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Merge branch '225-tolerate-non-unit-quaternion' into 'development' 

add option to normalize quaternions

Closes #225

See merge request damask/DAMASK!657
This commit is contained in:
Martin Diehl 2022-11-18 09:32:18 +00:00
parent 4e13a28ecf ce4b373f9c
commit d596de7a32
2 changed files with 181 additions and 187 deletions
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300
python/damask/_rotation.py
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@ -66,7 +66,7 @@ class Rotation:
__slots__ = ['quaternion'] __slots__ = ['quaternion']
def __init__(self, def __init__(self,
rotation: Union[FloatSequence, 'Rotation'] = np.array([1.0,0.0,0.0,0.0])): rotation: Union[FloatSequence, 'Rotation'] = np.array([1.,0.,0.,0.])):
""" """
New rotation. New rotation.
@ -82,7 +82,7 @@ class Rotation:
if isinstance(rotation,Rotation): if isinstance(rotation,Rotation):
self.quaternion = rotation.quaternion.copy() self.quaternion = rotation.quaternion.copy()
elif np.array(rotation).shape[-1] == 4: elif np.array(rotation).shape[-1] == 4:
self.quaternion = np.array(rotation) self.quaternion = np.array(rotation,dtype=float)
else: else:
raise TypeError('"rotation" is neither a Rotation nor a quaternion') raise TypeError('"rotation" is neither a Rotation nor a quaternion')
@ -141,7 +141,7 @@ class Rotation:
""" """
return NotImplemented if not isinstance(other, Rotation) else \ return NotImplemented if not isinstance(other, Rotation) else \
np.logical_or(np.all(self.quaternion == other.quaternion,axis=-1), np.logical_or(np.all(self.quaternion == other.quaternion,axis=-1),
np.all(self.quaternion == -1.0*other.quaternion,axis=-1)) np.all(self.quaternion == -1.*other.quaternion,axis=-1))
def __ne__(self, def __ne__(self,
@ -161,8 +161,8 @@ class Rotation:
def isclose(self: MyType, def isclose(self: MyType,
other: MyType, other: MyType,
rtol: float = 1e-5, rtol: float = 1.e-5,
atol: float = 1e-8, atol: float = 1.e-8,
equal_nan: bool = True) -> bool: equal_nan: bool = True) -> bool:
""" """
Report where values are approximately equal to corresponding ones of other Rotation. Report where values are approximately equal to corresponding ones of other Rotation.
@ -187,13 +187,13 @@ class Rotation:
s = self.quaternion s = self.quaternion
o = other.quaternion o = other.quaternion
return np.logical_or(np.all(np.isclose(s, o,rtol,atol,equal_nan),axis=-1), 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)) np.all(np.isclose(s,-1.*o,rtol,atol,equal_nan),axis=-1))
def allclose(self: MyType, def allclose(self: MyType,
other: MyType, other: MyType,
rtol: float = 1e-5, rtol: float = 1.e-5,
atol: float = 1e-8, atol: float = 1.e-8,
equal_nan: bool = True) -> Union[np.bool_, bool]: equal_nan: bool = True) -> Union[np.bool_, bool]:
""" """
Test whether all values are approximately equal to corresponding ones of other Rotation. Test whether all values are approximately equal to corresponding ones of other Rotation.
@ -250,7 +250,7 @@ class Rotation:
""" """
dup = self.copy() dup = self.copy()
dup.quaternion[...,1:] *= -1 dup.quaternion[...,1:] *= -1.
return dup return dup
@ -393,9 +393,9 @@ class Rotation:
if self.shape + (3,) == other.shape: if self.shape + (3,) == other.shape:
q_m = self.quaternion[...,0] q_m = self.quaternion[...,0]
p_m = self.quaternion[...,1:] p_m = self.quaternion[...,1:]
A = q_m**2.0 - np.einsum('...i,...i',p_m,p_m) A = q_m**2 - np.einsum('...i,...i',p_m,p_m)
B = 2.0 * np.einsum('...i,...i',p_m,other) B = 2. * np.einsum('...i,...i',p_m,other)
C = 2.0 * _P * q_m C = 2. * _P * q_m
return np.block([(A * other[...,i]).reshape(self.shape+(1,)) + return np.block([(A * other[...,i]).reshape(self.shape+(1,)) +
(B * p_m[...,i]).reshape(self.shape+(1,)) + (B * p_m[...,i]).reshape(self.shape+(1,)) +
(C * ( p_m[...,(i+1)%3]*other[...,(i+2)%3]\ (C * ( p_m[...,(i+1)%3]*other[...,(i+2)%3]\
@ -419,7 +419,7 @@ class Rotation:
def _standardize(self: MyType) -> MyType: def _standardize(self: MyType) -> MyType:
"""Standardize quaternion (ensure positive real hemisphere).""" """Standardize quaternion (ensure positive real hemisphere)."""
self.quaternion[self.quaternion[...,0] < 0.0] *= -1 self.quaternion[self.quaternion[...,0] < 0.] *= -1.
return self return self
@ -540,7 +540,7 @@ class Rotation:
weights_ = np.ones(self.shape,dtype=float) if weights is None else np.array(weights,float) weights_ = np.ones(self.shape,dtype=float) if weights is None else np.array(weights,float)
eig, vec = np.linalg.eig(np.sum(_M(self.quaternion) * weights_[...,np.newaxis,np.newaxis],axis=-3) \ 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)) /np.sum( weights_[...,np.newaxis,np.newaxis],axis=-3))
return self.copy(Rotation.from_quaternion(np.real( return self.copy(Rotation.from_quaternion(np.real(
@ -751,6 +751,7 @@ class Rotation:
@staticmethod @staticmethod
def from_quaternion(q: Union[Sequence[FloatSequence], np.ndarray], def from_quaternion(q: Union[Sequence[FloatSequence], np.ndarray],
accept_homomorph: bool = False, accept_homomorph: bool = False,
normalize: bool = False,
P: Literal[1, -1] = -1) -> 'Rotation': P: Literal[1, -1] = -1) -> 'Rotation':
""" """
Initialize from quaternion. Initialize from quaternion.
@ -762,6 +763,8 @@ class Rotation:
accept_homomorph : bool, optional accept_homomorph : bool, optional
Allow homomorphic variants, i.e. q_0 < 0 (negative real hemisphere). Allow homomorphic variants, i.e. q_0 < 0 (negative real hemisphere).
Defaults to False. Defaults to False.
normalize: bool, optional
Allow ǀqǀ ≠ 1. Defaults to False.
P : int ∈ {-1,1}, optional P : int ∈ {-1,1}, optional
Sign convention. Defaults to -1. Sign convention. Defaults to -1.
@ -773,11 +776,12 @@ class Rotation:
qu[...,1:4] *= -P qu[...,1:4] *= -P
if accept_homomorph: if accept_homomorph:
qu[qu[...,0] < 0.0] *= -1 qu[qu[...,0]<0.] *= -1.
elif np.any(qu[...,0] < 0.0): elif np.any(qu[...,0] < 0.):
raise ValueError('quaternion with negative first (real) component') raise ValueError('quaternion with negative first (real) component')
if normalize:
if not np.allclose(np.linalg.norm(qu,axis=-1), 1.0,rtol=1e-8): qu /= np.linalg.norm(qu,axis=-1,keepdims=True)
elif not np.allclose(np.linalg.norm(qu,axis=-1),1.,rtol=1.e-8):
raise ValueError('quaternion is not of unit length') raise ValueError('quaternion is not of unit length')
return Rotation(qu) return Rotation(qu)
@ -805,7 +809,7 @@ class Rotation:
if eu.shape[:-2:-1] != (3,): raise ValueError('invalid shape') if eu.shape[:-2:-1] != (3,): raise ValueError('invalid shape')
eu = np.radians(eu) if degrees else eu eu = np.radians(eu) if degrees else eu
if np.any(eu < 0.0) or np.any(eu > np.pi*np.array([2.0,1.0,2.0])): if np.any(eu < 0.) or np.any(eu > np.pi*np.array([2.,1.,2.])):
raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π]') raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π]')
return Rotation(Rotation._eu2qu(eu)) return Rotation(Rotation._eu2qu(eu))
@ -837,12 +841,12 @@ class Rotation:
ax[...,0:3] *= -P ax[...,0:3] *= -P
if degrees: ax[..., 3] = np.radians(ax[...,3]) if degrees: ax[..., 3] = np.radians(ax[...,3])
if np.any(ax[...,3] < 0.0) or np.any(ax[...,3] > np.pi): if np.any(ax[...,3] < 0.) or np.any(ax[...,3] > np.pi):
raise ValueError('axis–angle rotation angle outside of [0..π]') raise ValueError('axis–angle rotation angle outside of [0..π]')
if normalize: if normalize:
ax[...,0:3] /= np.linalg.norm(ax[...,0:3],axis=-1,keepdims=True) ax[...,0:3] /= np.linalg.norm(ax[...,0:3],axis=-1,keepdims=True)
elif not np.allclose(np.linalg.norm(ax[...,0:3],axis=-1), 1.0): elif not np.allclose(np.linalg.norm(ax[...,0:3],axis=-1),1.):
raise ValueError('axis–angle rotation axis is not of unit length') raise ValueError('axis–angle rotation axis is not of unit length')
return Rotation(Rotation._ax2qu(ax)) return Rotation(Rotation._ax2qu(ax))
@ -874,12 +878,12 @@ class Rotation:
if not orthonormal: if not orthonormal:
(U,S,Vh) = np.linalg.svd(om) # singular value decomposition (U,S,Vh) = np.linalg.svd(om) # singular value decomposition
om = np.einsum('...ij,...jl',U,Vh) om = np.einsum('...ij,...jl',U,Vh)
elif not np.allclose(np.einsum('...i,...i',om[...,0],om[...,1]), 0.0) \ elif not np.allclose(np.einsum('...i,...i',om[...,0],om[...,1]),0.) \
or not np.allclose(np.einsum('...i,...i',om[...,1],om[...,2]), 0.0) \ or not np.allclose(np.einsum('...i,...i',om[...,1],om[...,2]),0.) \
or not np.allclose(np.einsum('...i,...i',om[...,2],om[...,0]), 0.0): or not np.allclose(np.einsum('...i,...i',om[...,2],om[...,0]),0.):
raise ValueError('orientation matrix is not orthogonal') raise ValueError('orientation matrix is not orthogonal')
if not np.allclose(np.linalg.det(om),1.0): if not np.allclose(np.linalg.det(om),1.):
raise ValueError('orientation matrix has determinant ≠ 1') raise ValueError('orientation matrix has determinant ≠ 1')
return Rotation(Rotation._om2qu(om)) return Rotation(Rotation._om2qu(om))
@ -911,8 +915,8 @@ class Rotation:
Corresponding three-dimensional vectors of second basis. Corresponding three-dimensional vectors of second basis.
""" """
a_ = np.array(a) a_ = np.array(a,dtype=float)
b_ = np.array(b) b_ = np.array(b,dtype=float)
if a_.shape[-2:] != (2,3) or b_.shape[-2:] != (2,3) or a_.shape != b_.shape: if a_.shape[-2:] != (2,3) or b_.shape[-2:] != (2,3) or a_.shape != b_.shape:
raise ValueError('invalid shape') raise ValueError('invalid shape')
am = np.stack([ a_[...,0,:], am = np.stack([ a_[...,0,:],
@ -948,11 +952,11 @@ class Rotation:
if abs(P) != 1: raise ValueError('P ∉ {-1,1}') if abs(P) != 1: raise ValueError('P ∉ {-1,1}')
ro[...,0:3] *= -P ro[...,0:3] *= -P
if np.any(ro[...,3] < 0.0): raise ValueError('Rodrigues vector rotation angle is negative') if np.any(ro[...,3] < 0.): raise ValueError('Rodrigues vector rotation angle is negative')
if normalize: if normalize:
ro[...,0:3] /= np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True) ro[...,0:3] /= np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True)
elif not np.allclose(np.linalg.norm(ro[...,0:3],axis=-1), 1.0): elif not np.allclose(np.linalg.norm(ro[...,0:3],axis=-1),1.):
raise ValueError('Rodrigues vector rotation axis is not of unit length') raise ValueError('Rodrigues vector rotation axis is not of unit length')
return Rotation(Rotation._ro2qu(ro)) return Rotation(Rotation._ro2qu(ro))
@ -977,7 +981,7 @@ class Rotation:
ho *= -P ho *= -P
if np.any(np.linalg.norm(ho,axis=-1) >_R1+1e-9): if np.any(np.linalg.norm(ho,axis=-1) > _R1+1.e-9):
raise ValueError('homochoric coordinate outside of the sphere') raise ValueError('homochoric coordinate outside of the sphere')
return Rotation(Rotation._ho2qu(ho)) return Rotation(Rotation._ho2qu(ho))
@ -999,7 +1003,7 @@ class Rotation:
cu = np.array(x,dtype=float) cu = np.array(x,dtype=float)
if cu.shape[:-2:-1] != (3,): raise ValueError('invalid shape') if cu.shape[:-2:-1] != (3,): raise ValueError('invalid shape')
if abs(P) != 1: raise ValueError('P ∉ {-1,1}') if abs(P) != 1: raise ValueError('P ∉ {-1,1}')
if np.abs(np.max(cu)) > np.pi**(2./3.) * 0.5+1e-9: if np.abs(np.max(cu)) > np.pi**(2./3.) * 0.5+1.e-9:
raise ValueError('cubochoric coordinate outside of the cube') raise ValueError('cubochoric coordinate outside of the cube')
ho = -P * Rotation._cu2ho(cu) ho = -P * Rotation._cu2ho(cu)
@ -1026,11 +1030,11 @@ class Rotation:
r = rng.random(3 if shape is None else tuple(shape)+(3,) if hasattr(shape, '__iter__') else (shape,3)) # type: ignore r = rng.random(3 if shape is None else tuple(shape)+(3,) if hasattr(shape, '__iter__') else (shape,3)) # type: ignore
A = np.sqrt(r[...,2]) A = np.sqrt(r[...,2])
B = np.sqrt(1.0-r[...,2]) B = np.sqrt(1.-r[...,2])
q = np.stack([np.cos(2.0*np.pi*r[...,0])*A, q = np.stack([np.cos(2.*np.pi*r[...,0])*A,
np.sin(2.0*np.pi*r[...,1])*B, np.sin(2.*np.pi*r[...,1])*B,
np.cos(2.0*np.pi*r[...,1])*B, np.cos(2.*np.pi*r[...,1])*B,
np.sin(2.0*np.pi*r[...,0])*A],axis=-1) np.sin(2.*np.pi*r[...,0])*A],axis=-1)
return Rotation(q if shape is None else q.reshape(r.shape[:-1]+(4,)))._standardize() return Rotation(q if shape is None else q.reshape(r.shape[:-1]+(4,)))._standardize()
@ -1079,10 +1083,10 @@ class Rotation:
phi_sorted = eu[np.lexsort((eu[:,0],eu[:,1],eu[:,2]))] phi_sorted = eu[np.lexsort((eu[:,0],eu[:,1],eu[:,2]))]
steps,size,_ = grid_filters.cellsSizeOrigin_coordinates0_point(phi_sorted) steps,size,_ = grid_filters.cellsSizeOrigin_coordinates0_point(phi_sorted)
delta = np.radians(size/steps) if deg else size/steps 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]) return delta[0]*2.*np.sin(delta[1]/2.)*delta[2] / 8. / np.pi**2 * np.sin(np.radians(eu[:,1]) if deg else eu[:,1])
dg = 1.0 if fractions else _dg(phi,degrees) dg = 1. if fractions else _dg(phi,degrees)
dV_V = dg * np.maximum(0.0,weights.squeeze()) dV_V = dg * np.maximum(0.,weights.squeeze())
N = 1 if shape is None else np.prod(shape) N = 1 if shape is None else np.prod(shape)
return Rotation.from_Euler_angles(phi[util.hybrid_IA(dV_V,N,rng_seed)],degrees).reshape(() if shape is None else shape) return Rotation.from_Euler_angles(phi[util.hybrid_IA(dV_V,N,rng_seed)],degrees).reshape(() if shape is None else shape)
@ -1117,24 +1121,24 @@ class Rotation:
200 orientations: 200 orientations:
>>> import damask >>> import damask
>>> center = damask.Rotation.from_Euler_angles([35.0,45.0,0.0],degrees=True) >>> center = damask.Rotation.from_Euler_angles([35.,45.,0.],degrees=True)
>>> brass = damask.Rotation.from_spherical_component(center=center,sigma=3.0,shape=200,degrees=True) >>> brass = damask.Rotation.from_spherical_component(center=center,sigma=3.,shape=200,degrees=True)
Create a Goss texture consisting of Create a Goss texture consisting of
100 orientations: 100 orientations:
>>> import damask >>> import damask
>>> center = damask.Rotation.from_Euler_angles([0.0,45.0,0.0],degrees=True) >>> center = damask.Rotation.from_Euler_angles([0.,45.,0.],degrees=True)
>>> goss = damask.Rotation.from_spherical_component(center=center,sigma=3.0,shape=100,degrees=True) >>> goss = damask.Rotation.from_spherical_component(center=center,sigma=3.,shape=100,degrees=True)
""" """
rng = np.random.default_rng(rng_seed) rng = np.random.default_rng(rng_seed)
sigma = np.radians(sigma) if degrees else sigma sigma = np.radians(sigma) if degrees else sigma
N = 1 if shape is None else np.prod(shape) N = 1 if shape is None else np.prod(shape)
u,Theta = (rng.random((N,2)) * 2.0 * np.array([1,np.pi]) - np.array([1.0, 0])).T u,Theta = (rng.random((N,2)) * 2. * np.array([1.,np.pi]) - np.array([1.,0.])).T
omega = abs(rng.normal(scale=sigma,size=N)) omega = abs(rng.normal(scale=sigma,size=N))
p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta), p = np.column_stack([np.sqrt(1.-u**2)*np.cos(Theta),
np.sqrt(1-u**2)*np.sin(Theta), np.sqrt(1.-u**2)*np.sin(Theta),
u, omega]) u, omega])
return Rotation.from_axis_angle(p).reshape(() if shape is None else shape) * center return Rotation.from_axis_angle(p).reshape(() if shape is None else shape) * center
@ -1143,7 +1147,7 @@ class Rotation:
@staticmethod @staticmethod
def from_fiber_component(crystal: IntSequence, def from_fiber_component(crystal: IntSequence,
sample: IntSequence, sample: IntSequence,
sigma: float = 0.0, sigma: float = 0.,
shape: Union[int, IntSequence] = None, shape: Union[int, IntSequence] = None,
degrees: bool = False, degrees: bool = False,
rng_seed: NumpyRngSeed = None): rng_seed: NumpyRngSeed = None):
@ -1193,7 +1197,7 @@ class Rotation:
100 orientations: 100 orientations:
>>> import damask >>> import damask
>>> gamma = damask.Rotation.from_fiber_component([54.7,45.0],[0.,0.],shape=100,degrees=True) >>> gamma = damask.Rotation.from_fiber_component([54.7,45.],[0.,0.],shape=100,degrees=True)
""" """
rng = np.random.default_rng(rng_seed) rng = np.random.default_rng(rng_seed)
@ -1203,18 +1207,18 @@ class Rotation:
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_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])]) 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))) 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]) if np.isclose(ax_align[3],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 R_align = Rotation.from_axis_angle(ax_align if ax_align[3] > 0. else -ax_align,normalize=True) # rotate fiber axis from sample to crystal frame
N = 1 if shape is None else np.prod(shape) N = 1 if shape is None else np.prod(shape)
u,Theta = (rng.random((N,2)) * 2.0 * np.array([1,np.pi]) - np.array([1.0, 0])).T u,Theta = (rng.random((N,2)) * 2. * np.array([1.,np.pi]) - np.array([1.,0.])).T
omega = abs(rng.normal(scale=sigma_,size=N)) omega = abs(rng.normal(scale=sigma_,size=N))
p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta), p = np.column_stack([np.sqrt(1-u**2)*np.cos(Theta),
np.sqrt(1-u**2)*np.sin(Theta), np.sqrt(1-u**2)*np.sin(Theta),
u, omega]) u, omega])
p[:,:3] = np.einsum('ij,...j',np.eye(3)-np.outer(d_lab,d_lab),p[:,:3]) # remove component along fiber axis 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 = 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 f[::2,:3] *= -1. # flip half the rotation axes to negative sense
return (R_align.broadcast_to(N) return (R_align.broadcast_to(N)
* Rotation.from_axis_angle(p,normalize=True) * Rotation.from_axis_angle(p,normalize=True)
@ -1255,15 +1259,15 @@ class Rotation:
@staticmethod @staticmethod
def _qu2om(qu: np.ndarray) -> np.ndarray: def _qu2om(qu: np.ndarray) -> np.ndarray:
qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2) 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, om = np.block([qq + 2.*qu[...,1:2]**2,
2.0*(qu[...,2:3]*qu[...,1:2]-_P*qu[...,0:1]*qu[...,3:4]), 2.*(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.*(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]), 2.*(qu[...,1:2]*qu[...,2:3]+_P*qu[...,0:1]*qu[...,3:4]),
qq + 2.0*qu[...,2:3]**2, qq + 2.*qu[...,2:3]**2,
2.0*(qu[...,3:4]*qu[...,2:3]-_P*qu[...,0:1]*qu[...,1:2]), 2.*(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.*(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]), 2.*(qu[...,2:3]*qu[...,3:4]+_P*qu[...,0:1]*qu[...,1:2]),
qq + 2.0*qu[...,3:4]**2, qq + 2.*qu[...,3:4]**2,
]).reshape(qu.shape[:-1]+(3,3)) ]).reshape(qu.shape[:-1]+(3,3))
return om return om
@ -1278,22 +1282,20 @@ class Rotation:
q12_s = qu[...,1:2]**2+qu[...,2:3]**2 q12_s = qu[...,1:2]**2+qu[...,2:3]**2
chi = np.sqrt(q03_s*q12_s) chi = np.sqrt(q03_s*q12_s)
eu = np.where(np.abs(q12_s) < 1.0e-8, eu = np.where(np.abs(q12_s) < 1.e-8,
np.block([np.arctan2(-_P*2.0*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2), np.block([np.arctan2(-_P*2.*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2),
np.zeros(qu.shape[:-1]+(2,))]), np.zeros(qu.shape[:-1]+(2,))]),
np.where(np.abs(q03_s) < 1.0e-8, np.where(np.abs(q03_s) < 1.e-8,
np.block([np.arctan2( 2.0*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2), np.block([np.arctan2( 2.*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2),
np.broadcast_to(np.pi,qu[...,0:1].shape), np.broadcast_to(np.pi,qu[...,0:1].shape),
np.zeros(qu.shape[:-1]+(1,))]), np.zeros(qu.shape[:-1]+(1,))]),
np.block([np.arctan2((-_P*q02+q13)*chi, (-_P*q01-q23)*chi), np.block([np.arctan2((-_P*q02+q13)*chi, (-_P*q01-q23)*chi),
np.arctan2( 2.0*chi, q03_s-q12_s ), np.arctan2( 2.*chi, q03_s-q12_s ),
np.arctan2(( _P*q02+q13)*chi, (-_P*q01+q23)*chi)]) np.arctan2(( _P*q02+q13)*chi, (-_P*q01+q23)*chi)])
) )
) )
# reduce Euler angles to definition range eu[np.abs(eu) < 1.e-6] = 0.
eu[np.abs(eu)<1.e-6] = 0.0 return np.where(eu < 0., eu%(np.pi*np.array([2.,1.,2.])),eu)
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 @staticmethod
def _qu2ax(qu: np.ndarray) -> np.ndarray: def _qu2ax(qu: np.ndarray) -> np.ndarray:
@ -1304,11 +1306,11 @@ class Rotation:
""" """
with np.errstate(invalid='ignore',divide='ignore'): 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) 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)) omega = 2. * np.arccos(np.clip(qu[...,0:1],-1.,1.))
ax = np.where(np.broadcast_to(qu[...,0:1] < 1.0e-8,qu.shape), ax = np.where(np.broadcast_to(qu[...,0:1] < 1.e-8,qu.shape),
np.block([qu[...,1:4],np.broadcast_to(np.pi,qu[...,0:1].shape)]), np.block([qu[...,1:4],np.broadcast_to(np.pi,qu[...,0:1].shape)]),
np.block([qu[...,1:4]*s,omega])) np.block([qu[...,1:4]*s,omega]))
ax[np.isclose(qu[...,0],1.,rtol=0.0)] = [0.0, 0.0, 1.0, 0.0] ax[np.isclose(qu[...,0],1.,rtol=0.)] = np.array([0.,0.,1.,0.])
return ax return ax
@staticmethod @staticmethod
@ -1316,23 +1318,23 @@ class Rotation:
"""Quaternion to Rodrigues–Frank vector.""" """Quaternion to Rodrigues–Frank vector."""
with np.errstate(invalid='ignore',divide='ignore'): with np.errstate(invalid='ignore',divide='ignore'):
s = np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) 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), ro = np.where(np.broadcast_to(np.abs(qu[...,0:1]) < 1.e-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], 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.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))) np.tan(np.arccos(np.clip(qu[...,0:1],-1.,1.)))
]) ])
) )
ro[np.abs(s).squeeze(-1) < 1.0e-12] = [0.0,0.0,_P,0.0] ro[np.abs(s).squeeze(-1) < 1.e-12] = np.array([0.,0.,_P,0.])
return ro return ro
@staticmethod @staticmethod
def _qu2ho(qu: np.ndarray) -> np.ndarray: def _qu2ho(qu: np.ndarray) -> np.ndarray:
"""Quaternion to homochoric vector.""" """Quaternion to homochoric vector."""
with np.errstate(invalid='ignore'): with np.errstate(invalid='ignore'):
omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0)) omega = 2. * np.arccos(np.clip(qu[...,0:1],-1.,1.))
ho = np.where(np.abs(omega) < 1.0e-12, ho = np.where(np.abs(omega) < 1.e-12,
np.zeros(3), np.zeros(3),
qu[...,1:4]/np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) \ qu[...,1:4]/np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True)
* (0.75*(omega - np.sin(omega)))**(1./3.)) * (0.75*(omega - np.sin(omega)))**(1./3.))
return ho return ho
@ -1351,16 +1353,16 @@ class Rotation:
This formulation is from www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion. This formulation is from www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion.
The original formulation had issues. The original formulation had issues.
""" """
trace = om[...,0,0:1]+om[...,1,1:2]+om[...,2,2:3] trace = om[...,0,0:1] + om[...,1,1:2] + om[...,2,2:3]
with np.errstate(invalid='ignore',divide='ignore'): with np.errstate(invalid='ignore',divide='ignore'):
s = [ s = np.array([
0.5 / np.sqrt( 1.0 + trace), 0.5 / np.sqrt( 1. + trace),
2.0 * np.sqrt( 1.0 + om[...,0,0:1] - om[...,1,1:2] - om[...,2,2:3]), 2. * np.sqrt( 1. + 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. * np.sqrt( 1. + 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] ) 2. * np.sqrt( 1. + om[...,2,2:3] - om[...,0,0:1] - om[...,1,1:2] )
] ])
qu= np.where(trace>0, qu = np.where(trace>0,
np.block([0.25 / s[0], np.block([0.25 / s[0],
(om[...,2,1:2] - om[...,1,2:3] ) * s[0], (om[...,2,1:2] - om[...,1,2:3] ) * s[0],
(om[...,0,2:3] - om[...,2,0:1] ) * s[0], (om[...,0,2:3] - om[...,2,0:1] ) * s[0],
@ -1381,16 +1383,16 @@ class Rotation:
0.25 * s[3]]), 0.25 * s[3]]),
) )
) )
)*np.array([1,_P,_P,_P]) )*np.array([1.,_P,_P,_P])
qu[qu[...,0]<0] *=-1 qu[qu[...,0] < 0.] *= -1.
return qu return qu
@staticmethod @staticmethod
def _om2eu(om: np.ndarray) -> np.ndarray: def _om2eu(om: np.ndarray) -> np.ndarray:
"""Rotation matrix to Bunge Euler angles.""" """Rotation matrix to Bunge Euler angles."""
with np.errstate(invalid='ignore',divide='ignore'): with np.errstate(invalid='ignore',divide='ignore'):
zeta = 1.0/np.sqrt(1.0-om[...,2,2:3]**2) zeta = 1./np.sqrt(1.-om[...,2,2:3]**2)
eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.0,0.0), eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.,0.),
np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]), np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]),
np.pi*0.5*(1-om[...,2,2:3]), np.pi*0.5*(1-om[...,2,2:3]),
np.zeros(om.shape[:-2]+(1,)), np.zeros(om.shape[:-2]+(1,)),
@ -1400,9 +1402,8 @@ class Rotation:
np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta) np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta)
]) ])
) )
eu[np.abs(eu)<1.e-8] = 0.0 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 np.where(eu < 0., eu%(np.pi*np.array([2.,1.,2.])),eu)
return eu
@staticmethod @staticmethod
def _om2ax(om: np.ndarray) -> np.ndarray: def _om2ax(om: np.ndarray) -> np.ndarray:
@ -1411,18 +1412,18 @@ class Rotation:
om[...,2,0:1]-om[...,0,2:3], om[...,2,0:1]-om[...,0,2:3],
om[...,0,1:2]-om[...,1,0:1] om[...,0,1:2]-om[...,1,0:1]
]) ])
t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.0).reshape(om.shape[:-2]+(1,)) t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.).reshape(om.shape[:-2]+(1,))
w,vr = np.linalg.eig(om) w,vr = np.linalg.eig(om)
# mask duplicated real eigenvalues # mask duplicated real eigenvalues
w[np.isclose(w[...,0],1.0+0.0j),1:] = 0. w[np.isclose(w[...,0],1.+0.j),1:] = 0.
w[np.isclose(w[...,1],1.0+0.0j),2:] = 0. w[np.isclose(w[...,1],1.+0.j),2:] = 0.
vr = np.swapaxes(vr,-1,-2) vr = np.swapaxes(vr,-1,-2)
ax = np.where(np.abs(diag_delta)<1e-13, ax = np.where(np.abs(diag_delta)<1.e-13,
np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,)), np.real(vr[np.isclose(w,1.+0.j)]).reshape(om.shape[:-2]+(3,)),
np.abs(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.j)]).reshape(om.shape[:-2]+(3,)))
*np.sign(diag_delta)) *np.sign(diag_delta))
ax = np.block([ax,np.arccos(np.clip(t,-1.0,1.0))]) ax = np.block([ax,np.arccos(np.clip(t,-1.,1.))])
ax[np.abs(ax[...,3])<1.e-8] = [ 0.0, 0.0, 1.0, 0.0] ax[np.abs(ax[...,3]) < 1.e-8] = np.array([0.,0.,1.,0.])
return ax return ax
@staticmethod @staticmethod
@ -1452,7 +1453,7 @@ class Rotation:
-_P*sPhi*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*sPhi*np.sin(ee[...,0:1]-ee[...,2:3]),
-_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])]) -_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])])
qu[qu[...,0]<0.0]*=-1 qu[qu[...,0] < 0.] *= -1.
return qu return qu
@staticmethod @staticmethod
@ -1470,7 +1471,7 @@ class Rotation:
-c[...,0:1]*s[...,1:2], -c[...,0:1]*s[...,1:2],
+c[...,1:2] +c[...,1:2]
]).reshape(eu.shape[:-1]+(3,3)) ]).reshape(eu.shape[:-1]+(3,3))
om[np.abs(om)<1.e-12] = 0.0 om[np.abs(om) < 1.e-12] = 0.
return om return om
@staticmethod @staticmethod
@ -1480,16 +1481,16 @@ class Rotation:
sigma = 0.5*(eu[...,0:1]+eu[...,2:3]) sigma = 0.5*(eu[...,0:1]+eu[...,2:3])
delta = 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) 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))) alpha = np.where(np.abs(np.cos(sigma))<1.e-12,np.pi,2.*np.arctan(tau/np.cos(sigma)))
with np.errstate(invalid='ignore',divide='ignore'): with np.errstate(invalid='ignore',divide='ignore'):
ax = np.where(np.broadcast_to(np.abs(alpha)<1.0e-12,eu.shape[:-1]+(4,)), ax = np.where(np.broadcast_to(np.abs(alpha)<1.e-12,eu.shape[:-1]+(4,)),
[0.0,0.0,1.0,0.0], [0.,0.,1.,0.],
np.block([-_P/tau*t*np.cos(delta), np.block([-_P/tau*t*np.cos(delta),
-_P/tau*t*np.sin(delta), -_P/tau*t*np.sin(delta),
-_P/tau* np.sin(sigma), -_P/tau* np.sin(sigma),
alpha alpha
])) ]))
ax[(alpha<0.0).squeeze()] *=-1 ax[(alpha<0.).squeeze()] *= -1.
return ax return ax
@staticmethod @staticmethod
@ -1497,8 +1498,8 @@ class Rotation:
"""Bunge Euler angles to Rodrigues–Frank vector.""" """Bunge Euler angles to Rodrigues–Frank vector."""
ax = Rotation._eu2ax(eu) ax = Rotation._eu2ax(eu)
ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)]) ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)])
ro[ax[...,3]>=np.pi,3] = np.inf ro[ax[...,3] >= np.pi,3] = np.inf
ro[np.abs(ax[...,3])<1.e-16] = [ 0.0, 0.0, _P, 0.0 ] ro[np.abs(ax[...,3])<1.e-16] = np.array([0.,0.,_P,0.])
return ro return ro
@staticmethod @staticmethod
@ -1518,7 +1519,7 @@ class Rotation:
"""Axis–angle pair to quaternion.""" """Axis–angle pair to quaternion."""
c = np.cos(ax[...,3:4]*.5) c = np.cos(ax[...,3:4]*.5)
s = np.sin(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])) qu = np.where(np.abs(ax[...,3:4]) < 1.e-6,[1.,0.,0.,0.],np.block([c,ax[...,:3]*s]))
return qu return qu
@staticmethod @staticmethod
@ -1526,7 +1527,7 @@ class Rotation:
"""Axis-angle pair to rotation matrix.""" """Axis-angle pair to rotation matrix."""
c = np.cos(ax[...,3:4]) c = np.cos(ax[...,3:4])
s = np.sin(ax[...,3:4]) s = np.sin(ax[...,3:4])
omc = 1. -c omc = 1.-c
om = np.block([c+omc*ax[...,0:1]**2, 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[...,1:2] + s*ax[...,2:3],
omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2], omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2],
@ -1536,7 +1537,7 @@ class Rotation:
omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2], omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2],
omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1], omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1],
c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3)) c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3))
return om if _P < 0.0 else np.swapaxes(om,-1,-2) return om if _P < 0. else np.swapaxes(om,-1,-2)
@staticmethod @staticmethod
def _ax2eu(ax: np.ndarray) -> np.ndarray: def _ax2eu(ax: np.ndarray) -> np.ndarray:
@ -1551,15 +1552,14 @@ class Rotation:
np.inf, np.inf,
np.tan(ax[...,3:4]*0.5)) np.tan(ax[...,3:4]*0.5))
]) ])
ro[np.abs(ax[...,3])<1.e-6] = [.0,.0,_P,.0] ro[np.abs(ax[...,3]) < 1.e-6] = np.array([.0,.0,_P,.0])
return ro return ro
@staticmethod @staticmethod
def _ax2ho(ax: np.ndarray) -> np.ndarray: def _ax2ho(ax: np.ndarray) -> np.ndarray:
"""Axis–angle pair to homochoric vector.""" """Axis–angle pair to homochoric vector."""
f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1.0/3.0) f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1./3.)
ho = ax[...,:3] * f return ax[...,:3] * f
return ho
@staticmethod @staticmethod
def _ax2cu(ax: np.ndarray) -> np.ndarray: def _ax2cu(ax: np.ndarray) -> np.ndarray:
@ -1590,16 +1590,15 @@ class Rotation:
ax = np.where(np.isfinite(ro[...,3:4]), 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.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)])) 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 ]) ax[np.abs(ro[...,3]) < 1.e-8] = np.array([0.,0.,1.,0.])
return ax return ax
@staticmethod @staticmethod
def _ro2ho(ro: np.ndarray) -> np.ndarray: def _ro2ho(ro: np.ndarray) -> np.ndarray:
"""Rodrigues–Frank vector to homochoric vector.""" """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) f = np.where(np.isfinite(ro[...,3:4]),2.*np.arctan(ro[...,3:4]) -np.sin(2.*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), return np.where(np.broadcast_to(np.sum(ro[...,0:3]**2,axis=-1,keepdims=True) < 1.e-8,ro[...,0:3].shape),
np.zeros(3), ro[...,0:3]* (0.75*f)**(1.0/3.0)) np.zeros(3), ro[...,0:3]* (0.75*f)**(1./3.))
return ho
@staticmethod @staticmethod
def _ro2cu(ro: np.ndarray) -> np.ndarray: def _ro2cu(ro: np.ndarray) -> np.ndarray:
@ -1633,13 +1632,12 @@ class Rotation:
+9.528014229335313e-6, -5.660288876265125e-6, +1.2844901692764126e-6, +9.528014229335313e-6, -5.660288876265125e-6, +1.2844901692764126e-6,
+1.1255185726258763e-6, -1.3834391419956455e-6, +7.513691751164847e-7, +1.1255185726258763e-6, -1.3834391419956455e-6, +7.513691751164847e-7,
-2.401996891720091e-7, +4.386887017466388e-8, -3.5917775353564864e-9]) -2.401996891720091e-7, +4.386887017466388e-8, -3.5917775353564864e-9])
hmag_squared = np.sum(ho**2.,axis=-1,keepdims=True) hmag_squared = np.sum(ho**2,axis=-1,keepdims=True)
s = np.sum(tfit*hmag_squared**np.arange(len(tfit)),axis=-1,keepdims=True) s = np.sum(tfit*hmag_squared**np.arange(len(tfit)),axis=-1,keepdims=True)
with np.errstate(invalid='ignore'): with np.errstate(invalid='ignore'):
ax = np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-8,ho.shape[:-1]+(4,)), return np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-8,ho.shape[:-1]+(4,)),
[ 0.0, 0.0, 1.0, 0.0 ], [0.,0.,1.,0.],
np.block([ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))])) np.block([ho/np.sqrt(hmag_squared),2.*np.arccos(np.clip(s,-1.,1.))]))
return ax
@staticmethod @staticmethod
def _ho2ro(ho: np.ndarray) -> np.ndarray: def _ho2ro(ho: np.ndarray) -> np.ndarray:
@ -1663,27 +1661,25 @@ class Rotation:
with np.errstate(invalid='ignore',divide='ignore'): with np.errstate(invalid='ignore',divide='ignore'):
# inverse M_3 # inverse M_3
xyz2 = xyz3[...,0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[...,2:3])) ) xyz2 = xyz3[...,0:2] * np.sqrt( 2.*rs/(rs+np.abs(xyz3[...,2:3])) )
qxy = np.sum(xyz2**2,axis=-1,keepdims=True) qxy = np.sum(xyz2**2,axis=-1,keepdims=True)
q2 = qxy + np.max(np.abs(xyz2),axis=-1,keepdims=True)**2 q2 = qxy + np.max(np.abs(xyz2),axis=-1,keepdims=True)**2
sq2 = np.sqrt(q2) 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)) q = (_beta/np.sqrt(2.)/_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\ 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) +np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2)/np.sqrt(2.)/qxy,-1.,1.)
T_inv = np.where(np.abs(xyz2[...,1:2]) <= np.abs(xyz2[...,0:1]), 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.ones_like(tt),np.arccos(tt)/np.pi*12.]),
np.block([np.arccos(tt)/np.pi*12.0,np.ones_like(tt)]))*q np.block([np.arccos(tt)/np.pi*12.,np.ones_like(tt)]))*q
T_inv[xyz2<0.0] *= -1.0 T_inv[xyz2<0.] *= -1.
T_inv[np.broadcast_to(np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-12),T_inv.shape)] = 0.0 T_inv[np.broadcast_to(np.isclose(qxy,0.,rtol=0.,atol=1.e-12),T_inv.shape)] = 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])) \ cu = np.block([T_inv, np.where(xyz3[...,2:3]<0.,-np.ones_like(xyz3[...,2:3]),np.ones_like(xyz3[...,2:3])) \
* rs/np.sqrt(6.0/np.pi), * rs/np.sqrt(6./np.pi),
])/ _sc ])/ _sc
cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0 cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.,rtol=0.,atol=1.e-16)] = 0.
cu = np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1) return np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1)
return cu
#---------- Cubochoric ---------- #---------- Cubochoric ----------
@staticmethod @staticmethod
@ -1726,32 +1722,30 @@ class Rotation:
# get pyramid and scale by grid parameter ratio # get pyramid and scale by grid parameter ratio
XYZ = np.take_along_axis(cu,Rotation._get_pyramid_order(cu,'forward'),-1) * _sc 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]) 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]) \ q = np.pi/12. * np.where(order,XYZ[...,1:2],XYZ[...,0:1]) \
/ np.where(order,XYZ[...,0:1],XYZ[...,1:2]) / np.where(order,XYZ[...,0:1],XYZ[...,1:2])
c = np.cos(q) c = np.cos(q)
s = np.sin(q) s = np.sin(q)
q = _R1*2.0**0.25/_beta/ np.sqrt(np.sqrt(2.0)-c) \ q = _R1*2.**0.25/_beta/ np.sqrt(np.sqrt(2.)-c) \
* np.where(order,XYZ[...,0:1],XYZ[...,1:2]) * 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 T = np.block([(np.sqrt(2.)*c - 1.), np.sqrt(2.) * s]) * q
# transform to sphere grid (inverse Lambert) # transform to sphere grid (inverse Lambert)
c = np.sum(T**2,axis=-1,keepdims=True) c = np.sum(T**2,axis=-1,keepdims=True)
s = c * np.pi/24.0 /XYZ[...,2:3]**2 s = c * np.pi/24. /XYZ[...,2:3]**2
c = c * np.sqrt(np.pi/24.0)/XYZ[...,2:3] c = c * np.sqrt(np.pi/24.)/XYZ[...,2:3]
q = np.sqrt( 1.0 - s) q = np.sqrt( 1. - 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), ho = np.where(np.isclose(np.sum(np.abs(XYZ[...,0:2]),axis=-1,keepdims=True),0.,rtol=0.,atol=1.e-16),
np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6.0/np.pi) *XYZ[...,2:3]]), np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6./np.pi)*XYZ[...,2:3]]),
np.block([np.where(order,T[...,0:1],T[...,1:2])*q, np.block([np.where(order,T[...,0:1],T[...,1:2])*q,
np.where(order,T[...,1:2],T[...,0:1])*q, np.where(order,T[...,1:2],T[...,0:1])*q,
np.sqrt(6.0/np.pi) * XYZ[...,2:3] - c]) np.sqrt(6./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.isclose(np.sum(np.abs(cu),axis=-1),0.,rtol=0.,atol=1.e-16)] = 0.
ho = np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1) return np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1)
return ho
@staticmethod @staticmethod

8
python/tests/test_Rotation.py
View File

@ -775,12 +775,12 @@ class TestRotation:
@pytest.mark.parametrize('P',[1,-1]) @pytest.mark.parametrize('P',[1,-1])
@pytest.mark.parametrize('accept_homomorph',[True,False]) @pytest.mark.parametrize('accept_homomorph',[True,False])
# @pytest.mark.parametrize('normalize',[True,False]) @pytest.mark.parametrize('normalize',[True,False])
def test_quaternion(self,set_of_rotations,P,accept_homomorph): def test_quaternion(self,set_of_rotations,P,accept_homomorph,normalize):
c = np.array([1,P*-1,P*-1,P*-1]) * (-1 if accept_homomorph else 1) #* (0.9 if normalize else 1.0) c = np.array([1,P*-1,P*-1,P*-1]) * (-1 if accept_homomorph else 1) * (0.9 if normalize else 1.0)
for rot in set_of_rotations: for rot in set_of_rotations:
m = rot.as_cubochoric() m = rot.as_cubochoric()
o = Rotation.from_quaternion(rot.as_quaternion()*c,accept_homomorph,P).as_cubochoric() o = Rotation.from_quaternion(rot.as_quaternion()*c,accept_homomorph,normalize,P).as_cubochoric()
ok = np.allclose(m,o,atol=atol) ok = np.allclose(m,o,atol=atol)
if np.count_nonzero(np.isclose(np.abs(o),np.pi**(2./3.)*.5)): if np.count_nonzero(np.isclose(np.abs(o),np.pi**(2./3.)*.5)):
ok |= np.allclose(m*-1.,o,atol=atol) ok |= np.allclose(m*-1.,o,atol=atol)