import numpy as np from ._Lambert import ball_to_cube, cube_to_ball from . import mechanics _P = -1 def iszero(a): return np.isclose(a,0.0,atol=1.0e-12,rtol=0.0) 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: the real part of a quaternion is positive, Re(q) > 0 Convention 7: 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. """ self.quaternion = quaternion.copy() 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.""" if self.quaternion.shape != (4,): raise NotImplementedError('Support for multiple rotations missing') return '\n'.join([ 'Quaternion: (real={:.3f}, imag=<{:+.3f}, {:+.3f}, {:+.3f}>)'.format(*(self.quaternion)), 'Matrix:\n{}'.format(self.asMatrix()), 'Bunge Eulers / deg: ({:3.2f}, {:3.2f}, {:3.2f})'.format(*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 self.quaternion.shape != (4,): raise NotImplementedError('Support for multiple rotations missing') if isinstance(other, Rotation): # rotate a rotation self_q = self.quaternion[0] self_p = self.quaternion[1:] other_q = other.quaternion[0] other_p = other.quaternion[1:] R = self.__class__(np.append(self_q*other_q - np.dot(self_p,other_p), self_q*other_p + other_q*self_p + _P * np.cross(self_p,other_p))) return R.standardize() elif isinstance(other, (tuple,np.ndarray)): if isinstance(other,tuple) or other.shape == (3,): # rotate a single (3)-vector or meshgrid A = self.quaternion[0]**2.0 - np.dot(self.quaternion[1:],self.quaternion[1:]) B = 2.0 * ( self.quaternion[1]*other[0] + self.quaternion[2]*other[1] + self.quaternion[3]*other[2]) C = 2.0 * _P*self.quaternion[0] return np.array([ A*other[0] + B*self.quaternion[1] + C*(self.quaternion[2]*other[2] - self.quaternion[3]*other[1]), A*other[1] + B*self.quaternion[2] + C*(self.quaternion[3]*other[0] - self.quaternion[1]*other[2]), A*other[2] + B*self.quaternion[3] + C*(self.quaternion[1]*other[1] - self.quaternion[2]*other[0]), ]) 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('Support for rotation of 4th order tensors missing') else: return NotImplemented else: return NotImplemented def inverse(self): """In-place inverse rotation/backward rotation.""" self.quaternion[...,1:] *= -1 return self def inversed(self): """Inverse rotation/backward rotation.""" return self.copy().inverse() def standardize(self): """In-place quaternion representation with positive real part.""" self.quaternion[self.quaternion[...,0] < 0.0] *= -1 return self def standardized(self): """Quaternion representation with positive real part.""" 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. """ if self.quaternion.shape != (4,) or other.quaternion.shape != (4,): raise NotImplementedError('Support for multiple rotations missing') return Rotation.fromAverage([self,other]) ################################################################################################ # convert to different orientation representations (numpy arrays) def as_quaternion(self): """ Unit quaternion [q, p_1, p_2, p_3]. Parameters ---------- quaternion : bool, optional return quaternion as DAMASK object. """ return self.quaternion def as_Eulers(self, degrees = False): """ Bunge-Euler angles: (φ_1, ϕ, φ_2). Parameters ---------- degrees : bool, optional return angles in degrees. """ eu = Rotation.qu2eu(self.quaternion) if degrees: eu = np.degrees(eu) return eu def as_axis_angle(self, degrees = False, pair = False): """ Axis angle representation [n_1, n_2, n_3, ω] unless pair == True: ([n_1, n_2, n_3], ω). Parameters ---------- degrees : bool, optional return rotation angle in degrees. pair : bool, optional return tuple of axis and angle. """ ax = Rotation.qu2ax(self.quaternion) if degrees: ax[...,3] = np.degrees(ax[...,3]) return (ax[...,:3],ax[...,3]) if pair else ax def as_matrix(self): """Rotation matrix.""" return Rotation.qu2om(self.quaternion) def as_Rodrigues(self, vector = False): """ Rodrigues-Frank vector representation [n_1, n_2, n_3, tan(ω/2)] unless vector == True: [n_1, n_2, n_3] * tan(ω/2). Parameters ---------- vector : bool, optional return as actual Rodrigues--Frank vector, i.e. rotation axis scaled by tan(ω/2). """ ro = Rotation.qu2ro(self.quaternion) return ro[...,:3]*ro[...,3] if vector else ro def as_homochoric(self): """Homochoric vector: (h_1, h_2, h_3).""" return Rotation.qu2ho(self.quaternion) def asCubochoric(self): """Cubochoric vector: (c_1, c_2, c_3).""" return Rotation.qu2cu(self.quaternion) def M(self): # ToDo not sure about the name: as_M or M? we do not have a from_M """ 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 np.einsum('...i,...j',self.quaternion,self.quaternion) # for compatibility (old names do not follow convention) asM = M asQuaternion = as_quaternion asEulers = as_Eulers asAxisAngle = as_axis_angle asMatrix = as_matrix asRodrigues = as_Rodrigues asHomochoric = as_homochoric ################################################################################################ # Static constructors. The input data needs to follow the conventions, options allow to # relax the conventions. @staticmethod def from_quaternion(quaternion, acceptHomomorph = False, P = -1): qu = np.array(quaternion,dtype=float) if qu.shape[:-2:-1] != (4,): raise ValueError('Invalid shape.') if P > 0: qu[...,1:4] *= -1 # convert from P=1 to P=-1 if acceptHomomorph: 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)): raise ValueError('Quaternion is not of unit length.') return Rotation(qu) @staticmethod def from_Eulers(eulers, degrees = False): eu = np.array(eulers,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)) @staticmethod def from_axis_angle(axis_angle, degrees = False, normalise = False, P = -1): ax = np.array(axis_angle,dtype=float) if ax.shape[:-2:-1] != (4,): raise ValueError('Invalid shape.') 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],axis=-1) 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)): raise ValueError('Axis angle rotation axis is not of unit length.') return Rotation(Rotation.ax2qu(ax)) @staticmethod def from_basis(basis, orthonormal = True, reciprocal = False): om = np.array(basis,dtype=float) if om.shape[:-3:-1] != (3,3): raise ValueError('Invalid shape.') if reciprocal: om = np.linalg.inv(mechanics.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->...il',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)) @staticmethod def from_matrix(om): return Rotation.from_basis(om) @staticmethod def from_Rodrigues(rodrigues, normalise = False, P = -1): ro = np.array(rodrigues,dtype=float) if ro.shape[:-2:-1] != (4,): raise ValueError('Invalid shape.') 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],axis=-1) if np.any(ro[...,3] < 0.0): raise ValueError('Rodrigues vector rotation angle not positive.') 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)) @staticmethod def from_homochoric(homochoric, P = -1): ho = np.array(homochoric,dtype=float) if ho.shape[:-2:-1] != (3,): raise ValueError('Invalid shape.') if P > 0: ho *= -1 # convert from P=1 to P=-1 if np.any(np.linalg.norm(ho,axis=-1) > (3.*np.pi/4.)**(1./3.)+1e-9): raise ValueError('Homochoric coordinate outside of the sphere.') return Rotation(Rotation.ho2qu(ho)) @staticmethod def fromCubochoric(cubochoric, P = -1): cu = np.array(cubochoric,dtype=float) if cu.shape[:-2:-1] != (3,): raise ValueError('Invalid shape.') if np.abs(np.max(cu))>np.pi**(2./3.) * 0.5+1e-9: raise ValueError('Cubochoric coordinate outside of the cube: {} {} {}.'.format(*cu)) ho = Rotation.cu2ho(cu) if P > 0: ho *= -1 # convert from P=1 to P=-1 return Rotation(Rotation.ho2qu(ho)) @staticmethod def fromAverage(rotations,weights = None): """ 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 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 Rotation.fromQuaternion(np.real(vec.T[eig.argmax()]),acceptHomomorph = True) @staticmethod def from_random(shape=None): if shape is None: r = np.random.random(3) elif hasattr(shape, '__iter__'): r = np.random.random(tuple(shape)+(3,)) else: r = np.random.random((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.reshape(r.shape[:-1]+(4,)) if shape is not None else q).standardize() # for compatibility (old names do not follow convention) fromQuaternion = from_quaternion fromEulers = from_Eulers fromAxisAngle = from_axis_angle fromBasis = from_basis fromMatrix = from_matrix fromRodrigues = from_Rodrigues fromHomochoric = from_homochoric fromRandom = from_random #################################################################################################### # 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. #################################################################################################### #---------- Quaternion ---------- @staticmethod def qu2om(qu): if len(qu.shape) == 1: """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[0,1] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3]) om[1,0] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3]) om[1,2] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1]) om[2,1] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1]) om[2,0] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2]) om[0,2] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2]) else: 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]+qu[...,0:1]*qu[...,3:4]), 2.0*(qu[...,3:4]*qu[...,1:2]-qu[...,0:1]*qu[...,2:3]), 2.0*(qu[...,1:2]*qu[...,2:3]-qu[...,0:1]*qu[...,3:4]), qq + 2.0*qu[...,2:3]**2, 2.0*(qu[...,3:4]*qu[...,2:3]+qu[...,0:1]*qu[...,1:2]), 2.0*(qu[...,1:2]*qu[...,3:4]+qu[...,0:1]*qu[...,2:3]), 2.0*(qu[...,2:3]*qu[...,3:4]-qu[...,0:1]*qu[...,1:2]), qq + 2.0*qu[...,3:4]**2, ]).reshape(qu.shape[:-1]+(3,3)) return om if _P < 0.0 else np.swapaxes(om,(-1,-2)) @staticmethod def qu2eu(qu): """Quaternion to Bunge-Euler angles.""" if len(qu.shape) == 1: q03 = qu[0]**2+qu[3]**2 q12 = qu[1]**2+qu[2]**2 chi = np.sqrt(q03*q12) if np.abs(q12) < 1.e-8: eu = np.array([np.arctan2(-_P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0, 0.0]) elif np.abs(q03) < 1.e-8: eu = 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 )]) else: 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.shape[:-1]+(1,)), 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) return eu @staticmethod def qu2ax(qu): """ Quaternion to axis angle pair. Modified version of the original formulation, should be numerically more stable """ if len(qu.shape) == 1: if np.abs(np.sum(qu[1:4]**2)) < 1.e-6: # set axis to [001] if the angle is 0/360 ax = np.array([ 0.0, 0.0, 1.0, 0.0 ]) elif qu[0] > 1.e-6: 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 = ax = np.array([ qu[1]*s, qu[2]*s, qu[3]*s, omega ]) else: ax = ax = np.array([ qu[1], qu[2], qu[3], np.pi]) else: 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-6,qu.shape), np.block([qu[...,1:4],np.broadcast_to(np.pi,qu.shape[:-1]+(1,))]), np.block([qu[...,1:4]*s,omega])) ax[np.sum(np.abs(qu[...,1:4])**2,axis=-1) < 1.0e-6,] = [0.0, 0.0, 1.0, 0.0] return ax @staticmethod def qu2ro(qu): """Quaternion to Rodrigues-Frank vector.""" if len(qu.shape) == 1: if iszero(qu[0]): ro = np.array([qu[1], qu[2], qu[3], np.inf]) else: s = np.linalg.norm(qu[1:4]) ro = np.array([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)))]) else: 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.shape[:-1]+(1,))]), 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.""" if len(qu.shape) == 1: omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0)) if np.abs(omega) < 1.0e-12: ho = np.zeros(3) 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.) else: 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. The original formulation (direct conversion) had (numerical?) issues """ return Rotation.eu2qu(Rotation.om2eu(om)) @staticmethod def om2eu(om): """Rotation matrix to Bunge-Euler angles.""" if len(om.shape) == 2: if not np.isclose(np.abs(om[2,2]),1.0,1.e-4): 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 else: 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]),1.0,1e-4), 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-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) return eu @staticmethod def om2ax(om): """Rotation matrix to axis angle pair.""" if len(om.shape) == 2: 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 np.abs(ax[3])<1.e-6: ax = np.array([ 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 = -_P*np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]]) diagDelta[np.abs(diagDelta)<1.e-6] = 1.0 ax[0:3] = np.where(np.abs(diagDelta)<0, ax[0:3],np.abs(ax[0:3])*np.sign(diagDelta)) else: 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] ]) diag_delta[np.abs(diag_delta)<1.e-6] = 1.0 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)<0, 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-6] = [ 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.""" if len(eu.shape) == 1: 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 else: 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.""" if len(eu.shape) == 1: 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] ]]) else: 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.""" if len(eu.shape) == 1: 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 np.abs(alpha)<1.e-6: 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 else: 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.""" if len(eu.shape) == 1: ro = Rotation.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) else: 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.""" if len(ax.shape) == 1: if np.abs(ax[3])<1.e-6: 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 ]) else: 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.""" if len(ax.shape) == 1: 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]] else: 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.""" if len(ax.shape) == 1: if np.abs(ax[3])<1.e-6: 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)] ro = np.array(ro) else: 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.""" if len(ax.shape) == 1: f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0) ho = ax[0:3] * f else: 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.""" if len(ro.shape) == 1: if np.abs(ro[3]) < 1.e-6: ax = np.array([ 0.0, 0.0, 1.0, 0.0 ]) elif not np.isfinite(ro[3]): ax = np.array([ ro[0], ro[1], ro[2], np.pi ]) else: angle = 2.0*np.arctan(ro[3]) ta = np.linalg.norm(ro[0:3]) ax = np.array([ ro[0]*ta, ro[1]*ta, ro[2]*ta, angle ]) else: 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-6] = np.array([ 0.0, 0.0, 1.0, 0.0 ]) return ax @staticmethod def ro2ho(ro): """Rodrigues-Frank vector to homochoric vector.""" if len(ro.shape) == 1: if np.sum(ro[0:3]**2.0) < 1.e-6: ho = np.zeros(3) 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) else: 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-6,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]) if len(ho.shape) == 1: # 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))) else: 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-6,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.""" if len(ho.shape) == 1: return ball_to_cube(ho) else: raise NotImplementedError('Support for multiple rotations missing') #---------- 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.""" if len(cu.shape) == 1: return cube_to_ball(cu) else: raise NotImplementedError('Support for multiple rotations missing')