import math import numpy as np from . import Lambert from .quaternion import Quaternion from .quaternion import P #################################################################################################### class Rotation: u""" Orientation stored with functionality for conversion to different representations. References ---------- D. Rowenhorst et al., Modelling and Simulation in Materials Science and Engineering 23:083501, 2015 https://doi.org/10.1088/0965-0393/23/8/083501 Conventions ----------- 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). Usage ----- 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. Parameters ---------- quaternion : numpy.ndarray, optional Unit quaternion that follows the conventions. Use .fromQuaternion to perform a sanity check. """ if isinstance(quaternion,Quaternion): self.quaternion = quaternion.copy() else: self.quaternion = Quaternion(q=quaternion[0],p=quaternion[1:4]) def __copy__(self): """Copy.""" return self.__class__(self.quaternion) copy = __copy__ def __repr__(self): """Orientation displayed as unit quaternion, rotation matrix, and Bunge-Euler angles.""" 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)))) ), ]) def __mul__(self, other): """ Multiplication. Parameters ---------- other : numpy.ndarray or Rotation Vector, second or fourth order tensor, or rotation object that is rotated. Todo ---- Document details active/passive) considere rotation of (3,3,3,3)-matrix """ if isinstance(other, Rotation): # rotate a rotation return self.__class__(self.quaternion * other.quaternion).standardize() 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): """In-place inverse rotation/backward rotation.""" self.quaternion.conjugate() return self def inversed(self): """Inverse rotation/backward rotation.""" return self.copy().inverse() def standardize(self): """In-place quaternion representation with positive q.""" if self.quaternion.q < 0.0: self.quaternion.homomorph() return self def standardized(self): """Quaternion representation with positive q.""" return self.copy().standardize() def misorientation(self,other): """ Get Misorientation. Parameters ---------- other : Rotation Rotation to which the misorientation is computed. """ return other*self.inversed() def average(self,other): """ Calculate the average rotation. Parameters ---------- other : Rotation Rotation from which the average is rotated. """ return Rotation.fromAverage([self,other]) ################################################################################################ # 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). Parameters ---------- degrees : bool, optional return angles in degrees. """ 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). Parameters ---------- degrees : bool, optional return rotation angle in degrees. """ 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, vector=False): """ Rodrigues-Frank vector: ([n_1, n_2, n_3], tan(ω/2)). Parameters ---------- vector : bool, optional return as array of length 3, i.e. scale the unit vector giving the rotation axis. """ ro = qu2ro(self.quaternion.asArray()) return ro[:3]*ro[3] if vector else ro def asHomochoric(self): """Homochoric vector: (h_1, h_2, h_3).""" return qu2ho(self.quaternion.asArray()) def asCubochoric(self): """Cubochoric vector: (c_1, c_2, c_3).""" return qu2cu(self.quaternion.asArray()) def asM(self): """ Intermediate representation supporting quaternion averaging. References ---------- F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007 https://doi.org/10.2514/1.28949 """ return self.quaternion.asM() ################################################################################################ # 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) and quaternion.dtype == np.dtype(float) \ else np.array(quaternion,dtype=float) 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) and eulers.dtype == np.dtype(float) \ else np.array(eulers,dtype=float) 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) and angleAxis.dtype == np.dtype(float) \ else np.array(angleAxis,dtype=float) 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 fromBasis(cls, basis, orthonormal = True, reciprocal = False, ): om = basis if isinstance(basis, np.ndarray) else np.array(basis).reshape((3,3)) if reciprocal: om = np.linalg.inv(om.T/np.pi) # transform reciprocal basis set orthonormal = False # contains stretch if not orthonormal: (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 fromMatrix(cls, om, ): return cls.fromBasis(om) @classmethod def fromRodrigues(cls, rodrigues, normalise = False, P = -1): ro = rodrigues if isinstance(rodrigues, np.ndarray) and rodrigues.dtype == np.dtype(float) \ else np.array(rodrigues,dtype=float) 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) and homochoric.dtype == np.dtype(float) \ else np.array(homochoric,dtype=float) 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) and cubochoric.dtype == np.dtype(float) \ else np.array(cubochoric,dtype=float) ho = cu2ho(cu) if P > 0: ho *= -1 # convert from P=1 to P=-1 return cls(ho2qu(ho)) @classmethod def fromAverage(cls, rotations, weights = []): """ Average rotation. References ---------- F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007 https://doi.org/10.2514/1.28949 Parameters ---------- rotations : list of Rotations Rotations to average from weights : list of floats, optional Weights for each rotation used for averaging """ if not all(isinstance(item, Rotation) for item in rotations): raise TypeError("Only instances of Rotation can be averaged.") N = len(rotations) if weights == [] or not weights: weights = np.ones(N,dtype='i') for i,(r,n) in enumerate(zip(rotations,weights)): M = r.asM() * n if i == 0 \ else M + r.asM() * n # noqa add (multiples) of this rotation to average noqa eig, vec = np.linalg.eig(M/N) return cls.fromQuaternion(np.real(vec.T[eig.argmax()]),acceptHomomorph = True) @classmethod def fromRandom(cls): r = np.random.random(3) A = np.sqrt(r[2]) B = np.sqrt(1.0-r[2]) w = np.cos(2.0*np.pi*r[0])*A x = np.sin(2.0*np.pi*r[1])*B y = np.cos(2.0*np.pi*r[1])*B z = np.sin(2.0*np.pi*r[0])*A return cls.fromQuaternion([w,x,y,z],acceptHomomorph=True) # ****************************************************************************************** class Symmetry: """ Symmetry operations for lattice systems. References ---------- https://en.wikipedia.org/wiki/Crystal_system """ lattices = [None,'orthorhombic','tetragonal','hexagonal','cubic',] def __init__(self, symmetry = None): """ Symmetry Definition. Parameters ---------- symmetry : str, optional label of the crystal system """ if symmetry is not None and symmetry.lower() not in Symmetry.lattices: raise KeyError('Symmetry/crystal system "{}" is unknown'.format(symmetry)) self.lattice = symmetry.lower() if isinstance(symmetry,str) else symmetry 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. Parameters ---------- other : Symmetry Symmetry to check for equality. """ return self.lattice == other.lattice def __neq__(self, other): """ Not Equal to other. Parameters ---------- other : Symmetry Symmetry to check for inequality. """ return not self.__eq__(other) def __cmp__(self,other): """ Linear ordering. Parameters ---------- other : Symmetry Symmetry to check for for order. """ myOrder = Symmetry.lattices.index(self.lattice) otherOrder = Symmetry.lattices.index(other.lattice) return (myOrder > otherOrder) - (myOrder < otherOrder) def symmetryOperations(self,members=[]): """List (or single element) of symmetry operations as rotations.""" 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*np.sqrt(2), 0.5*np.sqrt(2) ], [ 0.0, 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2) ], [ 0.0, 0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2) ], [ 0.0, 0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2) ], [ 0.0, 0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0 ], [ 0.0, -0.5*np.sqrt(2),-0.5*np.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*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ], [ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ], [-0.5*np.sqrt(2), 0.0, 0.5*np.sqrt(2), 0.0 ], [-0.5*np.sqrt(2), 0.0, -0.5*np.sqrt(2), 0.0 ], [-0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0, 0.0 ], [-0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0, 0.0 ], ] elif self.lattice == 'hexagonal': symQuats = [ [ 1.0, 0.0, 0.0, 0.0 ], [-0.5*np.sqrt(3), 0.0, 0.0, -0.5 ], [ 0.5, 0.0, 0.0, 0.5*np.sqrt(3) ], [ 0.0, 0.0, 0.0, 1.0 ], [-0.5, 0.0, 0.0, 0.5*np.sqrt(3) ], [-0.5*np.sqrt(3), 0.0, 0.0, 0.5 ], [ 0.0, 1.0, 0.0, 0.0 ], [ 0.0, -0.5*np.sqrt(3), 0.5, 0.0 ], [ 0.0, 0.5, -0.5*np.sqrt(3), 0.0 ], [ 0.0, 0.0, 1.0, 0.0 ], [ 0.0, -0.5, -0.5*np.sqrt(3), 0.0 ], [ 0.0, 0.5*np.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*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ], [ 0.0, -0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0 ], [ 0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.sqrt(2) ], [-0.5*np.sqrt(2), 0.0, 0.0, 0.5*np.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 ], ] symOps = list(map(Rotation, np.array(symQuats)[np.atleast_1d(members) if members != [] else range(len(symQuats))])) try: iter(members) # asking for (even empty) list of members? except TypeError: return symOps[0] # no, return rotation object else: return symOps # yes, return list of rotations def inFZ(self,rodrigues): """ Check whether given Rodriques-Frank vector falls into fundamental zone of own symmetry. Fundamental zone in Rodrigues space is point symmetric around origin. """ if (len(rodrigues) != 3): raise ValueError('Input is not a Rodriques-Frank vector.\n') if np.any(rodrigues == np.inf): return False Rabs = abs(rodrigues) 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 Rodriques-Frank vector (of misorientation) falls into standard stereographic triangle of own symmetry. References ---------- A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991 https://doi.org/10.1107/S0108767391006864 """ if (len(rodrigues) != 3): raise ValueError('Input is not a Rodriques-Frank vector.\n') R = rodrigues 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. References ---------- 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): """ New lattice of given type. Parameters ---------- lattice : str Bravais 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:s587-s592, 2013 # also see K. Kitahara et al., Acta Materialia 54:1279-1288, 2006 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 39:72-81, 2006 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 39:72-81, 2006 GTprime = {'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 54:378-386, 2005 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 53:1179-1190, 2005 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 39:72-81, 2006 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): """ Crystallographic orientation relationships for phase transformations. References ---------- S. Morito et al., Journal of Alloys and Compounds 577:s587-s592, 2013 https://doi.org/10.1016/j.jallcom.2012.02.004 K. Kitahara et al., Acta Materialia 54(5):1279-1288, 2006 https://doi.org/10.1016/j.actamat.2005.11.001 Y. He et al., Journal of Applied Crystallography 39:72-81, 2006 https://doi.org/10.1107/S0021889805038276 H. Kitahara et al., Materials Characterization 54(4-5):378-386, 2005 https://doi.org/10.1016/j.matchar.2004.12.015 Y. He et al., Acta Materialia 53(4):1179-1190, 2005 https://doi.org/10.1016/j.actamat.2004.11.021 """ models={'KS':self.KS, 'GT':self.GT, "GT'":self.GTprime, 'NW':self.NW, 'Pitsch': self.Pitsch, 'Bain':self.Bain} try: relationship = models[model] except KeyError : 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): """ New orientation from rotation and lattice. Parameters ---------- rotation : Rotation Rotation specifying the lattice orientation. lattice : Lattice Lattice type of the crystal. """ 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, symmetries = False): """ 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.') mySymEqs = self.equivalentOrientations() if SST else self.equivalentOrientations([0]) # take all or only first sym operation otherSymEqs = other.equivalentOrientations() for i,sA in enumerate(mySymEqs): aInv = sA.rotation.inversed() for j,sB in enumerate(otherSymEqs): b = sB.rotation r = b*aInv for k in range(2): r.inverse() breaker = self.lattice.symmetry.inFZ(r.asRodrigues(vector=True)) \ and (not SST or other.lattice.symmetry.inDisorientationSST(r.asRodrigues(vector=True))) if breaker: break if breaker: break if breaker: break return (Orientation(r,self.lattice), i,j, k == 1) if symmetries else r # disorientation ... # ... own sym, other sym, # self-->other: True, self<--other: False def inFZ(self): return self.lattice.symmetry.inFZ(self.rotation.asRodrigues(vector=True)) def equivalentOrientations(self,members=[]): """List of orientations which are symmetrically equivalent.""" try: iter(members) # asking for (even empty) list of members? except TypeError: return self.__class__(self.lattice.symmetry.symmetryOperations(members)*self.rotation,self.lattice) # no, return rotation object else: return [self.__class__(q*self.rotation,self.lattice) \ for q in self.lattice.symmetry.symmetryOperations(members)] # yes, return list of rotations 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(vector=True)): 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 fromAverage(cls, orientations, weights = []): """Create orientation from average of list of orientations.""" if not all(isinstance(item, Orientation) for item in orientations): raise TypeError("Only instances of Orientation can be averaged.") closest = [] ref = orientations[0] for o in orientations: closest.append(o.equivalentOrientations( ref.disorientation(o, SST = False, # select (o[ther]'s) sym orientation symmetries = True)[2]).rotation) # with lowest misorientation return Orientation(Rotation.fromAverage(closest,weights),ref.lattice) def average(self,other): """Calculate the average rotation.""" return Orientation.fromAverage([self,other]) #################################################################################################### # 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) #---------- Quaternion ---------- def qu2om(qu): """Quaternion to rotation 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 qu2eu(qu): """Quaternion to Bunge-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 qu2ax(qu): """ Quaternion to axis angle pair. Modified version of the original formulation, should be numerically more stable """ if iszero(qu[1]**2+qu[2]**2+qu[3]**2): # set axis to [001] if the angle is 0/360 ax = [ 0.0, 0.0, 1.0, 0.0 ] elif not iszero(qu[0]): s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2) omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0)) 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 Rodriques-Frank 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 vector.""" 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 qu2cu(qu): """Quaternion to cubochoric vector.""" return ho2cu(qu2ho(qu)) #---------- Rotation matrix ---------- def om2qu(om): """ Rotation matrix to quaternion. The original formulation (direct conversion) had (numerical?) issues """ return eu2qu(om2eu(om)) def om2eu(om): """Rotation matrix to Bunge-Euler angles.""" if abs(om[2,2]) < 1.0: 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)]) else: 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 # 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 om2ax(om): """Rotation matrix to axis angle pair.""" 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 om2ro(om): """Rotation matrix to Rodriques-Frank vector.""" return eu2ro(om2eu(om)) def om2ho(om): """Rotation matrix to homochoric vector.""" return ax2ho(om2ax(om)) def om2cu(om): """Rotation matrix to cubochoric vector.""" return ho2cu(om2ho(om)) #---------- Bunge-Euler angles ---------- def eu2qu(eu): """Bunge-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*=-1 return qu def eu2om(eu): """Bunge-Euler angles to rotation 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): """Bunge-Euler angles to axis angle pair.""" 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): """Bunge-Euler angles to Rodriques-Frank vector.""" ro = eu2ax(eu) # convert to axis angle pair 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 eu2ho(eu): """Bunge-Euler angles to homochoric vector.""" return ax2ho(eu2ax(eu)) def eu2cu(eu): """Bunge-Euler angles to cubochoric vector.""" return ho2cu(eu2ho(eu)) #---------- Axis angle pair ---------- def ax2qu(ax): """Axis angle pair 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 ax2om(ax): """Axis angle pair to rotation 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 ax2eu(ax): """Rotation matrix to Bunge Euler angles.""" return om2eu(ax2om(ax)) def ax2ro(ax): """Axis angle pair to Rodriques-Frank 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 ax2ho(ax): """Axis angle pair to homochoric vector.""" f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0) ho = ax[0:3] * f return ho def ax2cu(ax): """Axis angle pair to cubochoric vector.""" return ho2cu(ax2ho(ax)) #---------- Rodrigues-Frank vector ---------- def ro2qu(ro): """Rodriques-Frank vector to quaternion.""" return ax2qu(ro2ax(ro)) def ro2om(ro): """Rodgrigues-Frank vector to rotation matrix.""" return ax2om(ro2ax(ro)) def ro2eu(ro): """Rodriques-Frank vector to Bunge-Euler angles.""" return om2eu(ro2om(ro)) def ro2ax(ro): """Rodriques-Frank vector to axis angle pair.""" 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 ro2ho(ro): """Rodriques-Frank vector to homochoric vector.""" 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 ro2cu(ro): """Rodriques-Frank vector to cubochoric vector.""" return ho2cu(ro2ho(ro)) #---------- Homochoric vector---------- def ho2qu(ho): """Homochoric vector to quaternion.""" return ax2qu(ho2ax(ho)) def ho2om(ho): """Homochoric vector to rotation matrix.""" return ax2om(ho2ax(ho)) def ho2eu(ho): """Homochoric vector to Bunge-Euler angles.""" return ax2eu(ho2ax(ho)) def ho2ax(ho): """Homochoric vector to axis angle pair.""" 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 ho2ro(ho): """Axis angle pair to Rodriques-Frank vector.""" return ax2ro(ho2ax(ho)) def ho2cu(ho): """Homochoric vector to cubochoric vector.""" return Lambert.BallToCube(ho) #---------- Cubochoric ---------- def cu2qu(cu): """Cubochoric vector to quaternion.""" return ho2qu(cu2ho(cu)) def cu2om(cu): """Cubochoric vector to rotation matrix.""" return ho2om(cu2ho(cu)) def cu2eu(cu): """Cubochoric vector to Bunge-Euler angles.""" return ho2eu(cu2ho(cu)) def cu2ax(cu): """Cubochoric vector to axis angle pair.""" return ho2ax(cu2ho(cu)) def cu2ro(cu): """Cubochoric vector to Rodriques-Frank vector.""" return ho2ro(cu2ho(cu)) def cu2ho(cu): """Cubochoric vector to homochoric vector.""" return Lambert.CubeToBall(cu)