Source code for damask._rotation

import numpy as np

from . import tensor
from . import util
from . import grid_filters

_P = -1

# parameters for conversion from/to cubochoric
_sc   = np.pi**(1./6.)/6.**(1./6.)
_beta = np.pi**(5./6.)/6.**(1./6.)/2.
_R1   = (3.*np.pi/4.)**(1./3.)

[docs]class Rotation: u""" Rotation with functionality for conversion between different representations. The following conventions apply: - Coordinate frames are right-handed. - 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. - Rotations will be interpreted in the passive sense. - Euler angle triplets are implemented using the Bunge convention, with angular ranges of [0,2π], [0,π], [0,2π]. - The rotation angle ω is limited to the interval [0,π]. - The real part of a quaternion is positive, Re(q) ≥ 0 - P = -1 (as default). Examples -------- Rotate vector 'a' (defined in coordinate system 'A') to coordinates 'b' expressed in system 'B': >>> import damask >>> import numpy as np >>> Q = damask.Rotation.from_random() >>> a = np.random.rand(3) >>> b = Q @ a >>> np.allclose(np.dot(Q.as_matrix(),a),b) True Compound rotations R1 (first) and R2 (second): >>> import damask >>> import numpy as np >>> R1 = damask.Rotation.from_random() >>> R2 = damask.Rotation.from_random() >>> R = R2 * R1 >>> np.allclose(R.as_matrix(), np.dot(R2.as_matrix(),R1.as_matrix())) True 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 """ __slots__ = ['quaternion'] def __init__(self,rotation = np.array([1.0,0.0,0.0,0.0])): """ New rotation. Parameters ---------- rotation : list, numpy.ndarray, Rotation, optional Unit quaternion in positive real hemisphere. Use .from_quaternion to perform a sanity check. Defaults to no rotation. """ if isinstance(rotation,Rotation): self.quaternion = rotation.quaternion.copy() elif np.array(rotation).shape[-1] == 4: self.quaternion = np.array(rotation) else: raise TypeError('"rotation" is neither a Rotation nor a quaternion') def __repr__(self): """Represent rotation as unit quaternion(s).""" return f'Quaternion{" " if self.quaternion.shape == (4,) else "s of shape "+str(self.quaternion.shape[:-1])+chr(10)}'\ + str(self.quaternion) def __copy__(self,**kwargs): """Create deep copy.""" return self.__class__(rotation=kwargs['rotation'] if 'rotation' in kwargs else self.quaternion) copy = __copy__ def __getitem__(self,item): """Return slice according to item.""" return self.copy() \ if self.shape == () else \ self.copy(rotation=self.quaternion[item+(slice(None),)] if isinstance(item,tuple) else self.quaternion[item]) def __eq__(self,other): """ Equal to other. Parameters ---------- other : Rotation Rotation to check for equality. """ return np.logical_or(np.all(self.quaternion == other.quaternion,axis=-1), np.all(self.quaternion == -1.0*other.quaternion,axis=-1)) def __ne__(self,other): """ Not equal to other. Parameters ---------- other : Rotation Rotation to check for inequality. """ return np.logical_not(self==other)
[docs] def isclose(self,other,rtol=1e-5,atol=1e-8,equal_nan=True): """ Report where values are approximately equal to corresponding ones of other Rotation. Parameters ---------- other : Rotation Rotation to compare against. rtol : float, optional Relative tolerance of equality. atol : float, optional Absolute tolerance of equality. equal_nan : bool, optional Consider matching NaN values as equal. Defaults to True. Returns ------- mask : numpy.ndarray bool Mask indicating where corresponding rotations are close. """ s = self.quaternion o = other.quaternion return np.logical_or(np.all(np.isclose(s, o,rtol,atol,equal_nan),axis=-1), np.all(np.isclose(s,-1.0*o,rtol,atol,equal_nan),axis=-1))
[docs] def allclose(self,other,rtol=1e-5,atol=1e-8,equal_nan=True): """ Test whether all values are approximately equal to corresponding ones of other Rotation. Parameters ---------- other : Rotation Rotation to compare against. rtol : float, optional Relative tolerance of equality. atol : float, optional Absolute tolerance of equality. equal_nan : bool, optional Consider matching NaN values as equal. Defaults to True. Returns ------- answer : bool Whether all values are close between both rotations. """ return np.all(self.isclose(other,rtol,atol,equal_nan))
def __array__(self): """Initializer for numpy.""" return self.quaternion @property def size(self): return self.quaternion[...,0].size @property def shape(self): return self.quaternion[...,0].shape def __len__(self): """Length of leading/leftmost dimension of array.""" return 0 if self.shape == () else self.shape[0] def __invert__(self): """Inverse rotation (backward rotation).""" dup = self.copy() dup.quaternion[...,1:] *= -1 return dup def __pow__(self,exp): """ Perform the rotation 'exp' times. Parameters ---------- exp : float Exponent. """ phi = np.arccos(self.quaternion[...,0:1]) p = self.quaternion[...,1:]/np.linalg.norm(self.quaternion[...,1:],axis=-1,keepdims=True) return self.copy(rotation=Rotation(np.block([np.cos(exp*phi),np.sin(exp*phi)*p]))._standardize()) def __ipow__(self,exp): """ Perform the rotation 'exp' times (in-place). Parameters ---------- exp : float Exponent. """ return self**exp def __mul__(self,other): """ Compose with other. Parameters ---------- other : Rotation of shape (self.shape) Rotation for composition. Returns ------- composition : Rotation Compound rotation self*other, i.e. first other then self rotation. """ if isinstance(other,Rotation): q_m = self.quaternion[...,0:1] p_m = self.quaternion[...,1:] q_o = other.quaternion[...,0:1] p_o = other.quaternion[...,1:] q = (q_m*q_o - np.einsum('...i,...i',p_m,p_o).reshape(self.shape+(1,))) p = q_m*p_o + q_o*p_m + _P * np.cross(p_m,p_o) return Rotation(np.block([q,p]))._standardize() else: raise TypeError('Use "R@b", i.e. matmul, to apply rotation "R" to object "b"') def __imul__(self,other): """ Compose with other (in-place). Parameters ---------- other : Rotation of shape (self.shape) Rotation for composition. """ return self*other def __truediv__(self,other): """ Compose with inverse of other. Parameters ---------- other : damask.Rotation of shape (self.shape) Rotation to invert for composition. Returns ------- composition : Rotation Compound rotation self*(~other), i.e. first inverse of other then self rotation. """ if isinstance(other,Rotation): return self*~other else: raise TypeError('Use "R@b", i.e. matmul, to apply rotation "R" to object "b"') def __itruediv__(self,other): """ Compose with inverse of other (in-place). Parameters ---------- other : Rotation of shape (self.shape) Rotation to invert for composition. """ return self/other def __matmul__(self,other): """ Rotate vector, second order tensor, or fourth order tensor. Parameters ---------- other : numpy.ndarray of shape (...,3), (...,3,3), or (...,3,3,3,3) Vector or tensor on which to apply the rotation. Returns ------- rotated : numpy.ndarray of shape (...,3), (...,3,3), or (...,3,3,3,3) Rotated vector or tensor, i.e. transformed to frame defined by rotation. """ if isinstance(other,np.ndarray): if self.shape + (3,) == other.shape: q_m = self.quaternion[...,0] p_m = self.quaternion[...,1:] A = q_m**2.0 - np.einsum('...i,...i',p_m,p_m) B = 2.0 * np.einsum('...i,...i',p_m,other) C = 2.0 * _P * q_m return np.block([(A * other[...,i]).reshape(self.shape+(1,)) + (B * p_m[...,i]).reshape(self.shape+(1,)) + (C * ( p_m[...,(i+1)%3]*other[...,(i+2)%3]\ - p_m[...,(i+2)%3]*other[...,(i+1)%3])).reshape(self.shape+(1,)) for i in [0,1,2]]) if self.shape + (3,3) == other.shape: R = self.as_matrix() return np.einsum('...im,...jn,...mn',R,R,other) if self.shape + (3,3,3,3) == other.shape: R = self.as_matrix() return np.einsum('...im,...jn,...ko,...lp,...mnop',R,R,R,R,other) else: raise ValueError('Can only rotate vectors, 2nd order tensors, and 4th order tensors') elif isinstance(other,Rotation): raise TypeError('Use "R1*R2", i.e. multiplication, to compose rotations "R1" and "R2"') else: raise TypeError(f'Cannot rotate {type(other)}') apply = __matmul__ def _standardize(self): """Standardize quaternion (ensure positive real hemisphere).""" self.quaternion[self.quaternion[...,0] < 0.0] *= -1 return self
[docs] def append(self,other): """ Extend array along first dimension with other array(s). Parameters ---------- other : damask.Rotation """ return self.copy(rotation=np.vstack(tuple(map(lambda x:x.quaternion, [self]+other if isinstance(other,list) else [self,other]))))
[docs] def flatten(self,order = 'C'): """ Flatten array. Returns ------- flattened : damask.Rotation Rotation flattened to single dimension. """ return self.copy(rotation=self.quaternion.reshape((-1,4),order=order))
[docs] def reshape(self,shape,order = 'C'): """ Reshape array. Returns ------- reshaped : damask.Rotation Rotation of given shape. """ if isinstance(shape,(int,np.integer)): shape = (shape,) return self.copy(rotation=self.quaternion.reshape(tuple(shape)+(4,),order=order))
[docs] def broadcast_to(self,shape,mode = 'right'): """ Broadcast array. Parameters ---------- shape : tuple Shape of broadcasted array. mode : str, optional Where to preferentially locate missing dimensions. Either 'left' or 'right' (default). Returns ------- broadcasted : damask.Rotation Rotation broadcasted to given shape. """ if isinstance(shape,(int,np.integer)): shape = (shape,) return self.copy(rotation=np.broadcast_to(self.quaternion.reshape(util.shapeshifter(self.shape,shape,mode)+(4,)), shape+(4,)))
[docs] def average(self,weights = None): """ Average along last array dimension. Parameters ---------- weights : list of floats, optional Relative weight of each rotation. Returns ------- average : damask.Rotation Weighted average of original Rotation field. References ---------- F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007 https://doi.org/10.2514/1.28949 """ def _M(quat): """Intermediate representation supporting quaternion averaging.""" return np.einsum('...i,...j',quat,quat) if weights is None: weights = np.ones(self.shape,dtype=float) eig, vec = np.linalg.eig(np.sum(_M(self.quaternion) * weights[...,np.newaxis,np.newaxis],axis=-3) \ /np.sum( weights[...,np.newaxis,np.newaxis],axis=-3)) return Rotation.from_quaternion(np.real( np.squeeze( np.take_along_axis(vec, eig.argmax(axis=-1)[...,np.newaxis,np.newaxis], axis=-1), axis=-1)), accept_homomorph = True)
[docs] def misorientation(self,other): """ Calculate misorientation to other Rotation. Parameters ---------- other : damask.Rotation Rotation to which the misorientation is computed. Returns ------- g : damask.Rotation Misorientation. """ return other/self
################################################################################################ # convert to different orientation representations (numpy arrays)
[docs] def as_quaternion(self): """ Represent as unit quaternion. Returns ------- q : numpy.ndarray of shape (...,4) Unit quaternion (q_0, q_1, q_2, q_3) in positive real hemisphere, i.e. ǀqǀ = 1, q_0 ≥ 0. """ return self.quaternion.copy()
[docs] def as_Euler_angles(self, degrees = False): """ Represent as Bunge Euler angles. Parameters ---------- degrees : bool, optional Return angles in degrees. Returns ------- phi : numpy.ndarray of shape (...,3) Bunge Euler angles (φ_1 ∈ [0,2π], ϕ ∈ [0,π], φ_2 ∈ [0,2π]) or (φ_1 ∈ [0,360], ϕ ∈ [0,180], φ_2 ∈ [0,360]) if degrees == True. Notes ----- Bunge Euler angles correspond to a rotation axis sequence of z–x'–z''. Examples -------- Cube orientation as Bunge Euler angles. >>> import damask >>> import numpy as np >>> damask.Rotation(np.array([1,0,0,0])).as_Euler_angles() array([0., 0., 0.]) """ eu = Rotation._qu2eu(self.quaternion) if degrees: eu = np.degrees(eu) return eu
[docs] def as_axis_angle(self, degrees = False, pair = False): """ Represent as axis–angle pair. Parameters ---------- degrees : bool, optional Return rotation angle in degrees. Defaults to False. pair : bool, optional Return tuple of axis and angle. Defaults to False. Returns ------- axis_angle : numpy.ndarray of shape (...,4) or tuple ((...,3), (...)) if pair == True Axis and angle [n_1, n_2, n_3, ω] with ǀnǀ = 1 and ω ∈ [0,π] or ω ∈ [0,180] if degrees == True. Examples -------- Cube orientation as axis–angle pair. >>> import damask >>> import numpy as np >>> damask.Rotation(np.array([1,0,0,0])).as_axis_angle(pair=True) (array([0., 0., 1.]), array(0.)) """ ax = Rotation._qu2ax(self.quaternion) if degrees: ax[...,3] = np.degrees(ax[...,3]) return (ax[...,:3],ax[...,3]) if pair else ax
[docs] def as_matrix(self): """ Represent as rotation matrix. Returns ------- R : numpy.ndarray of shape (...,3,3) Rotation matrix R with det(R) = 1, R.T ∙ R = I. Examples -------- Cube orientation as rotation matrix. >>> import damask >>> import numpy as np >>> damask.Rotation(np.array([1,0,0,0])).as_matrix() array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]) """ return Rotation._qu2om(self.quaternion)
[docs] def as_Rodrigues_vector(self, compact = False): """ Represent as Rodrigues–Frank vector with separate axis and angle argument. Parameters ---------- compact : bool, optional Return three-component Rodrigues–Frank vector, i.e. axis and angle argument are not separated. Returns ------- rho : numpy.ndarray of shape (...,4) or (...,3) if compact == True Rodrigues–Frank vector [n_1, n_2, n_3, tan(ω/2)] with ǀnǀ = 1 and ω ∈ [0,π] or [n_1, n_2, n_3] with ǀnǀ = tan(ω/2) and ω ∈ [0,π] if compact == True. Examples -------- Cube orientation as three-component Rodrigues–Frank vector. >>> import damask >>> import numpy as np >>> damask.Rotation(np.array([1,0,0,0])).as_Rodrigues_vector(compact=True) array([ 0., 0., 0.]) """ ro = Rotation._qu2ro(self.quaternion) if compact: with np.errstate(invalid='ignore'): return ro[...,:3]*ro[...,3:4] else: return ro
[docs] def as_homochoric(self): """ Represent as homochoric vector. Returns ------- h : numpy.ndarray of shape (...,3) Homochoric vector (h_1, h_2, h_3) with ǀhǀ < (3/4*π)^(1/3). Examples -------- Cube orientation as homochoric vector. >>> import damask >>> import numpy as np >>> damask.Rotation(np.array([1,0,0,0])).as_homochoric() array([0., 0., 0.]) """ return Rotation._qu2ho(self.quaternion)
[docs] def as_cubochoric(self): """ Represent as cubochoric vector. Returns ------- x : numpy.ndarray of shape (...,3) Cubochoric vector (x_1, x_2, x_3) with max(x_i) < 1/2*π^(2/3). Examples -------- Cube orientation as cubochoric vector. >>> import damask >>> import numpy as np >>> damask.Rotation(np.array([1,0,0,0])).as_cubochoric() array([0., 0., 0.]) """ return Rotation._qu2cu(self.quaternion)
################################################################################################ # Static constructors. The input data needs to follow the conventions, options allow to # relax the conventions.
[docs] @staticmethod def from_quaternion(q, accept_homomorph = False, P = -1): """ Initialize from quaternion. Parameters ---------- q : numpy.ndarray of shape (...,4) Unit quaternion (q_0, q_1, q_2, q_3) in positive real hemisphere, i.e. ǀqǀ = 1, q_0 ≥ 0. accept_homomorph : boolean, optional Allow homomorphic variants, i.e. q_0 < 0 (negative real hemisphere). Defaults to False. P : int ∈ {-1,1}, optional Sign convention. Defaults to -1. """ qu = np.array(q,dtype=float) if qu.shape[:-2:-1] != (4,): raise ValueError('Invalid shape.') if abs(P) != 1: raise ValueError('P ∉ {-1,1}') qu[...,1:4] *= -P if accept_homomorph: qu[qu[...,0] < 0.0] *= -1 else: if np.any(qu[...,0] < 0.0): raise ValueError('Quaternion with negative first (real) component.') if not np.all(np.isclose(np.linalg.norm(qu,axis=-1), 1.0,rtol=0.0)): raise ValueError('Quaternion is not of unit length.') return Rotation(qu)
[docs] @staticmethod def from_Euler_angles(phi, degrees = False): """ Initialize from Bunge Euler angles. Parameters ---------- phi : numpy.ndarray of shape (...,3) Euler angles (φ_1 ∈ [0,2π], ϕ ∈ [0,π], φ_2 ∈ [0,2π]) or (φ_1 ∈ [0,360], ϕ ∈ [0,180], φ_2 ∈ [0,360]) if degrees == True. degrees : boolean, optional Euler angles are given in degrees. Defaults to False. Notes ----- Bunge Euler angles correspond to a rotation axis sequence of z–x'–z''. """ eu = np.array(phi,dtype=float) if eu.shape[:-2:-1] != (3,): raise ValueError('Invalid shape.') eu = np.radians(eu) if degrees else eu if np.any(eu < 0.0) or np.any(eu > 2.0*np.pi) or np.any(eu[...,1] > np.pi): # ToDo: No separate check for PHI raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].') return Rotation(Rotation._eu2qu(eu))
[docs] @staticmethod def from_axis_angle(axis_angle, degrees = False, normalize = False, P = -1): """ Initialize from Axis angle pair. Parameters ---------- axis_angle : numpy.ndarray of shape (...,4) Axis and angle (n_1, n_2, n_3, ω) with ǀnǀ = 1 and ω ∈ [0,π] or ω ∈ [0,180] if degrees == True. degrees : boolean, optional Angle ω is given in degrees. Defaults to False. normalize: boolean, optional Allow ǀnǀ ≠ 1. Defaults to False. P : int ∈ {-1,1}, optional Sign convention. Defaults to -1. """ ax = np.array(axis_angle,dtype=float) if ax.shape[:-2:-1] != (4,): raise ValueError('Invalid shape.') if abs(P) != 1: raise ValueError('P ∉ {-1,1}') ax[...,0:3] *= -P if degrees: ax[..., 3] = np.radians(ax[...,3]) if normalize: ax[...,0:3] /= np.linalg.norm(ax[...,0:3],axis=-1,keepdims=True) if np.any(ax[...,3] < 0.0) or np.any(ax[...,3] > np.pi): raise ValueError('Axis–angle rotation angle outside of [0..π].') if not np.all(np.isclose(np.linalg.norm(ax[...,0:3],axis=-1), 1.0)): print(np.linalg.norm(ax[...,0:3],axis=-1)) raise ValueError('Axis–angle rotation axis is not of unit length.') return Rotation(Rotation._ax2qu(ax))
[docs] @staticmethod def from_basis(basis, orthonormal = True, reciprocal = False): """ Initialize from lattice basis vectors. Parameters ---------- basis : numpy.ndarray of shape (...,3,3) Three three-dimensional lattice basis vectors. orthonormal : boolean, optional Basis is strictly orthonormal, i.e. is free of stretch components. Defaults to True. reciprocal : boolean, optional Basis vectors are given in reciprocal (instead of real) space. Defaults to False. """ om = np.array(basis,dtype=float) if om.shape[-2:] != (3,3): raise ValueError('Invalid shape.') if reciprocal: om = np.linalg.inv(tensor.transpose(om)/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.einsum('...ij,...jl',U,Vh) if not np.all(np.isclose(np.linalg.det(om),1.0)): raise ValueError('Orientation matrix has determinant ≠ 1.') if not np.all(np.isclose(np.einsum('...i,...i',om[...,0],om[...,1]), 0.0)) \ or not np.all(np.isclose(np.einsum('...i,...i',om[...,1],om[...,2]), 0.0)) \ or not np.all(np.isclose(np.einsum('...i,...i',om[...,2],om[...,0]), 0.0)): raise ValueError('Orientation matrix is not orthogonal.') return Rotation(Rotation._om2qu(om))
[docs] @staticmethod def from_matrix(R): """ Initialize from rotation matrix. Parameters ---------- R : numpy.ndarray of shape (...,3,3) Rotation matrix with det(R) = 1, R.T ∙ R = I. """ return Rotation.from_basis(R)
[docs] @staticmethod def from_parallel(a,b, **kwargs): """ Initialize from pairs of two orthogonal lattice basis vectors. Parameters ---------- a : numpy.ndarray of shape (...,2,3) Two three-dimensional lattice vectors of first orthogonal basis. b : numpy.ndarray of shape (...,2,3) Corresponding three-dimensional lattice vectors of second basis. """ a_ = np.array(a) b_ = np.array(b) if a_.shape[-2:] != (2,3) or b_.shape[-2:] != (2,3) or a_.shape != b_.shape: raise ValueError('Invalid shape.') am = np.stack([ a_[...,0,:], a_[...,1,:], np.cross(a_[...,0,:],a_[...,1,:]) ],axis=-2) bm = np.stack([ b_[...,0,:], b_[...,1,:], np.cross(b_[...,0,:],b_[...,1,:]) ],axis=-2) return Rotation.from_basis(np.swapaxes(am/np.linalg.norm(am,axis=-1,keepdims=True),-1,-2))\ .misorientation(Rotation.from_basis(np.swapaxes(bm/np.linalg.norm(bm,axis=-1,keepdims=True),-1,-2)))
[docs] @staticmethod def from_Rodrigues_vector(rho, normalize = False, P = -1): """ Initialize from Rodrigues–Frank vector (angle separated from axis). Parameters ---------- rho : numpy.ndarray of shape (...,4) Rodrigues–Frank vector (n_1, n_2, n_3, tan(ω/2)) with ǀnǀ = 1 and ω ∈ [0,π]. normalize : boolean, optional Allow ǀnǀ ≠ 1. Defaults to False. P : int ∈ {-1,1}, optional Sign convention. Defaults to -1. """ ro = np.array(rho,dtype=float) if ro.shape[:-2:-1] != (4,): raise ValueError('Invalid shape.') if abs(P) != 1: raise ValueError('P ∉ {-1,1}') ro[...,0:3] *= -P if normalize: ro[...,0:3] /= np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True) if np.any(ro[...,3] < 0.0): raise ValueError('Rodrigues vector rotation angle is negative.') if not np.all(np.isclose(np.linalg.norm(ro[...,0:3],axis=-1), 1.0)): raise ValueError('Rodrigues vector rotation axis is not of unit length.') return Rotation(Rotation._ro2qu(ro))
[docs] @staticmethod def from_homochoric(h, P = -1): """ Initialize from homochoric vector. Parameters ---------- h : numpy.ndarray of shape (...,3) Homochoric vector (h_1, h_2, h_3) with ǀhǀ < (3/4*π)^(1/3). P : int ∈ {-1,1}, optional Sign convention. Defaults to -1. """ ho = np.array(h,dtype=float) if ho.shape[:-2:-1] != (3,): raise ValueError('Invalid shape.') if abs(P) != 1: raise ValueError('P ∉ {-1,1}') ho *= -P if np.any(np.linalg.norm(ho,axis=-1) >_R1+1e-9): raise ValueError('Homochoric coordinate outside of the sphere.') return Rotation(Rotation._ho2qu(ho))
[docs] @staticmethod def from_cubochoric(x, P = -1): """ Initialize from cubochoric vector. Parameters ---------- x : numpy.ndarray of shape (...,3) Cubochoric vector (x_1, x_2, x_3) with max(x_i) < 1/2*π^(2/3). P : int ∈ {-1,1}, optional Sign convention. Defaults to -1. """ cu = np.array(x,dtype=float) if cu.shape[:-2:-1] != (3,): raise ValueError('Invalid shape.') if abs(P) != 1: raise ValueError('P ∉ {-1,1}') if np.abs(np.max(cu)) > np.pi**(2./3.) * 0.5+1e-9: raise ValueError('Cubochoric coordinate outside of the cube.') ho = -P * Rotation._cu2ho(cu) return Rotation(Rotation._ho2qu(ho))
[docs] @staticmethod def from_random(shape = None, rng_seed = None): """ Initialize with random rotation. Rotations are uniformly distributed. Parameters ---------- shape : tuple of ints, optional Shape of the sample. Defaults to None, which gives a single rotation. rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy will be pulled from the OS. """ rng = np.random.default_rng(rng_seed) r = rng.random(3 if shape is None else tuple(shape)+(3,) if hasattr(shape, '__iter__') else (shape,3)) A = np.sqrt(r[...,2]) B = np.sqrt(1.0-r[...,2]) q = np.stack([np.cos(2.0*np.pi*r[...,0])*A, np.sin(2.0*np.pi*r[...,1])*B, np.cos(2.0*np.pi*r[...,1])*B, np.sin(2.0*np.pi*r[...,0])*A],axis=-1) return Rotation(q if shape is None else q.reshape(r.shape[:-1]+(4,)))._standardize()
[docs] @staticmethod def from_ODF(weights, phi, N = 500, degrees = True, fractions = True, rng_seed = None, **kwargs): """ Sample discrete values from a binned orientation distribution function (ODF). Parameters ---------- weights : numpy.ndarray of shape (n) Texture intensity values (probability density or volume fraction) at Euler space grid points. phi : numpy.ndarray of shape (n,3) Grid coordinates in Euler space at which weights are defined. N : integer, optional Number of discrete orientations to be sampled from the given ODF. Defaults to 500. degrees : boolean, optional Euler space grid coordinates are in degrees. Defaults to True. fractions : boolean, optional ODF values correspond to volume fractions, not probability densities. Defaults to True. rng_seed: {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy will be pulled from the OS. Returns ------- samples : damask.Rotation of shape (N) Array of sampled rotations closely representing the input ODF. Notes ----- Due to the distortion of Euler space in the vicinity of ϕ = 0, probability densities, p, defined on grid points with ϕ = 0 will never result in reconstructed orientations as their dV/V = p dγ = p × 0. Hence, it is recommended to transform any such dataset to cell centers that avoid grid points at ϕ = 0. References ---------- P. Eisenlohr and F. Roters, Computational Materials Science 42(4):670-678, 2008 https://doi.org/10.1016/j.commatsci.2007.09.015 """ def _dg(eu,deg): """Return infinitesimal Euler space volume of bin(s).""" phi_sorted = eu[np.lexsort((eu[:,0],eu[:,1],eu[:,2]))] steps,size,_ = grid_filters.cellsSizeOrigin_coordinates0_point(phi_sorted) delta = np.radians(size/steps) if deg else size/steps return delta[0]*2.0*np.sin(delta[1]/2.0)*delta[2] / 8.0 / np.pi**2 * np.sin(np.radians(eu[:,1]) if deg else eu[:,1]) dg = 1.0 if fractions else _dg(phi,degrees) dV_V = dg * np.maximum(0.0,weights.squeeze()) return Rotation.from_Euler_angles(phi[util.hybrid_IA(dV_V,N,rng_seed)],degrees)
[docs] @staticmethod def from_spherical_component(center, sigma, N = 500, degrees = True, rng_seed = None): """ Calculate set of rotations with Gaussian distribution around center. Parameters ---------- center : Rotation Central Rotation. sigma : float Standard deviation of (Gaussian) misorientation distribution. N : int, optional Number of samples. Defaults to 500. degrees : boolean, optional sigma is given in degrees. Defaults to True. rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy will be pulled from the OS. """ rng = np.random.default_rng(rng_seed) sigma = np.radians(sigma) if degrees else sigma u,Theta = (rng.random((N,2)) * 2.0 * np.array([1,np.pi]) - np.array([1.0, 0])).T omega = abs(rng.normal(scale=sigma,size=N)) p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta), np.sqrt(1-u**2)*np.sin(Theta), u, omega]) return Rotation.from_axis_angle(p) * center
[docs] @staticmethod def from_fiber_component(alpha, beta, sigma = 0.0, N = 500, degrees = True, rng_seed = None): """ Calculate set of rotations with Gaussian distribution around direction. Parameters ---------- alpha : numpy.ndarray of shape (2) Polar coordinates (phi from x, theta from z) of fiber direction in crystal frame. beta : numpy.ndarray of shape (2) Polar coordinates (phi from x, theta from z) of fiber direction in sample frame. sigma : float, optional Standard deviation of (Gaussian) misorientation distribution. Defaults to 0. N : int, optional Number of samples. Defaults to 500. degrees : boolean, optional sigma, alpha, and beta are given in degrees. rng_seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional A seed to initialize the BitGenerator. Defaults to None, i.e. unpredictable entropy will be pulled from the OS. """ rng = np.random.default_rng(rng_seed) sigma_,alpha_,beta_ = map(np.radians,(sigma,alpha,beta)) if degrees else (sigma,alpha,beta) d_cr = np.array([np.sin(alpha_[0])*np.cos(alpha_[1]), np.sin(alpha_[0])*np.sin(alpha_[1]), np.cos(alpha_[0])]) d_lab = np.array([np.sin( beta_[0])*np.cos( beta_[1]), np.sin( beta_[0])*np.sin( beta_[1]), np.cos( beta_[0])]) ax_align = np.append(np.cross(d_lab,d_cr), np.arccos(np.dot(d_lab,d_cr))) if np.isclose(ax_align[3],0.0): ax_align[:3] = np.array([1,0,0]) R_align = Rotation.from_axis_angle(ax_align if ax_align[3] > 0.0 else -ax_align,normalize=True) # rotate fiber axis from sample to crystal frame u,Theta = (rng.random((N,2)) * 2.0 * np.array([1,np.pi]) - np.array([1.0, 0])).T omega = abs(rng.normal(scale=sigma_,size=N)) p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta), np.sqrt(1-u**2)*np.sin(Theta), u, omega]) p[:,:3] = np.einsum('ij,...j',np.eye(3)-np.outer(d_lab,d_lab),p[:,:3]) # remove component along fiber axis f = np.column_stack((np.broadcast_to(d_lab,(N,3)),rng.random(N)*np.pi)) f[::2,:3] *= -1 # flip half the rotation axes to negative sense return R_align.broadcast_to(N) \ * Rotation.from_axis_angle(p,normalize=True) \ * Rotation.from_axis_angle(f)
#################################################################################################### # Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations #################################################################################################### # Copyright (c) 2017-2020, 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. #################################################################################################### #---------- Quaternion ---------- @staticmethod def _qu2om(qu): qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2) om = np.block([qq + 2.0*qu[...,1:2]**2, 2.0*(qu[...,2:3]*qu[...,1:2]-_P*qu[...,0:1]*qu[...,3:4]), 2.0*(qu[...,3:4]*qu[...,1:2]+_P*qu[...,0:1]*qu[...,2:3]), 2.0*(qu[...,1:2]*qu[...,2:3]+_P*qu[...,0:1]*qu[...,3:4]), qq + 2.0*qu[...,2:3]**2, 2.0*(qu[...,3:4]*qu[...,2:3]-_P*qu[...,0:1]*qu[...,1:2]), 2.0*(qu[...,1:2]*qu[...,3:4]-_P*qu[...,0:1]*qu[...,2:3]), 2.0*(qu[...,2:3]*qu[...,3:4]+_P*qu[...,0:1]*qu[...,1:2]), qq + 2.0*qu[...,3:4]**2, ]).reshape(qu.shape[:-1]+(3,3)) return om @staticmethod def _qu2eu(qu): """Quaternion to Bunge Euler angles.""" q02 = qu[...,0:1]*qu[...,2:3] q13 = qu[...,1:2]*qu[...,3:4] q01 = qu[...,0:1]*qu[...,1:2] q23 = qu[...,2:3]*qu[...,3:4] q03_s = qu[...,0:1]**2+qu[...,3:4]**2 q12_s = qu[...,1:2]**2+qu[...,2:3]**2 chi = np.sqrt(q03_s*q12_s) eu = np.where(np.abs(q12_s) < 1.0e-8, np.block([np.arctan2(-_P*2.0*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2), np.zeros(qu.shape[:-1]+(2,))]), np.where(np.abs(q03_s) < 1.0e-8, np.block([np.arctan2( 2.0*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2), np.broadcast_to(np.pi,qu[...,0:1].shape), np.zeros(qu.shape[:-1]+(1,))]), np.block([np.arctan2((-_P*q02+q13)*chi, (-_P*q01-q23)*chi), np.arctan2( 2.0*chi, q03_s-q12_s ), np.arctan2(( _P*q02+q13)*chi, (-_P*q01+q23)*chi)]) ) ) # reduce Euler angles to definition range eu[np.abs(eu)<1.e-6] = 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) # needed? return eu @staticmethod def _qu2ax(qu): """ Quaternion to axis–angle pair. Modified version of the original formulation, should be numerically more stable. """ with np.errstate(invalid='ignore',divide='ignore'): s = np.sign(qu[...,0:1])/np.sqrt(qu[...,1:2]**2+qu[...,2:3]**2+qu[...,3:4]**2) omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0)) ax = np.where(np.broadcast_to(qu[...,0:1] < 1.0e-8,qu.shape), np.block([qu[...,1:4],np.broadcast_to(np.pi,qu[...,0:1].shape)]), np.block([qu[...,1:4]*s,omega])) ax[np.isclose(qu[...,0],1.,rtol=0.0)] = [0.0, 0.0, 1.0, 0.0] return ax @staticmethod def _qu2ro(qu): """Quaternion to Rodrigues–Frank vector.""" with np.errstate(invalid='ignore',divide='ignore'): s = np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) ro = np.where(np.broadcast_to(np.abs(qu[...,0:1]) < 1.0e-12,qu.shape), np.block([qu[...,1:2], qu[...,2:3], qu[...,3:4], np.broadcast_to(np.inf,qu[...,0:1].shape)]), np.block([qu[...,1:2]/s,qu[...,2:3]/s,qu[...,3:4]/s, np.tan(np.arccos(np.clip(qu[...,0:1],-1.0,1.0))) ]) ) ro[np.abs(s).squeeze(-1) < 1.0e-12] = [0.0,0.0,_P,0.0] return ro @staticmethod def _qu2ho(qu): """Quaternion to homochoric vector.""" with np.errstate(invalid='ignore'): omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0)) ho = np.where(np.abs(omega) < 1.0e-12, np.zeros(3), qu[...,1:4]/np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) \ * (0.75*(omega - np.sin(omega)))**(1./3.)) return ho @staticmethod def _qu2cu(qu): """Quaternion to cubochoric vector.""" return Rotation._ho2cu(Rotation._qu2ho(qu)) #---------- Rotation matrix ---------- @staticmethod def _om2qu(om): """ Rotation matrix to quaternion. This formulation is from www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion. The original formulation had issues. """ trace = om[...,0,0:1]+om[...,1,1:2]+om[...,2,2:3] with np.errstate(invalid='ignore',divide='ignore'): s = [ 0.5 / np.sqrt( 1.0 + trace), 2.0 * np.sqrt( 1.0 + om[...,0,0:1] - om[...,1,1:2] - om[...,2,2:3]), 2.0 * np.sqrt( 1.0 + om[...,1,1:2] - om[...,2,2:3] - om[...,0,0:1]), 2.0 * np.sqrt( 1.0 + om[...,2,2:3] - om[...,0,0:1] - om[...,1,1:2] ) ] qu= np.where(trace>0, np.block([0.25 / s[0], (om[...,2,1:2] - om[...,1,2:3] ) * s[0], (om[...,0,2:3] - om[...,2,0:1] ) * s[0], (om[...,1,0:1] - om[...,0,1:2] ) * s[0]]), np.where(om[...,0,0:1] > np.maximum(om[...,1,1:2],om[...,2,2:3]), np.block([(om[...,2,1:2] - om[...,1,2:3]) / s[1], 0.25 * s[1], (om[...,0,1:2] + om[...,1,0:1]) / s[1], (om[...,0,2:3] + om[...,2,0:1]) / s[1]]), np.where(om[...,1,1:2] > om[...,2,2:3], np.block([(om[...,0,2:3] - om[...,2,0:1]) / s[2], (om[...,0,1:2] + om[...,1,0:1]) / s[2], 0.25 * s[2], (om[...,1,2:3] + om[...,2,1:2]) / s[2]]), np.block([(om[...,1,0:1] - om[...,0,1:2]) / s[3], (om[...,0,2:3] + om[...,2,0:1]) / s[3], (om[...,1,2:3] + om[...,2,1:2]) / s[3], 0.25 * s[3]]), ) ) )*np.array([1,_P,_P,_P]) qu[qu[...,0]<0] *=-1 return qu @staticmethod def _om2eu(om): """Rotation matrix to Bunge Euler angles.""" with np.errstate(invalid='ignore',divide='ignore'): zeta = 1.0/np.sqrt(1.0-om[...,2,2:3]**2) eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.0,0.0), np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]), np.pi*0.5*(1-om[...,2,2:3]), np.zeros(om.shape[:-2]+(1,)), ]), np.block([np.arctan2(om[...,2,0:1]*zeta,-om[...,2,1:2]*zeta), np.arccos( om[...,2,2:3]), np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta) ]) ) eu[np.abs(eu)<1.e-8] = 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 @staticmethod def _om2ax(om): """Rotation matrix to axis–angle pair.""" diag_delta = -_P*np.block([om[...,1,2:3]-om[...,2,1:2], om[...,2,0:1]-om[...,0,2:3], om[...,0,1:2]-om[...,1,0:1] ]) t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.0).reshape(om.shape[:-2]+(1,)) w,vr = np.linalg.eig(om) # mask duplicated real eigenvalues w[np.isclose(w[...,0],1.0+0.0j),1:] = 0. w[np.isclose(w[...,1],1.0+0.0j),2:] = 0. vr = np.swapaxes(vr,-1,-2) ax = np.where(np.abs(diag_delta)<1e-12, np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,)), np.abs(np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,))) \ *np.sign(diag_delta)) ax = np.block([ax,np.arccos(np.clip(t,-1.0,1.0))]) ax[np.abs(ax[...,3])<1.e-8] = [ 0.0, 0.0, 1.0, 0.0] return ax @staticmethod def _om2ro(om): """Rotation matrix to Rodrigues–Frank vector.""" return Rotation._eu2ro(Rotation._om2eu(om)) @staticmethod def _om2ho(om): """Rotation matrix to homochoric vector.""" return Rotation._ax2ho(Rotation._om2ax(om)) @staticmethod def _om2cu(om): """Rotation matrix to cubochoric vector.""" return Rotation._ho2cu(Rotation._om2ho(om)) #---------- Bunge Euler angles ---------- @staticmethod def _eu2qu(eu): """Bunge Euler angles to quaternion.""" ee = 0.5*eu cPhi = np.cos(ee[...,1:2]) sPhi = np.sin(ee[...,1:2]) qu = np.block([ cPhi*np.cos(ee[...,0:1]+ee[...,2:3]), -_P*sPhi*np.cos(ee[...,0:1]-ee[...,2:3]), -_P*sPhi*np.sin(ee[...,0:1]-ee[...,2:3]), -_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])]) qu[qu[...,0]<0.0]*=-1 return qu @staticmethod def _eu2om(eu): """Bunge Euler angles to rotation matrix.""" c = np.cos(eu) s = np.sin(eu) om = np.block([+c[...,0:1]*c[...,2:3]-s[...,0:1]*s[...,2:3]*c[...,1:2], +s[...,0:1]*c[...,2:3]+c[...,0:1]*s[...,2:3]*c[...,1:2], +s[...,2:3]*s[...,1:2], -c[...,0:1]*s[...,2:3]-s[...,0:1]*c[...,2:3]*c[...,1:2], -s[...,0:1]*s[...,2:3]+c[...,0:1]*c[...,2:3]*c[...,1:2], +c[...,2:3]*s[...,1:2], +s[...,0:1]*s[...,1:2], -c[...,0:1]*s[...,1:2], +c[...,1:2] ]).reshape(eu.shape[:-1]+(3,3)) om[np.abs(om)<1.e-12] = 0.0 return om @staticmethod def _eu2ax(eu): """Bunge Euler angles to axis–angle pair.""" t = np.tan(eu[...,1:2]*0.5) sigma = 0.5*(eu[...,0:1]+eu[...,2:3]) delta = 0.5*(eu[...,0:1]-eu[...,2:3]) tau = np.linalg.norm(np.block([t,np.sin(sigma)]),axis=-1,keepdims=True) alpha = np.where(np.abs(np.cos(sigma))<1.e-12,np.pi,2.0*np.arctan(tau/np.cos(sigma))) with np.errstate(invalid='ignore',divide='ignore'): ax = np.where(np.broadcast_to(np.abs(alpha)<1.0e-12,eu.shape[:-1]+(4,)), [0.0,0.0,1.0,0.0], np.block([-_P/tau*t*np.cos(delta), -_P/tau*t*np.sin(delta), -_P/tau* np.sin(sigma), alpha ])) ax[(alpha<0.0).squeeze()] *=-1 return ax @staticmethod def _eu2ro(eu): """Bunge Euler angles to Rodrigues–Frank vector.""" ax = Rotation._eu2ax(eu) ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)]) ro[ax[...,3]>=np.pi,3] = np.inf ro[np.abs(ax[...,3])<1.e-16] = [ 0.0, 0.0, _P, 0.0 ] return ro @staticmethod def _eu2ho(eu): """Bunge Euler angles to homochoric vector.""" return Rotation._ax2ho(Rotation._eu2ax(eu)) @staticmethod def _eu2cu(eu): """Bunge Euler angles to cubochoric vector.""" return Rotation._ho2cu(Rotation._eu2ho(eu)) #---------- Axis angle pair ---------- @staticmethod def _ax2qu(ax): """Axis–angle pair to quaternion.""" c = np.cos(ax[...,3:4]*.5) s = np.sin(ax[...,3:4]*.5) qu = np.where(np.abs(ax[...,3:4])<1.e-6,[1.0, 0.0, 0.0, 0.0],np.block([c, ax[...,:3]*s])) return qu @staticmethod def _ax2om(ax): """Axis-angle pair to rotation matrix.""" c = np.cos(ax[...,3:4]) s = np.sin(ax[...,3:4]) omc = 1. -c om = np.block([c+omc*ax[...,0:1]**2, omc*ax[...,0:1]*ax[...,1:2] + s*ax[...,2:3], omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2], omc*ax[...,0:1]*ax[...,1:2] - s*ax[...,2:3], c+omc*ax[...,1:2]**2, omc*ax[...,1:2]*ax[...,2:3] + s*ax[...,0:1], omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2], omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1], c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3)) return om if _P < 0.0 else np.swapaxes(om,-1,-2) @staticmethod def _ax2eu(ax): """Rotation matrix to Bunge Euler angles.""" return Rotation._om2eu(Rotation._ax2om(ax)) @staticmethod def _ax2ro(ax): """Axis–angle pair to Rodrigues–Frank vector.""" ro = np.block([ax[...,:3], np.where(np.isclose(ax[...,3:4],np.pi,atol=1.e-15,rtol=.0), np.inf, np.tan(ax[...,3:4]*0.5)) ]) ro[np.abs(ax[...,3])<1.e-6] = [.0,.0,_P,.0] return ro @staticmethod def _ax2ho(ax): """Axis–angle pair to homochoric vector.""" f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1.0/3.0) ho = ax[...,:3] * f return ho @staticmethod def _ax2cu(ax): """Axis–angle pair to cubochoric vector.""" return Rotation._ho2cu(Rotation._ax2ho(ax)) #---------- Rodrigues-Frank vector ---------- @staticmethod def _ro2qu(ro): """Rodrigues–Frank vector to quaternion.""" return Rotation._ax2qu(Rotation._ro2ax(ro)) @staticmethod def _ro2om(ro): """Rodgrigues–Frank vector to rotation matrix.""" return Rotation._ax2om(Rotation._ro2ax(ro)) @staticmethod def _ro2eu(ro): """Rodrigues–Frank vector to Bunge Euler angles.""" return Rotation._om2eu(Rotation._ro2om(ro)) @staticmethod def _ro2ax(ro): """Rodrigues–Frank vector to axis–angle pair.""" with np.errstate(invalid='ignore',divide='ignore'): ax = np.where(np.isfinite(ro[...,3:4]), np.block([ro[...,0:3]*np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True),2.*np.arctan(ro[...,3:4])]), np.block([ro[...,0:3],np.broadcast_to(np.pi,ro[...,3:4].shape)])) ax[np.abs(ro[...,3]) < 1.e-8] = np.array([ 0.0, 0.0, 1.0, 0.0 ]) return ax @staticmethod def _ro2ho(ro): """Rodrigues–Frank vector to homochoric vector.""" f = np.where(np.isfinite(ro[...,3:4]),2.0*np.arctan(ro[...,3:4]) -np.sin(2.0*np.arctan(ro[...,3:4])),np.pi) ho = np.where(np.broadcast_to(np.sum(ro[...,0:3]**2.0,axis=-1,keepdims=True) < 1.e-8,ro[...,0:3].shape), np.zeros(3), ro[...,0:3]* (0.75*f)**(1.0/3.0)) return ho @staticmethod def _ro2cu(ro): """Rodrigues–Frank vector to cubochoric vector.""" return Rotation._ho2cu(Rotation._ro2ho(ro)) #---------- Homochoric vector---------- @staticmethod def _ho2qu(ho): """Homochoric vector to quaternion.""" return Rotation._ax2qu(Rotation._ho2ax(ho)) @staticmethod def _ho2om(ho): """Homochoric vector to rotation matrix.""" return Rotation._ax2om(Rotation._ho2ax(ho)) @staticmethod def _ho2eu(ho): """Homochoric vector to Bunge Euler angles.""" return Rotation._ax2eu(Rotation._ho2ax(ho)) @staticmethod 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]) hmag_squared = np.sum(ho**2.,axis=-1,keepdims=True) hm = hmag_squared.copy() s = tfit[0] + tfit[1] * hmag_squared for i in range(2,16): hm *= hmag_squared s += tfit[i] * hm with np.errstate(invalid='ignore'): ax = np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-8,ho.shape[:-1]+(4,)), [ 0.0, 0.0, 1.0, 0.0 ], np.block([ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))])) return ax @staticmethod def _ho2ro(ho): """Axis–angle pair to Rodrigues–Frank vector.""" return Rotation._ax2ro(Rotation._ho2ax(ho)) @staticmethod def _ho2cu(ho): """ Homochoric vector to cubochoric vector. References ---------- D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014 https://doi.org/10.1088/0965-0393/22/7/075013 """ rs = np.linalg.norm(ho,axis=-1,keepdims=True) xyz3 = np.take_along_axis(ho,Rotation._get_pyramid_order(ho,'forward'),-1) with np.errstate(invalid='ignore',divide='ignore'): # inverse M_3 xyz2 = xyz3[...,0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[...,2:3])) ) qxy = np.sum(xyz2**2,axis=-1,keepdims=True) q2 = qxy + np.max(np.abs(xyz2),axis=-1,keepdims=True)**2 sq2 = np.sqrt(q2) q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2)) tt = np.clip((np.min(np.abs(xyz2),axis=-1,keepdims=True)**2\ +np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0) T_inv = np.where(np.abs(xyz2[...,1:2]) <= np.abs(xyz2[...,0:1]), np.block([np.ones_like(tt),np.arccos(tt)/np.pi*12.0]), np.block([np.arccos(tt)/np.pi*12.0,np.ones_like(tt)]))*q T_inv[xyz2<0.0] *= -1.0 T_inv[np.broadcast_to(np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-12),T_inv.shape)] = 0.0 cu = np.block([T_inv, np.where(xyz3[...,2:3]<0.0,-np.ones_like(xyz3[...,2:3]),np.ones_like(xyz3[...,2:3])) \ * rs/np.sqrt(6.0/np.pi), ])/ _sc cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0 cu = np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1) return cu #---------- Cubochoric ---------- @staticmethod def _cu2qu(cu): """Cubochoric vector to quaternion.""" return Rotation._ho2qu(Rotation._cu2ho(cu)) @staticmethod def _cu2om(cu): """Cubochoric vector to rotation matrix.""" return Rotation._ho2om(Rotation._cu2ho(cu)) @staticmethod def _cu2eu(cu): """Cubochoric vector to Bunge Euler angles.""" return Rotation._ho2eu(Rotation._cu2ho(cu)) @staticmethod def _cu2ax(cu): """Cubochoric vector to axis–angle pair.""" return Rotation._ho2ax(Rotation._cu2ho(cu)) @staticmethod def _cu2ro(cu): """Cubochoric vector to Rodrigues–Frank vector.""" return Rotation._ho2ro(Rotation._cu2ho(cu)) @staticmethod def _cu2ho(cu): """ Cubochoric vector to homochoric vector. References ---------- D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014 https://doi.org/10.1088/0965-0393/22/7/075013 """ with np.errstate(invalid='ignore',divide='ignore'): # get pyramide and scale by grid parameter ratio XYZ = np.take_along_axis(cu,Rotation._get_pyramid_order(cu,'forward'),-1) * _sc order = np.abs(XYZ[...,1:2]) <= np.abs(XYZ[...,0:1]) q = np.pi/12.0 * np.where(order,XYZ[...,1:2],XYZ[...,0:1]) \ / np.where(order,XYZ[...,0:1],XYZ[...,1:2]) c = np.cos(q) s = np.sin(q) q = _R1*2.0**0.25/_beta/ np.sqrt(np.sqrt(2.0)-c) \ * np.where(order,XYZ[...,0:1],XYZ[...,1:2]) T = np.block([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q # transform to sphere grid (inverse Lambert) c = np.sum(T**2,axis=-1,keepdims=True) s = c * np.pi/24.0 /XYZ[...,2:3]**2 c = c * np.sqrt(np.pi/24.0)/XYZ[...,2:3] q = np.sqrt( 1.0 - s) ho = np.where(np.isclose(np.sum(np.abs(XYZ[...,0:2]),axis=-1,keepdims=True),0.0,rtol=0.0,atol=1.0e-16), np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6.0/np.pi) *XYZ[...,2:3]]), np.block([np.where(order,T[...,0:1],T[...,1:2])*q, np.where(order,T[...,1:2],T[...,0:1])*q, np.sqrt(6.0/np.pi) * XYZ[...,2:3] - c]) ) ho[np.isclose(np.sum(np.abs(cu),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0 ho = np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1) return ho @staticmethod def _get_pyramid_order(xyz,direction=None): """ Get order of the coordinates. Depending on the pyramid in which the point is located, the order need to be adjusted. Parameters ---------- xyz : numpy.ndarray Coordinates of a point on a uniform refinable grid on a ball or in a uniform refinable cubical grid. References ---------- D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014 https://doi.org/10.1088/0965-0393/22/7/075013 """ order = {'forward': np.array([[0,1,2],[1,2,0],[2,0,1]]), 'backward':np.array([[0,1,2],[2,0,1],[1,2,0]])} p = np.where(np.maximum(np.abs(xyz[...,0]),np.abs(xyz[...,1])) <= np.abs(xyz[...,2]),0, np.where(np.maximum(np.abs(xyz[...,1]),np.abs(xyz[...,2])) <= np.abs(xyz[...,0]),1,2)) return order[direction][p]