder) def symmetryOperations(self): """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 [Rotation(q) for q in symQuats] def inFZ(self,R): """ Check whether given Rodrigues vector falls into fundamental zone of own symmetry. Fundamental zone in Rodrigues space is point symmetric around origin. """ Rabs = abs(R[0:3]*R[3]) if self.lattice == 'cubic': return math.sqrt(2.0)-1.0 >= Rabs[0] \ and math.sqrt(2.0)-1.0 >= Rabs[1] \ and math.sqrt(2.0)-1.0 >= Rabs[2] \ and 1.0 >= Rabs[0] + Rabs[1] + Rabs[2] elif self.lattice == 'hexagonal': return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] and 1.0 >= Rabs[2] \ and 2.0 >= math.sqrt(3)*Rabs[0] + Rabs[1] \ and 2.0 >= math.sqrt(3)*Rabs[1] + Rabs[0] \ and 2.0 >= math.sqrt(3) + Rabs[2] elif self.lattice == 'tetragonal': return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] \ and math.sqrt(2.0) >= Rabs[0] + Rabs[1] \ and math.sqrt(2.0) >= Rabs[2] + 1.0 elif self.lattice == 'orthorhombic': return 1.0 >= Rabs[0] and 1.0 >= Rabs[1] and 1.0 >= Rabs[2] else: return True def inDisorientationSST(self,rodrigues): """ 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(rodrigues, Quaternion): R = rodrigues.asRodrigues() # translate accidentially passed quaternion else: R = rodrigues if R.shape[0]==4: # transition old (length not stored separately) to new R = (R[0:3]*R[3]) 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. Bases are computed from 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.)]).T), # 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.)]).T), # 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.)]).T), # direction of blue 'orthorhombic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red [1.,0.,0.], # direction of green [0.,1.,0.]]).T), # direction of blue } """ if self.lattice == 'cubic': basis = {'improper':np.array([ [-1. , 0. , 1. ], [ np.sqrt(2.) , -np.sqrt(2.) , 0. ], [ 0. , np.sqrt(3.) , 0. ] ]), 'proper':np.array([ [ 0. , -1. , 1. ], [-np.sqrt(2.) , np.sqrt(2.) , 0. ], [ np.sqrt(3.) , 0. , 0. ] ]), } elif self.lattice == 'hexagonal': basis = {'improper':np.array([ [ 0. , 0. , 1. ], [ 1. , -np.sqrt(3.) , 0. ], [ 0. , 2. , 0. ] ]), 'proper':np.array([ [ 0. , 0. , 1. ], [-1. , np.sqrt(3.) , 0. ], [ np.sqrt(3.) , -1. , 0. ] ]), } elif self.lattice == 'tetragonal': basis = {'improper':np.array([ [ 0. , 0. , 1. ], [ 1. , -1. , 0. ], [ 0. , np.sqrt(2.) , 0. ] ]), 'proper':np.array([ [ 0. , 0. , 1. ], [-1. , 1. , 0. ], [ np.sqrt(2.) , 0. , 0. ] ]), } elif self.lattice == 'orthorhombic': basis = {'improper':np.array([ [ 0., 0., 1.], [ 1., 0., 0.], [ 0., 1., 0.] ]), 'proper':np.array([ [ 0., 0., 1.], [-1., 0., 0.], [ 0., 1., 0.] ]), } else: # direct exit for unspecified symmetry if color: return (True,np.zeros(3,'d')) else: return True v = np.array(vector,dtype=float) if proper: # check both improper ... theComponents = np.dot(basis['improper'],v) inSST = np.all(theComponents >= 0.0) if not inSST: # ... and proper SST theComponents = np.dot(basis['proper'],v) inSST = np.all(theComponents >= 0.0) else: v[2] = abs(v[2]) # z component projects identical theComponents = np.dot(basis['improper'],v) # for positive and negative values inSST = np.all(theComponents >= 0.0) if color: # have to return color array if inSST: rgb = np.power(theComponents/np.linalg.norm(theComponents),0.5) # smoothen color ramps rgb = np.minimum(np.ones(3,dtype=float),rgb) # limit to maximum intensity rgb /= max(rgb) # normalize to (HS)V = 1 else: rgb = np.zeros(3,dtype=float) return (inSST,rgb) else: return inSST # code derived from https://github.com/ezag/pyeuclid # suggested reading: http://web.mit.edu/2.998/www/QuaternionReport1.pdf # ****************************************************************************************** class Lattice: """ Lattice system Currently, this contains only a mapping from Bravais lattice to symmetry and orientation relationships. It could include twin and slip systems. https://en.wikipedia.org/wiki/Bravais_lattice """ lattices = { 'triclinic':{'symmetry':None}, 'bct':{'symmetry':'tetragonal'}, 'hex':{'symmetry':'hexagonal'}, 'fcc':{'symmetry':'cubic','c/a':1.0}, 'bcc':{'symmetry':'cubic','c/a':1.0}, } def __init__(self, lattice): self.lattice = lattice self.symmetry = Symmetry(self.lattices[lattice]['symmetry']) def __repr__(self): """Report basic lattice information""" return 'Bravais lattice {} ({} symmetry)'.format(self.lattice,self.symmetry) # Kurdjomov--Sachs orientation relationship for fcc <-> bcc transformation # from S. Morito et al. Journal of Alloys and Compounds 577 (2013) 587-S592 # also see K. Kitahara et al. Acta Materialia 54 (2006) 1279-1288 KS = {'mapping':{'fcc':0,'bcc':1}, 'planes': 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]]],dtype='float'), 'directions': 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]]],dtype='float')} # Greninger--Troiano orientation relationship for fcc <-> bcc transformation # from Y. He et al. Journal of Applied Crystallography 39 (2006) 72-81 GT = {'mapping':{'fcc':0,'bcc':1}, 'planes': 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]]],dtype='float'), 'directions': 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]]],dtype='float')} # Greninger--Troiano' orientation relationship for fcc <-> bcc transformation # from Y. He et al. Journal of Applied Crystallography 39 (2006) 72-81 GTdash = {'mapping':{'fcc':0,'bcc':1}, 'planes': 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]]],dtype='float'), 'directions': 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]]],dtype='float')} # Nishiyama--Wassermann orientation relationship for fcc <-> bcc transformation # from H. Kitahara et al. Materials Characterization 54 (2005) 378-386 NW = {'mapping':{'fcc':0,'bcc':1}, 'planes': 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]]],dtype='float'), 'directions': 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]]],dtype='float')} # Pitsch orientation relationship for fcc <-> bcc transformation # from Y. He et al. Acta Materialia 53 (2005) 1179-1190 Pitsch = {'mapping':{'fcc':0,'bcc':1}, 'planes': 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]]],dtype='float'), 'directions': 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]]],dtype='float')} # Bain orientation relationship for fcc <-> bcc transformation # from Y. He et al. Journal of Applied Crystallography 39 (2006) 72-81 Bain = {'mapping':{'fcc':0,'bcc':1}, 'planes': np.array([ [[ 1, 0, 0],[ 1, 0, 0]], [[ 0, 1, 0],[ 0, 1, 0]], [[ 0, 0, 1],[ 0, 0, 1]]],dtype='float'), 'directions': np.array([ [[ 0, 1, 0],[ 0, 1, 1]], [[ 0, 0, 1],[ 1, 0, 1]], [[ 1, 0, 0],[ 1, 1, 0]]],dtype='float')} def relationOperations(self,model): models={'KS':self.KS, 'GT':self.GT, "GT'":self.GTdash, 'NW':self.NW, 'Pitsch': self.Pitsch, 'Bain':self.Bain} try: relationship = models[model] except: raise KeyError('Orientation relationship "{}" is unknown'.format(model)) if self.lattice not in relationship['mapping']: raise ValueError('Relationship "{}" not supported for lattice "{}"'.format(model,self.lattice)) r = {'lattice':Lattice((set(relationship['mapping'])-{self.lattice}).pop()), # target lattice 'rotations':[] } myPlane_id = relationship['mapping'][self.lattice] otherPlane_id = (myPlane_id+1)%2 myDir_id = myPlane_id +2 otherDir_id = otherPlane_id +2 for miller in np.hstack((relationship['planes'],relationship['directions'])): myPlane = miller[myPlane_id]/ np.linalg.norm(miller[myPlane_id]) myDir = miller[myDir_id]/ np.linalg.norm(miller[myDir_id]) myMatrix = np.array([myDir,np.cross(myPlane,myDir),myPlane]).T otherPlane = miller[otherPlane_id]/ np.linalg.norm(miller[otherPlane_id]) otherDir = miller[otherDir_id]/ np.linalg.norm(miller[otherDir_id]) otherMatrix = np.array([otherDir,np.cross(otherPlane,otherDir),otherPlane]).T r['rotations'].append(Rotation.fromMatrix(np.dot(otherMatrix,myMatrix.T))) return r class Orientation: """ Crystallographic orientation A crystallographic orientation contains a rotation and a lattice """ __slots__ = ['rotation','lattice'] def __repr__(self): """Report lattice type and orientation""" return self.lattice.__repr__()+'\n'+self.rotation.__repr__() def __init__(self, rotation, lattice): if isinstance(lattice, Lattice): self.lattice = lattice else: self.lattice = Lattice(lattice) # assume string if isinstance(rotation, Rotation): self.rotation = rotation else: self.rotation = Rotation(rotation) # assume quaternion 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.lattice.symmetry != other.lattice.symmetry: # raise NotImplementedError('disorientation between different symmetry classes not supported yet.') mis = other.rotation*self.rotation.inversed() mySymEqs = self.equivalentOrientations() if SST else self.equivalentOrientations()[:1] # take all or only first sym operation otherSymEqs = other.equivalentOrientations() for i,sA in enumerate(mySymEqs): for j,sB in enumerate(otherSymEqs): r = sB.rotation*mis*sA.rotation.inversed() for k in range(2): r.inversed() breaker = self.lattice.symmetry.inFZ(r.asRodrigues()) \ and (not SST or other.lattice.symmetry.inDisorientationSST(r.asRodrigues())) if breaker: break if breaker: break if breaker: break return r def inFZ(self): return self.lattice.symmetry.inFZ(self.rotation.asRodrigues()) def equivalentOrientations(self): """List of orientations which are symmetrically equivalent""" return [self.__class__(q*self.rotation,self.lattice) \ for q in self.lattice.symmetry.symmetryOperations()] def relatedOrientations(self,model): """List of orientations related by the given orientation relationship""" r = self.lattice.relationOperations(model) return [self.__class__(self.rotation*o,r['lattice']) for o in r['rotations']] def reduced(self): """Transform orientation to fall into fundamental zone according to symmetry""" for me in self.equivalentOrientations(): if self.lattice.symmetry.inFZ(me.rotation.asRodrigues()): break return self.__class__(me.rotation,self.lattice) 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,o in enumerate(self.equivalentOrientations()): # test all symmetric equivalent quaternions pole = o.rotation*axis # align crystal direction to axis if self.lattice.symmetry.inSST(pole,proper): break # found SST version else: pole = self.rotation*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 o in self.equivalentOrientations(): pole = o.rotation*axis # align crystal direction to axis inSST,color = self.lattice.symmetry.inSST(pole,color=True) if inSST: break return color # @classmethod # def average(cls, # orientations, # multiplicity = []): # """ # Average orientation # ref: F. Landis Markley, Yang Cheng, John Lucas Crassidis, and Yaakov Oshman. # Averaging Quaternions, # Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4 (2007), pp. 1193-1197. # doi: 10.2514/1.28949 # usage: # a = Orientation(Eulers=np.radians([10, 10, 0]), symmetry='hexagonal') # b = Orientation(Eulers=np.radians([20, 0, 0]), symmetry='hexagonal') # avg = Orientation.average([a,b]) # """ # if not all(isinstance(item, Orientation) for item in orientations): # raise TypeError("Only instances of Orientation can be averaged.") # N = len(orientations) # if multiplicity == [] or not multiplicity: # multiplicity = np.ones(N,dtype='i') # reference = orientations[0] # take first as reference # for i,(o,n) in enumerate(zip(orientations,multiplicity)): # closest = o.equivalentOrientations(reference.disorientation(o,SST = False)[2])[0] # select sym orientation with lowest misorientation # M = closest.quaternion.asM() * n if i == 0 else M + closest.quaternion.asM() * n # noqa add (multiples) of this orientation to average noqa # eig, vec = np.linalg.eig(M/N) # return Orientation(quaternion = Quaternion(quat = np.real(vec.T[eig.argmax()])), # symmetry = reference.symmetry.lattice) #################################################################################################### # Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations #################################################################################################### # Copyright (c) 2017-2019, Martin Diehl/Max-Planck-Institut für Eisenforschung GmbH # Copyright (c) 2013-2014, Marc De Graef/Carnegie Mellon University # All rights reserved. # # Redistribution and use in source and binary forms, with or without modification, are # permitted provided that the following conditions are met: # # - Redistributions of source code must retain the above copyright notice, this list # of conditions and the following disclaimer. # - Redistributions in binary form must reproduce the above copyright notice, this # list of conditions and the following disclaimer in the documentation and/or # other materials provided with the distribution. # - Neither the names of Marc De Graef, Carnegie Mellon University nor the names # of its contributors may be used to endorse or promote products derived from # this software without specific prior written permission. # # THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" # AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE # IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE # ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE # LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL # DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR # SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER # CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, # OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE # USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. #################################################################################################### def isone(a): return np.isclose(a,1.0,atol=1.0e-7,rtol=0.0) def iszero(a): return np.isclose(a,0.0,atol=1.0e-12,rtol=0.0) def eu2om(eu): """Euler angles to orientation matrix""" c = np.cos(eu) s = np.sin(eu) om = np.array([[+c[0]*c[2]-s[0]*s[2]*c[1], +s[0]*c[2]+c[0]*s[2]*c[1], +s[2]*s[1]], [-c[0]*s[2]-s[0]*c[2]*c[1], -s[0]*s[2]+c[0]*c[2]*c[1], +c[2]*s[1]], [+s[0]*s[1], -c[0]*s[1], +c[1] ]]) om[np.where(iszero(om))] = 0.0 return om def eu2ax(eu): """Euler angles to axis angle""" t = np.tan(eu[1]*0.5) sigma = 0.5*(eu[0]+eu[2]) delta = 0.5*(eu[0]-eu[2]) tau = np.linalg.norm([t,np.sin(sigma)]) alpha = np.pi if iszero(np.cos(sigma)) else \ 2.0*np.arctan(tau/np.cos(sigma)) if iszero(alpha): ax = np.array([ 0.0, 0.0, 1.0, 0.0 ]) else: ax = -P/tau * np.array([ t*np.cos(delta), t*np.sin(delta), np.sin(sigma) ]) # passive axis-angle pair so a minus sign in front ax = np.append(ax,alpha) if alpha < 0.0: ax *= -1.0 # ensure alpha is positive return ax def eu2ro(eu): """Euler angles to Rodrigues vector""" ro = eu2ax(eu) # convert to axis angle representation if ro[3] >= np.pi: # Differs from original implementation. check convention 5 ro[3] = np.inf elif iszero(ro[3]): ro = np.array([ 0.0, 0.0, P, 0.0 ]) else: ro[3] = np.tan(ro[3]*0.5) return ro def eu2qu(eu): """Euler angles to quaternion""" ee = 0.5*eu cPhi = np.cos(ee[1]) sPhi = np.sin(ee[1]) qu = np.array([ cPhi*np.cos(ee[0]+ee[2]), -P*sPhi*np.cos(ee[0]-ee[2]), -P*sPhi*np.sin(ee[0]-ee[2]), -P*cPhi*np.sin(ee[0]+ee[2]) ]) #if qu[0] < 0.0: qu.homomorph() !ToDo: Check with original return qu def om2eu(om): """Euler angles to orientation matrix""" if isone(om[2,2]**2): eu = np.array([np.arctan2( om[0,1],om[0,0]), np.pi*0.5*(1-om[2,2]),0.0]) # following the paper, not the reference implementation else: zeta = 1.0/np.sqrt(1.0-om[2,2]**2) eu = np.array([np.arctan2(om[2,0]*zeta,-om[2,1]*zeta), np.arccos(om[2,2]), np.arctan2(om[0,2]*zeta, om[1,2]*zeta)]) # reduce Euler angles to definition range, i.e a lower limit of 0.0 eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu) return eu def ax2om(ax): """Axis angle to orientation matrix""" c = np.cos(ax[3]) s = np.sin(ax[3]) omc = 1.0-c om=np.diag(ax[0:3]**2*omc + c) for idx in [[0,1,2],[1,2,0],[2,0,1]]: q = omc*ax[idx[0]] * ax[idx[1]] om[idx[0],idx[1]] = q + s*ax[idx[2]] om[idx[1],idx[0]] = q - s*ax[idx[2]] return om if P < 0.0 else om.T def qu2eu(qu): """Quaternion to Euler angles""" q03 = qu[0]**2+qu[3]**2 q12 = qu[1]**2+qu[2]**2 chi = np.sqrt(q03*q12) if iszero(chi): eu = np.array([np.arctan2(-P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0, 0.0]) if iszero(q12) else \ np.array([np.arctan2(2.0*qu[1]*qu[2],qu[1]**2-qu[2]**2), np.pi, 0.0]) else: eu = np.array([np.arctan2((-P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-P*qu[0]*qu[1]-qu[2]*qu[3])*chi ), np.arctan2( 2.0*chi, q03-q12 ), np.arctan2(( P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-P*qu[0]*qu[1]+qu[2]*qu[3])*chi )]) # reduce Euler angles to definition range, i.e a lower limit of 0.0 eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu) return eu def ax2ho(ax): """Axis angle to homochoric""" f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0) ho = ax[0:3] * f return ho def ho2ax(ho): """Homochoric to axis angle""" tfit = np.array([+1.0000000000018852, -0.5000000002194847, -0.024999992127593126, -0.003928701544781374, -0.0008152701535450438, -0.0002009500426119712, -0.00002397986776071756, -0.00008202868926605841, +0.00012448715042090092, -0.0001749114214822577, +0.0001703481934140054, -0.00012062065004116828, +0.000059719705868660826, -0.00001980756723965647, +0.000003953714684212874, -0.00000036555001439719544]) # normalize h and store the magnitude hmag_squared = np.sum(ho**2.) if iszero(hmag_squared): ax = np.array([ 0.0, 0.0, 1.0, 0.0 ]) else: hm = hmag_squared # convert the magnitude to the rotation angle s = tfit[0] + tfit[1] * hmag_squared for i in range(2,16): hm *= hmag_squared s += tfit[i] * hm ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))) return ax def om2ax(om): """Orientation matrix to axis angle""" ax=np.empty(4) # first get the rotation angle t = 0.5*(om.trace() -1.0) ax[3] = np.arccos(np.clip(t,-1.0,1.0)) if iszero(ax[3]): ax = [ 0.0, 0.0, 1.0, 0.0] else: w,vr = np.linalg.eig(om) # next, find the eigenvalue (1,0j) i = np.where(np.isclose(w,1.0+0.0j))[0][0] ax[0:3] = np.real(vr[0:3,i]) diagDelta = np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]]) ax[0:3] = np.where(iszero(diagDelta), ax[0:3],np.abs(ax[0:3])*np.sign(-P*diagDelta)) return np.array(ax) def ro2ax(ro): """Rodrigues vector to axis angle""" ta = ro[3] if iszero(ta): ax = [ 0.0, 0.0, 1.0, 0.0 ] elif not np.isfinite(ta): ax = [ ro[0], ro[1], ro[2], np.pi ] else: angle = 2.0*np.arctan(ta) ta = 1.0/np.linalg.norm(ro[0:3]) ax = [ ro[0]/ta, ro[1]/ta, ro[2]/ta, angle ] return np.array(ax) def ax2ro(ax): """Axis angle to Rodrigues vector""" if iszero(ax[3]): ro = [ 0.0, 0.0, P, 0.0 ] else: ro = [ax[0], ax[1], ax[2]] # 180 degree case ro += [np.inf] if np.isclose(ax[3],np.pi,atol=1.0e-15,rtol=0.0) else \ [np.tan(ax[3]*0.5)] return np.array(ro) def ax2qu(ax): """Axis angle to quaternion""" if iszero(ax[3]): qu = np.array([ 1.0, 0.0, 0.0, 0.0 ]) else: c = np.cos(ax[3]*0.5) s = np.sin(ax[3]*0.5) qu = np.array([ c, ax[0]*s, ax[1]*s, ax[2]*s ]) return qu def ro2ho(ro): """Rodrigues vector to homochoric""" if iszero(np.sum(ro[0:3]**2.0)): ho = [ 0.0, 0.0, 0.0 ] else: f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi ho = ro[0:3] * (0.75*f)**(1.0/3.0) return np.array(ho) def qu2om(qu): """Quaternion to orientation matrix""" qq = qu[0]**2-(qu[1]**2 + qu[2]**2 + qu[3]**2) om = np.diag(qq + 2.0*np.array([qu[1],qu[2],qu[3]])**2) om[1,0] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3]) om[0,1] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3]) om[2,1] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1]) om[1,2] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1]) om[0,2] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2]) om[2,0] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2]) return om if P > 0.0 else om.T def qu2ax(qu): """ Quaternion to axis angle Modified version of the original formulation, should be numerically more stable """ if isone(abs(qu[0])): # set axis to [001] if the angle is 0/360 ax = [ 0.0, 0.0, 1.0, 0.0 ] elif not iszero(qu[0]): omega = 2.0 * np.arccos(qu[0]) s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2) ax = [ qu[1]*s, qu[2]*s, qu[3]*s, omega ] else: ax = [ qu[1], qu[2], qu[3], np.pi] return np.array(ax) def qu2ro(qu): """Quaternion to Rodrigues vector""" if iszero(qu[0]): ro = [qu[1], qu[2], qu[3], np.inf] else: s = np.linalg.norm([qu[1],qu[2],qu[3]]) ro = [0.0,0.0,P,0.0] if iszero(s) else \ [ qu[1]/s, qu[2]/s, qu[3]/s, np.tan(np.arccos(np.clip(qu[0],-1.0,1.0)))] # avoid numerical difficulties return np.array(ro) def qu2ho(qu): """Quaternion to homochoric""" omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0)) # avoid numerical difficulties if iszero(omega): ho = np.array([ 0.0, 0.0, 0.0 ]) else: ho = np.array([qu[1], qu[2], qu[3]]) f = 0.75 * ( omega - np.sin(omega) ) ho = ho/np.linalg.norm(ho) * f**(1./3.) return ho def ho2cu(ho): """Homochoric to cubochoric""" return Lambert.BallToCube(ho) def cu2ho(cu): """Cubochoric to homochoric""" return Lambert.CubeToBall(cu) def ro2eu(ro): """Rodrigues vector to orientation matrix""" return om2eu(ro2om(ro)) def eu2ho(eu): """Euler angles to homochoric""" return ax2ho(eu2ax(eu)) def om2ro(om): """Orientation matrix to Rodriques vector""" return eu2ro(om2eu(om)) def om2ho(om): """Orientation matrix to homochoric""" return ax2ho(om2ax(om)) def ax2eu(ax): """Orientation matrix to Euler angles""" return om2eu(ax2om(ax)) def ro2om(ro): """Rodgrigues vector to orientation matrix""" return ax2om(ro2ax(ro)) def ro2qu(ro): """Rodrigues vector to quaternion""" return ax2qu(ro2ax(ro)) def ho2eu(ho): """Homochoric to Euler angles""" return ax2eu(ho2ax(ho)) def ho2om(ho): """Homochoric to orientation matrix""" return ax2om(ho2ax(ho)) def ho2ro(ho): """Axis angle to Rodriques vector""" return ax2ro(ho2ax(ho)) def ho2qu(ho): """Homochoric to quaternion""" return ax2qu(ho2ax(ho)) def eu2cu(eu): """Euler angles to cubochoric""" return ho2cu(eu2ho(eu)) def om2cu(om): """Orientation matrix to cubochoric""" return ho2cu(om2ho(om)) def om2qu(om): """ Orientation matrix to quaternion The original formulation (direct conversion) had numerical issues """ return ax2qu(om2ax(om)) def ax2cu(ax): """Axis angle to cubochoric""" return ho2cu(ax2ho(ax)) def ro2cu(ro): """Rodrigues vector to cubochoric""" return ho2cu(ro2ho(ro)) def qu2cu(qu): """Quaternion to cubochoric""" return ho2cu(qu2ho(qu)) def cu2eu(cu): """Cubochoric to Euler angles""" return ho2eu(cu2ho(cu)) def cu2om(cu): """Cubochoric to orientation matrix""" return ho2om(cu2ho(cu)) def cu2ax(cu): """Cubochoric to axis angle""" return ho2ax(cu2ho(cu)) def cu2ro(cu): """Cubochoric to Rodrigues vector""" return ho2ro(cu2ho(cu)) def cu2qu(cu): """Cubochoric to quaternion""" return ho2qu(cu2ho(cu))