store all test data together
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Martin Diehl
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####################################################################################################
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# Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations
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####################################################################################################
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# Copyright (c) 2017-2019, Martin Diehl/Max-Planck-Institut für Eisenforschung GmbH
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# Copyright (c) 2013-2014, Marc De Graef/Carnegie Mellon University
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# All rights reserved.
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#
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# Redistribution and use in source and binary forms, with or without modification, are
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# permitted provided that the following conditions are met:
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#
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# - Redistributions of source code must retain the above copyright notice, this list
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# of conditions and the following disclaimer.
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# - Redistributions in binary form must reproduce the above copyright notice, this
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# list of conditions and the following disclaimer in the documentation and/or
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# other materials provided with the distribution.
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# - Neither the names of Marc De Graef, Carnegie Mellon University nor the names
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# of its contributors may be used to endorse or promote products derived from
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# this software without specific prior written permission.
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#
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# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
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# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
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# USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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####################################################################################################
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import numpy as np
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from damask import _rotation
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_P = _rotation._P
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# parameters for conversion from/to cubochoric
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_sc = _rotation._sc
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_beta = _rotation._beta
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_R1 = _rotation._R1
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def iszero(a):
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return np.isclose(a,0.0,atol=1.0e-12,rtol=0.0)
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#---------- Quaternion ----------
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def qu2om(qu):
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"""Quaternion to rotation matrix."""
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qq = qu[0]**2-(qu[1]**2 + qu[2]**2 + qu[3]**2)
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om = np.diag(qq + 2.0*np.array([qu[1],qu[2],qu[3]])**2)
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om[0,1] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3])
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om[1,0] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3])
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om[1,2] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1])
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om[2,1] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1])
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om[2,0] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2])
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om[0,2] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2])
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return om if _P < 0.0 else np.swapaxes(om,-1,-2)
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def qu2eu(qu):
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"""Quaternion to Bunge-Euler angles."""
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q03 = qu[0]**2+qu[3]**2
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q12 = qu[1]**2+qu[2]**2
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chi = np.sqrt(q03*q12)
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if np.abs(q12) < 1.e-8:
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eu = np.array([np.arctan2(-_P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0, 0.0])
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elif np.abs(q03) < 1.e-8:
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eu = np.array([np.arctan2( 2.0*qu[1]*qu[2],qu[1]**2-qu[2]**2), np.pi, 0.0])
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else:
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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 ),
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np.arctan2( 2.0*chi, q03-q12 ),
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np.arctan2(( _P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]+qu[2]*qu[3])*chi )])
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# reduce Euler angles to definition range
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eu[np.abs(eu)<1.e-6] = 0.0
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eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
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return eu
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def qu2ax(qu):
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"""
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Quaternion to axis angle pair.
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Modified version of the original formulation, should be numerically more stable
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"""
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if np.isclose(qu[0],1.,rtol=0.0): # set axis to [001] if the angle is 0/360
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ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
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elif qu[0] > 1.e-8:
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s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2)
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omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
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ax = ax = np.array([ qu[1]*s, qu[2]*s, qu[3]*s, omega ])
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else:
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ax = ax = np.array([ qu[1], qu[2], qu[3], np.pi])
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return ax
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def qu2ro(qu):
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"""Quaternion to Rodrigues-Frank vector."""
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if iszero(qu[0]):
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ro = np.array([qu[1], qu[2], qu[3], np.inf])
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else:
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s = np.linalg.norm(qu[1:4])
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ro = np.array([0.0,0.0,_P,0.0] if iszero(s) else \
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[ qu[1]/s, qu[2]/s, qu[3]/s, np.tan(np.arccos(np.clip(qu[0],-1.0,1.0)))])
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return ro
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def qu2ho(qu):
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"""Quaternion to homochoric vector."""
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omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
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if np.abs(omega) < 1.0e-12:
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ho = np.zeros(3)
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else:
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ho = np.array([qu[1], qu[2], qu[3]])
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f = 0.75 * ( omega - np.sin(omega) )
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ho = ho/np.linalg.norm(ho) * f**(1./3.)
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return ho
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#---------- Rotation matrix ----------
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def om2qu(a):
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trace = a[0,0] + a[1,1] + a[2,2]
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if trace > 0:
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s = 0.5 / np.sqrt(trace+ 1.0)
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qu = np.array([0.25 / s,( a[2,1] - a[1,2] ) * s,( a[0,2] - a[2,0] ) * s,( a[1,0] - a[0,1] ) * s])
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else:
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if ( a[0,0] > a[1,1] and a[0,0] > a[2,2] ):
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s = 2.0 * np.sqrt( 1.0 + a[0,0] - a[1,1] - a[2,2])
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qu = np.array([ (a[2,1] - a[1,2]) / s,0.25 * s,(a[0,1] + a[1,0]) / s,(a[0,2] + a[2,0]) / s])
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elif (a[1,1] > a[2,2]):
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s = 2.0 * np.sqrt( 1.0 + a[1,1] - a[0,0] - a[2,2])
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qu = np.array([ (a[0,2] - a[2,0]) / s,(a[0,1] + a[1,0]) / s,0.25 * s,(a[1,2] + a[2,1]) / s])
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else:
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s = 2.0 * np.sqrt( 1.0 + a[2,2] - a[0,0] - a[1,1] )
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qu = np.array([ (a[1,0] - a[0,1]) / s,(a[0,2] + a[2,0]) / s,(a[1,2] + a[2,1]) / s,0.25 * s])
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if qu[0]<0: qu*=-1
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return qu*np.array([1.,_P,_P,_P])
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def om2eu(om):
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"""Rotation matrix to Bunge-Euler angles."""
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if not np.isclose(np.abs(om[2,2]),1.0,1.e-9):
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zeta = 1.0/np.sqrt(1.0-om[2,2]**2)
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eu = np.array([np.arctan2(om[2,0]*zeta,-om[2,1]*zeta),
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np.arccos(om[2,2]),
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np.arctan2(om[0,2]*zeta, om[1,2]*zeta)])
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else:
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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
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eu[np.abs(eu)<1.e-8] = 0.0
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eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
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return eu
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def om2ax(om):
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"""Rotation matrix to axis angle pair."""
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#return qu2ax(om2qu(om)) # HOTFIX
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ax=np.empty(4)
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# first get the rotation angle
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t = 0.5*(om.trace() -1.0)
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ax[3] = np.arccos(np.clip(t,-1.0,1.0))
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if np.abs(ax[3])<1.e-8:
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ax = np.array([ 0.0, 0.0, 1.0, 0.0])
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else:
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w,vr = np.linalg.eig(om)
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# next, find the eigenvalue (1,0j)
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i = np.where(np.isclose(w,1.0+0.0j))[0][0]
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ax[0:3] = np.real(vr[0:3,i])
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diagDelta = -_P*np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]])
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ax[0:3] = np.where(np.abs(diagDelta)<1e-12, ax[0:3],np.abs(ax[0:3])*np.sign(diagDelta))
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return ax
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#---------- Bunge-Euler angles ----------
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def eu2qu(eu):
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"""Bunge-Euler angles to quaternion."""
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ee = 0.5*eu
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cPhi = np.cos(ee[1])
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sPhi = np.sin(ee[1])
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qu = np.array([ cPhi*np.cos(ee[0]+ee[2]),
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-_P*sPhi*np.cos(ee[0]-ee[2]),
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-_P*sPhi*np.sin(ee[0]-ee[2]),
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-_P*cPhi*np.sin(ee[0]+ee[2]) ])
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if qu[0] < 0.0: qu*=-1
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return qu
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def eu2om(eu):
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"""Bunge-Euler angles to rotation matrix."""
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c = np.cos(eu)
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s = np.sin(eu)
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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]],
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[-c[0]*s[2]-s[0]*c[2]*c[1], -s[0]*s[2]+c[0]*c[2]*c[1], +c[2]*s[1]],
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[+s[0]*s[1], -c[0]*s[1], +c[1] ]])
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om[np.abs(om)<1.e-12] = 0.0
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return om
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def eu2ax(eu):
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"""Bunge-Euler angles to axis angle pair."""
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t = np.tan(eu[1]*0.5)
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sigma = 0.5*(eu[0]+eu[2])
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delta = 0.5*(eu[0]-eu[2])
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tau = np.linalg.norm([t,np.sin(sigma)])
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alpha = np.pi if iszero(np.cos(sigma)) else \
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2.0*np.arctan(tau/np.cos(sigma))
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if np.abs(alpha)<1.e-6:
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ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
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else:
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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
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ax = np.append(ax,alpha)
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if alpha < 0.0: ax *= -1.0 # ensure alpha is positive
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return ax
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def eu2ro(eu):
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"""Bunge-Euler angles to Rodrigues-Frank vector."""
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ro = eu2ax(eu) # convert to axis angle pair representation
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if ro[3] >= np.pi: # Differs from original implementation. check convention 5
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ro[3] = np.inf
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elif iszero(ro[3]):
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ro = np.array([ 0.0, 0.0, _P, 0.0 ])
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else:
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ro[3] = np.tan(ro[3]*0.5)
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return ro
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#---------- Axis angle pair ----------
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def ax2qu(ax):
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"""Axis angle pair to quaternion."""
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if np.abs(ax[3])<1.e-6:
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qu = np.array([ 1.0, 0.0, 0.0, 0.0 ])
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else:
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c = np.cos(ax[3]*0.5)
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s = np.sin(ax[3]*0.5)
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qu = np.array([ c, ax[0]*s, ax[1]*s, ax[2]*s ])
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return qu
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def ax2om(ax):
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"""Axis angle pair to rotation matrix."""
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c = np.cos(ax[3])
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s = np.sin(ax[3])
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omc = 1.0-c
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om=np.diag(ax[0:3]**2*omc + c)
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for idx in [[0,1,2],[1,2,0],[2,0,1]]:
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q = omc*ax[idx[0]] * ax[idx[1]]
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om[idx[0],idx[1]] = q + s*ax[idx[2]]
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om[idx[1],idx[0]] = q - s*ax[idx[2]]
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return om if _P < 0.0 else om.T
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def ax2ro(ax):
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"""Axis angle pair to Rodrigues-Frank vector."""
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if np.abs(ax[3])<1.e-6:
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ro = [ 0.0, 0.0, _P, 0.0 ]
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else:
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ro = [ax[0], ax[1], ax[2]]
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# 180 degree case
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ro += [np.inf] if np.isclose(ax[3],np.pi,atol=1.0e-15,rtol=0.0) else \
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[np.tan(ax[3]*0.5)]
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ro = np.array(ro)
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return ro
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def ax2ho(ax):
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"""Axis angle pair to homochoric vector."""
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f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0)
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ho = ax[0:3] * f
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return ho
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#---------- Rodrigues-Frank vector ----------
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def ro2ax(ro):
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"""Rodrigues-Frank vector to axis angle pair."""
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if np.abs(ro[3]) < 1.e-8:
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ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
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elif not np.isfinite(ro[3]):
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ax = np.array([ ro[0], ro[1], ro[2], np.pi ])
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else:
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angle = 2.0*np.arctan(ro[3])
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ta = np.linalg.norm(ro[0:3])
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ax = np.array([ ro[0]*ta, ro[1]*ta, ro[2]*ta, angle ])
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return ax
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def ro2ho(ro):
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"""Rodrigues-Frank vector to homochoric vector."""
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if np.sum(ro[0:3]**2.0) < 1.e-8:
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ho = np.zeros(3)
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else:
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f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
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ho = ro[0:3] * (0.75*f)**(1.0/3.0)
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return ho
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#---------- Homochoric vector----------
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def ho2ax(ho):
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"""Homochoric vector to axis angle pair."""
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tfit = np.array([+1.0000000000018852, -0.5000000002194847,
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-0.024999992127593126, -0.003928701544781374,
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-0.0008152701535450438, -0.0002009500426119712,
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-0.00002397986776071756, -0.00008202868926605841,
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+0.00012448715042090092, -0.0001749114214822577,
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+0.0001703481934140054, -0.00012062065004116828,
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+0.000059719705868660826, -0.00001980756723965647,
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+0.000003953714684212874, -0.00000036555001439719544])
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# normalize h and store the magnitude
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hmag_squared = np.sum(ho**2.)
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if iszero(hmag_squared):
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ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
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else:
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hm = hmag_squared
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# convert the magnitude to the rotation angle
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s = tfit[0] + tfit[1] * hmag_squared
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for i in range(2,16):
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hm *= hmag_squared
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s += tfit[i] * hm
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ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0)))
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return ax
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def ho2cu(ho):
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"""
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Homochoric vector to cubochoric vector.
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References
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----------
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D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
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https://doi.org/10.1088/0965-0393/22/7/075013
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"""
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rs = np.linalg.norm(ho)
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if np.allclose(ho,0.0,rtol=0.0,atol=1.0e-16):
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cu = np.zeros(3)
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else:
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xyz3 = ho[_get_pyramid_order(ho,'forward')]
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# inverse M_3
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xyz2 = xyz3[0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[2])) )
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# inverse M_2
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qxy = np.sum(xyz2**2)
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if np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-16):
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Tinv = np.zeros(2)
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else:
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q2 = qxy + np.max(np.abs(xyz2))**2
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sq2 = np.sqrt(q2)
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q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2))*sq2))
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tt = np.clip((np.min(np.abs(xyz2))**2+np.max(np.abs(xyz2))*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
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Tinv = np.array([1.0,np.arccos(tt)/np.pi*12.0]) if np.abs(xyz2[1]) <= np.abs(xyz2[0]) else \
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np.array([np.arccos(tt)/np.pi*12.0,1.0])
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Tinv = q * np.where(xyz2<0.0,-Tinv,Tinv)
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# inverse M_1
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cu = np.array([ Tinv[0], Tinv[1], (-1.0 if xyz3[2] < 0.0 else 1.0) * rs / np.sqrt(6.0/np.pi) ]) /_sc
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cu = cu[_get_pyramid_order(ho,'backward')]
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return cu
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#---------- Cubochoric ----------
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def cu2ho(cu):
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"""
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Cubochoric vector to homochoric vector.
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||||
|
||||
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
|
||||
|
||||
"""
|
||||
# transform to the sphere grid via the curved square, and intercept the zero point
|
||||
if np.allclose(cu,0.0,rtol=0.0,atol=1.0e-16):
|
||||
ho = np.zeros(3)
|
||||
else:
|
||||
# get pyramide and scale by grid parameter ratio
|
||||
XYZ = cu[_get_pyramid_order(cu,'forward')] * _sc
|
||||
|
||||
# intercept all the points along the z-axis
|
||||
if np.allclose(XYZ[0:2],0.0,rtol=0.0,atol=1.0e-16):
|
||||
ho = np.array([0.0, 0.0, np.sqrt(6.0/np.pi) * XYZ[2]])
|
||||
else:
|
||||
order = [1,0] if np.abs(XYZ[1]) <= np.abs(XYZ[0]) else [0,1]
|
||||
q = np.pi/12.0 * XYZ[order[0]]/XYZ[order[1]]
|
||||
c = np.cos(q)
|
||||
s = np.sin(q)
|
||||
q = _R1*2.0**0.25/_beta * XYZ[order[1]] / np.sqrt(np.sqrt(2.0)-c)
|
||||
T = np.array([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
|
||||
|
||||
# transform to sphere grid (inverse Lambert)
|
||||
# note that there is no need to worry about dividing by zero, since XYZ[2] can not become zero
|
||||
c = np.sum(T**2)
|
||||
s = c * np.pi/24.0 /XYZ[2]**2
|
||||
c = c * np.sqrt(np.pi/24.0)/XYZ[2]
|
||||
|
||||
q = np.sqrt( 1.0 - s )
|
||||
ho = np.array([ T[order[1]] * q, T[order[0]] * q, np.sqrt(6.0/np.pi) * XYZ[2] - c ])
|
||||
|
||||
ho = ho[_get_pyramid_order(cu,'backward')]
|
||||
return ho
|
||||
|
||||
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]])}
|
||||
if np.maximum(abs(xyz[0]),abs(xyz[1])) <= xyz[2] or \
|
||||
np.maximum(abs(xyz[0]),abs(xyz[1])) <=-xyz[2]:
|
||||
p = 0
|
||||
elif np.maximum(abs(xyz[1]),abs(xyz[2])) <= xyz[0] or \
|
||||
np.maximum(abs(xyz[1]),abs(xyz[2])) <=-xyz[0]:
|
||||
p = 1
|
||||
elif np.maximum(abs(xyz[2]),abs(xyz[0])) <= xyz[1] or \
|
||||
np.maximum(abs(xyz[2]),abs(xyz[0])) <=-xyz[1]:
|
||||
p = 2
|
||||
return order[direction][p]
|
||||
|
|
@ -4,7 +4,9 @@ import pytest
|
|||
import numpy as np
|
||||
|
||||
from damask import Rotation
|
||||
import rotation_conversion
|
||||
from damask import _rotation
|
||||
|
||||
|
||||
|
||||
n = 1100
|
||||
atol=1.e-4
|
||||
|
|
@ -13,68 +15,69 @@ scatter=1.e-2
|
|||
@pytest.fixture
|
||||
def default():
|
||||
"""A set of n random rotations."""
|
||||
specials = np.array(
|
||||
[np.array([ 1.0, 0.0, 0.0, 0.0]),
|
||||
#-----------------------------------------------
|
||||
np.array([0.0, 1.0, 0.0, 0.0]),
|
||||
np.array([0.0, 0.0, 1.0, 0.0]),
|
||||
np.array([0.0, 0.0, 0.0, 1.0]),
|
||||
np.array([0.0,-1.0, 0.0, 0.0]),
|
||||
np.array([0.0, 0.0,-1.0, 0.0]),
|
||||
np.array([0.0, 0.0, 0.0,-1.0]),
|
||||
#-----------------------------------------------
|
||||
np.array([1.0, 1.0, 0.0, 0.0])/np.sqrt(2.),
|
||||
np.array([1.0, 0.0, 1.0, 0.0])/np.sqrt(2.),
|
||||
np.array([1.0, 0.0, 0.0, 1.0])/np.sqrt(2.),
|
||||
np.array([0.0, 1.0, 1.0, 0.0])/np.sqrt(2.),
|
||||
np.array([0.0, 1.0, 0.0, 1.0])/np.sqrt(2.),
|
||||
np.array([0.0, 0.0, 1.0, 1.0])/np.sqrt(2.),
|
||||
#-----------------------------------------------
|
||||
np.array([1.0,-1.0, 0.0, 0.0])/np.sqrt(2.),
|
||||
np.array([1.0, 0.0,-1.0, 0.0])/np.sqrt(2.),
|
||||
np.array([1.0, 0.0, 0.0,-1.0])/np.sqrt(2.),
|
||||
np.array([0.0, 1.0,-1.0, 0.0])/np.sqrt(2.),
|
||||
np.array([0.0, 1.0, 0.0,-1.0])/np.sqrt(2.),
|
||||
np.array([0.0, 0.0, 1.0,-1.0])/np.sqrt(2.),
|
||||
#-----------------------------------------------
|
||||
np.array([0.0, 1.0,-1.0, 0.0])/np.sqrt(2.),
|
||||
np.array([0.0, 1.0, 0.0,-1.0])/np.sqrt(2.),
|
||||
np.array([0.0, 0.0, 1.0,-1.0])/np.sqrt(2.),
|
||||
#-----------------------------------------------
|
||||
np.array([0.0,-1.0,-1.0, 0.0])/np.sqrt(2.),
|
||||
np.array([0.0,-1.0, 0.0,-1.0])/np.sqrt(2.),
|
||||
np.array([0.0, 0.0,-1.0,-1.0])/np.sqrt(2.),
|
||||
#-----------------------------------------------
|
||||
np.array([1.0, 1.0, 1.0, 0.0])/np.sqrt(3.),
|
||||
np.array([1.0, 1.0, 0.0, 1.0])/np.sqrt(3.),
|
||||
np.array([1.0, 0.0, 1.0, 1.0])/np.sqrt(3.),
|
||||
np.array([1.0,-1.0, 1.0, 0.0])/np.sqrt(3.),
|
||||
np.array([1.0,-1.0, 0.0, 1.0])/np.sqrt(3.),
|
||||
np.array([1.0, 0.0,-1.0, 1.0])/np.sqrt(3.),
|
||||
np.array([1.0, 1.0,-1.0, 0.0])/np.sqrt(3.),
|
||||
np.array([1.0, 1.0, 0.0,-1.0])/np.sqrt(3.),
|
||||
np.array([1.0, 0.0, 1.0,-1.0])/np.sqrt(3.),
|
||||
np.array([1.0,-1.0,-1.0, 0.0])/np.sqrt(3.),
|
||||
np.array([1.0,-1.0, 0.0,-1.0])/np.sqrt(3.),
|
||||
np.array([1.0, 0.0,-1.0,-1.0])/np.sqrt(3.),
|
||||
#-----------------------------------------------
|
||||
np.array([0.0, 1.0, 1.0, 1.0])/np.sqrt(3.),
|
||||
np.array([0.0, 1.0,-1.0, 1.0])/np.sqrt(3.),
|
||||
np.array([0.0, 1.0, 1.0,-1.0])/np.sqrt(3.),
|
||||
np.array([0.0,-1.0, 1.0, 1.0])/np.sqrt(3.),
|
||||
np.array([0.0,-1.0,-1.0, 1.0])/np.sqrt(3.),
|
||||
np.array([0.0,-1.0, 1.0,-1.0])/np.sqrt(3.),
|
||||
np.array([0.0,-1.0,-1.0,-1.0])/np.sqrt(3.),
|
||||
#-----------------------------------------------
|
||||
np.array([1.0, 1.0, 1.0, 1.0])/2.,
|
||||
np.array([1.0,-1.0, 1.0, 1.0])/2.,
|
||||
np.array([1.0, 1.0,-1.0, 1.0])/2.,
|
||||
np.array([1.0, 1.0, 1.0,-1.0])/2.,
|
||||
np.array([1.0,-1.0,-1.0, 1.0])/2.,
|
||||
np.array([1.0,-1.0, 1.0,-1.0])/2.,
|
||||
np.array([1.0, 1.0,-1.0,-1.0])/2.,
|
||||
np.array([1.0,-1.0,-1.0,-1.0])/2.,
|
||||
])
|
||||
specials = np.array([
|
||||
[1.0, 0.0, 0.0, 0.0],
|
||||
#----------------------
|
||||
[0.0, 1.0, 0.0, 0.0],
|
||||
[0.0, 0.0, 1.0, 0.0],
|
||||
[0.0, 0.0, 0.0, 1.0],
|
||||
[0.0,-1.0, 0.0, 0.0],
|
||||
[0.0, 0.0,-1.0, 0.0],
|
||||
[0.0, 0.0, 0.0,-1.0],
|
||||
#----------------------
|
||||
[1.0, 1.0, 0.0, 0.0],
|
||||
[1.0, 0.0, 1.0, 0.0],
|
||||
[1.0, 0.0, 0.0, 1.0],
|
||||
[0.0, 1.0, 1.0, 0.0],
|
||||
[0.0, 1.0, 0.0, 1.0],
|
||||
[0.0, 0.0, 1.0, 1.0],
|
||||
#----------------------
|
||||
[1.0,-1.0, 0.0, 0.0],
|
||||
[1.0, 0.0,-1.0, 0.0],
|
||||
[1.0, 0.0, 0.0,-1.0],
|
||||
[0.0, 1.0,-1.0, 0.0],
|
||||
[0.0, 1.0, 0.0,-1.0],
|
||||
[0.0, 0.0, 1.0,-1.0],
|
||||
#----------------------
|
||||
[0.0, 1.0,-1.0, 0.0],
|
||||
[0.0, 1.0, 0.0,-1.0],
|
||||
[0.0, 0.0, 1.0,-1.0],
|
||||
#----------------------
|
||||
[0.0,-1.0,-1.0, 0.0],
|
||||
[0.0,-1.0, 0.0,-1.0],
|
||||
[0.0, 0.0,-1.0,-1.0],
|
||||
#----------------------
|
||||
[1.0, 1.0, 1.0, 0.0],
|
||||
[1.0, 1.0, 0.0, 1.0],
|
||||
[1.0, 0.0, 1.0, 1.0],
|
||||
[1.0,-1.0, 1.0, 0.0],
|
||||
[1.0,-1.0, 0.0, 1.0],
|
||||
[1.0, 0.0,-1.0, 1.0],
|
||||
[1.0, 1.0,-1.0, 0.0],
|
||||
[1.0, 1.0, 0.0,-1.0],
|
||||
[1.0, 0.0, 1.0,-1.0],
|
||||
[1.0,-1.0,-1.0, 0.0],
|
||||
[1.0,-1.0, 0.0,-1.0],
|
||||
[1.0, 0.0,-1.0,-1.0],
|
||||
#----------------------
|
||||
[0.0, 1.0, 1.0, 1.0],
|
||||
[0.0, 1.0,-1.0, 1.0],
|
||||
[0.0, 1.0, 1.0,-1.0],
|
||||
[0.0,-1.0, 1.0, 1.0],
|
||||
[0.0,-1.0,-1.0, 1.0],
|
||||
[0.0,-1.0, 1.0,-1.0],
|
||||
[0.0,-1.0,-1.0,-1.0],
|
||||
#----------------------
|
||||
[1.0, 1.0, 1.0, 1.0],
|
||||
[1.0,-1.0, 1.0, 1.0],
|
||||
[1.0, 1.0,-1.0, 1.0],
|
||||
[1.0, 1.0, 1.0,-1.0],
|
||||
[1.0,-1.0,-1.0, 1.0],
|
||||
[1.0,-1.0, 1.0,-1.0],
|
||||
[1.0, 1.0,-1.0,-1.0],
|
||||
[1.0,-1.0,-1.0,-1.0],
|
||||
])
|
||||
specials /= np.linalg.norm(specials,axis=1).reshape(-1,1)
|
||||
specials_scatter = specials + np.broadcast_to(np.random.rand(4)*scatter,specials.shape)
|
||||
specials_scatter /= np.linalg.norm(specials_scatter,axis=1).reshape(-1,1)
|
||||
specials_scatter[specials_scatter[:,0]<0]*=-1
|
||||
|
|
@ -89,6 +92,425 @@ def reference_dir(reference_dir_base):
|
|||
return os.path.join(reference_dir_base,'Rotation')
|
||||
|
||||
|
||||
####################################################################################################
|
||||
# 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.
|
||||
####################################################################################################
|
||||
_P = _rotation._P
|
||||
|
||||
# parameters for conversion from/to cubochoric
|
||||
_sc = _rotation._sc
|
||||
_beta = _rotation._beta
|
||||
_R1 = _rotation._R1
|
||||
|
||||
def iszero(a):
|
||||
return np.isclose(a,0.0,atol=1.0e-12,rtol=0.0)
|
||||
|
||||
#---------- Quaternion ----------
|
||||
def qu2om(qu):
|
||||
"""Quaternion to rotation matrix."""
|
||||
qq = qu[0]**2-(qu[1]**2 + qu[2]**2 + qu[3]**2)
|
||||
om = np.diag(qq + 2.0*np.array([qu[1],qu[2],qu[3]])**2)
|
||||
|
||||
om[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])
|
||||
return om if _P < 0.0 else np.swapaxes(om,-1,-2)
|
||||
|
||||
def qu2eu(qu):
|
||||
"""Quaternion to Bunge-Euler angles."""
|
||||
q03 = qu[0]**2+qu[3]**2
|
||||
q12 = qu[1]**2+qu[2]**2
|
||||
chi = np.sqrt(q03*q12)
|
||||
if 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 )])
|
||||
# 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
|
||||
|
||||
def qu2ax(qu):
|
||||
"""
|
||||
Quaternion to axis angle pair.
|
||||
|
||||
Modified version of the original formulation, should be numerically more stable
|
||||
"""
|
||||
if np.isclose(qu[0],1.,rtol=0.0): # 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-8:
|
||||
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])
|
||||
return ax
|
||||
|
||||
def qu2ro(qu):
|
||||
"""Quaternion to Rodrigues-Frank vector."""
|
||||
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)))])
|
||||
return ro
|
||||
|
||||
def qu2ho(qu):
|
||||
"""Quaternion to homochoric vector."""
|
||||
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.)
|
||||
return ho
|
||||
|
||||
|
||||
#---------- Rotation matrix ----------
|
||||
def om2qu(a):
|
||||
trace = a[0,0] + a[1,1] + a[2,2]
|
||||
if trace > 0:
|
||||
s = 0.5 / np.sqrt(trace+ 1.0)
|
||||
qu = np.array([0.25 / s,( a[2,1] - a[1,2] ) * s,( a[0,2] - a[2,0] ) * s,( a[1,0] - a[0,1] ) * s])
|
||||
else:
|
||||
if ( a[0,0] > a[1,1] and a[0,0] > a[2,2] ):
|
||||
s = 2.0 * np.sqrt( 1.0 + a[0,0] - a[1,1] - a[2,2])
|
||||
qu = np.array([ (a[2,1] - a[1,2]) / s,0.25 * s,(a[0,1] + a[1,0]) / s,(a[0,2] + a[2,0]) / s])
|
||||
elif (a[1,1] > a[2,2]):
|
||||
s = 2.0 * np.sqrt( 1.0 + a[1,1] - a[0,0] - a[2,2])
|
||||
qu = np.array([ (a[0,2] - a[2,0]) / s,(a[0,1] + a[1,0]) / s,0.25 * s,(a[1,2] + a[2,1]) / s])
|
||||
else:
|
||||
s = 2.0 * np.sqrt( 1.0 + a[2,2] - a[0,0] - a[1,1] )
|
||||
qu = np.array([ (a[1,0] - a[0,1]) / s,(a[0,2] + a[2,0]) / s,(a[1,2] + a[2,1]) / s,0.25 * s])
|
||||
if qu[0]<0: qu*=-1
|
||||
return qu*np.array([1.,_P,_P,_P])
|
||||
|
||||
def om2eu(om):
|
||||
"""Rotation matrix to Bunge-Euler angles."""
|
||||
if not np.isclose(np.abs(om[2,2]),1.0,1.e-9):
|
||||
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
|
||||
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
|
||||
|
||||
def om2ax(om):
|
||||
"""Rotation matrix to axis angle pair."""
|
||||
#return qu2ax(om2qu(om)) # HOTFIX
|
||||
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-8:
|
||||
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]])
|
||||
ax[0:3] = np.where(np.abs(diagDelta)<1e-12, ax[0:3],np.abs(ax[0:3])*np.sign(diagDelta))
|
||||
return ax
|
||||
|
||||
#---------- Bunge-Euler angles ----------
|
||||
def eu2qu(eu):
|
||||
"""Bunge-Euler angles to quaternion."""
|
||||
ee = 0.5*eu
|
||||
cPhi = np.cos(ee[1])
|
||||
sPhi = np.sin(ee[1])
|
||||
qu = np.array([ cPhi*np.cos(ee[0]+ee[2]),
|
||||
-_P*sPhi*np.cos(ee[0]-ee[2]),
|
||||
-_P*sPhi*np.sin(ee[0]-ee[2]),
|
||||
-_P*cPhi*np.sin(ee[0]+ee[2]) ])
|
||||
if qu[0] < 0.0: qu*=-1
|
||||
return qu
|
||||
|
||||
def eu2om(eu):
|
||||
"""Bunge-Euler angles to rotation matrix."""
|
||||
c = np.cos(eu)
|
||||
s = np.sin(eu)
|
||||
|
||||
om = np.array([[+c[0]*c[2]-s[0]*s[2]*c[1], +s[0]*c[2]+c[0]*s[2]*c[1], +s[2]*s[1]],
|
||||
[-c[0]*s[2]-s[0]*c[2]*c[1], -s[0]*s[2]+c[0]*c[2]*c[1], +c[2]*s[1]],
|
||||
[+s[0]*s[1], -c[0]*s[1], +c[1] ]])
|
||||
om[np.abs(om)<1.e-12] = 0.0
|
||||
return om
|
||||
|
||||
def eu2ax(eu):
|
||||
"""Bunge-Euler angles to axis angle pair."""
|
||||
t = np.tan(eu[1]*0.5)
|
||||
sigma = 0.5*(eu[0]+eu[2])
|
||||
delta = 0.5*(eu[0]-eu[2])
|
||||
tau = np.linalg.norm([t,np.sin(sigma)])
|
||||
alpha = np.pi if iszero(np.cos(sigma)) else \
|
||||
2.0*np.arctan(tau/np.cos(sigma))
|
||||
|
||||
if 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
|
||||
return ax
|
||||
|
||||
def eu2ro(eu):
|
||||
"""Bunge-Euler angles to Rodrigues-Frank vector."""
|
||||
ro = eu2ax(eu) # convert to axis angle pair representation
|
||||
if ro[3] >= np.pi: # Differs from original implementation. check convention 5
|
||||
ro[3] = np.inf
|
||||
elif iszero(ro[3]):
|
||||
ro = np.array([ 0.0, 0.0, _P, 0.0 ])
|
||||
else:
|
||||
ro[3] = np.tan(ro[3]*0.5)
|
||||
return ro
|
||||
|
||||
#---------- Axis angle pair ----------
|
||||
def ax2qu(ax):
|
||||
"""Axis angle pair to quaternion."""
|
||||
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 ])
|
||||
return qu
|
||||
|
||||
def ax2om(ax):
|
||||
"""Axis angle pair to rotation matrix."""
|
||||
c = np.cos(ax[3])
|
||||
s = np.sin(ax[3])
|
||||
omc = 1.0-c
|
||||
om=np.diag(ax[0:3]**2*omc + c)
|
||||
|
||||
for idx in [[0,1,2],[1,2,0],[2,0,1]]:
|
||||
q = omc*ax[idx[0]] * ax[idx[1]]
|
||||
om[idx[0],idx[1]] = q + s*ax[idx[2]]
|
||||
om[idx[1],idx[0]] = q - s*ax[idx[2]]
|
||||
return om if _P < 0.0 else om.T
|
||||
|
||||
def ax2ro(ax):
|
||||
"""Axis angle pair to Rodrigues-Frank vector."""
|
||||
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)
|
||||
return ro
|
||||
|
||||
def ax2ho(ax):
|
||||
"""Axis angle pair to homochoric vector."""
|
||||
f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0)
|
||||
ho = ax[0:3] * f
|
||||
return ho
|
||||
|
||||
|
||||
#---------- Rodrigues-Frank vector ----------
|
||||
def ro2ax(ro):
|
||||
"""Rodrigues-Frank vector to axis angle pair."""
|
||||
if np.abs(ro[3]) < 1.e-8:
|
||||
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 ])
|
||||
return ax
|
||||
|
||||
def ro2ho(ro):
|
||||
"""Rodrigues-Frank vector to homochoric vector."""
|
||||
if np.sum(ro[0:3]**2.0) < 1.e-8:
|
||||
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)
|
||||
return ho
|
||||
|
||||
#---------- Homochoric vector----------
|
||||
def ho2ax(ho):
|
||||
"""Homochoric vector to axis angle pair."""
|
||||
tfit = np.array([+1.0000000000018852, -0.5000000002194847,
|
||||
-0.024999992127593126, -0.003928701544781374,
|
||||
-0.0008152701535450438, -0.0002009500426119712,
|
||||
-0.00002397986776071756, -0.00008202868926605841,
|
||||
+0.00012448715042090092, -0.0001749114214822577,
|
||||
+0.0001703481934140054, -0.00012062065004116828,
|
||||
+0.000059719705868660826, -0.00001980756723965647,
|
||||
+0.000003953714684212874, -0.00000036555001439719544])
|
||||
# normalize h and store the magnitude
|
||||
hmag_squared = np.sum(ho**2.)
|
||||
if iszero(hmag_squared):
|
||||
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
||||
else:
|
||||
hm = hmag_squared
|
||||
|
||||
# convert the magnitude to the rotation angle
|
||||
s = tfit[0] + tfit[1] * hmag_squared
|
||||
for i in range(2,16):
|
||||
hm *= hmag_squared
|
||||
s += tfit[i] * hm
|
||||
ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0)))
|
||||
return ax
|
||||
|
||||
def 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)
|
||||
|
||||
if np.allclose(ho,0.0,rtol=0.0,atol=1.0e-16):
|
||||
cu = np.zeros(3)
|
||||
else:
|
||||
xyz3 = ho[_get_pyramid_order(ho,'forward')]
|
||||
|
||||
# inverse M_3
|
||||
xyz2 = xyz3[0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[2])) )
|
||||
|
||||
# inverse M_2
|
||||
qxy = np.sum(xyz2**2)
|
||||
|
||||
if np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-16):
|
||||
Tinv = np.zeros(2)
|
||||
else:
|
||||
q2 = qxy + np.max(np.abs(xyz2))**2
|
||||
sq2 = np.sqrt(q2)
|
||||
q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2))*sq2))
|
||||
tt = np.clip((np.min(np.abs(xyz2))**2+np.max(np.abs(xyz2))*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
|
||||
Tinv = np.array([1.0,np.arccos(tt)/np.pi*12.0]) if np.abs(xyz2[1]) <= np.abs(xyz2[0]) else \
|
||||
np.array([np.arccos(tt)/np.pi*12.0,1.0])
|
||||
Tinv = q * np.where(xyz2<0.0,-Tinv,Tinv)
|
||||
|
||||
# inverse M_1
|
||||
cu = np.array([ Tinv[0], Tinv[1], (-1.0 if xyz3[2] < 0.0 else 1.0) * rs / np.sqrt(6.0/np.pi) ]) /_sc
|
||||
cu = cu[_get_pyramid_order(ho,'backward')]
|
||||
return cu
|
||||
|
||||
#---------- Cubochoric ----------
|
||||
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
|
||||
|
||||
"""
|
||||
# transform to the sphere grid via the curved square, and intercept the zero point
|
||||
if np.allclose(cu,0.0,rtol=0.0,atol=1.0e-16):
|
||||
ho = np.zeros(3)
|
||||
else:
|
||||
# get pyramide and scale by grid parameter ratio
|
||||
XYZ = cu[_get_pyramid_order(cu,'forward')] * _sc
|
||||
|
||||
# intercept all the points along the z-axis
|
||||
if np.allclose(XYZ[0:2],0.0,rtol=0.0,atol=1.0e-16):
|
||||
ho = np.array([0.0, 0.0, np.sqrt(6.0/np.pi) * XYZ[2]])
|
||||
else:
|
||||
order = [1,0] if np.abs(XYZ[1]) <= np.abs(XYZ[0]) else [0,1]
|
||||
q = np.pi/12.0 * XYZ[order[0]]/XYZ[order[1]]
|
||||
c = np.cos(q)
|
||||
s = np.sin(q)
|
||||
q = _R1*2.0**0.25/_beta * XYZ[order[1]] / np.sqrt(np.sqrt(2.0)-c)
|
||||
T = np.array([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
|
||||
|
||||
# transform to sphere grid (inverse Lambert)
|
||||
# note that there is no need to worry about dividing by zero, since XYZ[2] can not become zero
|
||||
c = np.sum(T**2)
|
||||
s = c * np.pi/24.0 /XYZ[2]**2
|
||||
c = c * np.sqrt(np.pi/24.0)/XYZ[2]
|
||||
|
||||
q = np.sqrt( 1.0 - s )
|
||||
ho = np.array([ T[order[1]] * q, T[order[0]] * q, np.sqrt(6.0/np.pi) * XYZ[2] - c ])
|
||||
|
||||
ho = ho[_get_pyramid_order(cu,'backward')]
|
||||
return ho
|
||||
|
||||
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]])}
|
||||
if np.maximum(abs(xyz[0]),abs(xyz[1])) <= xyz[2] or \
|
||||
np.maximum(abs(xyz[0]),abs(xyz[1])) <=-xyz[2]:
|
||||
p = 0
|
||||
elif np.maximum(abs(xyz[1]),abs(xyz[2])) <= xyz[0] or \
|
||||
np.maximum(abs(xyz[1]),abs(xyz[2])) <=-xyz[0]:
|
||||
p = 1
|
||||
elif np.maximum(abs(xyz[2]),abs(xyz[0])) <= xyz[1] or \
|
||||
np.maximum(abs(xyz[2]),abs(xyz[0])) <=-xyz[1]:
|
||||
p = 2
|
||||
return order[direction][p]
|
||||
|
||||
####################################################################################################
|
||||
####################################################################################################
|
||||
|
||||
|
||||
class TestRotation:
|
||||
|
||||
@pytest.mark.parametrize('forward,backward',[(Rotation._qu2om,Rotation._om2qu),
|
||||
|
|
@ -289,11 +711,11 @@ class TestRotation:
|
|||
with pytest.raises(ValueError):
|
||||
function(invalid)
|
||||
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._qu2om,rotation_conversion.qu2om),
|
||||
(Rotation._qu2eu,rotation_conversion.qu2eu),
|
||||
(Rotation._qu2ax,rotation_conversion.qu2ax),
|
||||
(Rotation._qu2ro,rotation_conversion.qu2ro),
|
||||
(Rotation._qu2ho,rotation_conversion.qu2ho)])
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._qu2om,qu2om),
|
||||
(Rotation._qu2eu,qu2eu),
|
||||
(Rotation._qu2ax,qu2ax),
|
||||
(Rotation._qu2ro,qu2ro),
|
||||
(Rotation._qu2ho,qu2ho)])
|
||||
def test_quaternion_vectorization(self,default,vectorized,single):
|
||||
qu = np.array([rot.as_quaternion() for rot in default])
|
||||
vectorized(qu.reshape(qu.shape[0]//2,-1,4))
|
||||
|
|
@ -303,9 +725,9 @@ class TestRotation:
|
|||
assert np.allclose(single(q),c) and np.allclose(single(q),vectorized(q))
|
||||
|
||||
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._om2qu,rotation_conversion.om2qu),
|
||||
(Rotation._om2eu,rotation_conversion.om2eu),
|
||||
(Rotation._om2ax,rotation_conversion.om2ax)])
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._om2qu,om2qu),
|
||||
(Rotation._om2eu,om2eu),
|
||||
(Rotation._om2ax,om2ax)])
|
||||
def test_matrix_vectorization(self,default,vectorized,single):
|
||||
om = np.array([rot.as_matrix() for rot in default])
|
||||
vectorized(om.reshape(om.shape[0]//2,-1,3,3))
|
||||
|
|
@ -314,10 +736,10 @@ class TestRotation:
|
|||
print(o,c)
|
||||
assert np.allclose(single(o),c) and np.allclose(single(o),vectorized(o))
|
||||
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._eu2qu,rotation_conversion.eu2qu),
|
||||
(Rotation._eu2om,rotation_conversion.eu2om),
|
||||
(Rotation._eu2ax,rotation_conversion.eu2ax),
|
||||
(Rotation._eu2ro,rotation_conversion.eu2ro)])
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._eu2qu,eu2qu),
|
||||
(Rotation._eu2om,eu2om),
|
||||
(Rotation._eu2ax,eu2ax),
|
||||
(Rotation._eu2ro,eu2ro)])
|
||||
def test_Euler_vectorization(self,default,vectorized,single):
|
||||
eu = np.array([rot.as_Eulers() for rot in default])
|
||||
vectorized(eu.reshape(eu.shape[0]//2,-1,3))
|
||||
|
|
@ -326,10 +748,10 @@ class TestRotation:
|
|||
print(e,c)
|
||||
assert np.allclose(single(e),c) and np.allclose(single(e),vectorized(e))
|
||||
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._ax2qu,rotation_conversion.ax2qu),
|
||||
(Rotation._ax2om,rotation_conversion.ax2om),
|
||||
(Rotation._ax2ro,rotation_conversion.ax2ro),
|
||||
(Rotation._ax2ho,rotation_conversion.ax2ho)])
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._ax2qu,ax2qu),
|
||||
(Rotation._ax2om,ax2om),
|
||||
(Rotation._ax2ro,ax2ro),
|
||||
(Rotation._ax2ho,ax2ho)])
|
||||
def test_axisAngle_vectorization(self,default,vectorized,single):
|
||||
ax = np.array([rot.as_axis_angle() for rot in default])
|
||||
vectorized(ax.reshape(ax.shape[0]//2,-1,4))
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|
|
@ -339,8 +761,8 @@ class TestRotation:
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assert np.allclose(single(a),c) and np.allclose(single(a),vectorized(a))
|
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|
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|
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@pytest.mark.parametrize('vectorized, single',[(Rotation._ro2ax,rotation_conversion.ro2ax),
|
||||
(Rotation._ro2ho,rotation_conversion.ro2ho)])
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._ro2ax,ro2ax),
|
||||
(Rotation._ro2ho,ro2ho)])
|
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def test_Rodrigues_vectorization(self,default,vectorized,single):
|
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ro = np.array([rot.as_Rodrigues() for rot in default])
|
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vectorized(ro.reshape(ro.shape[0]//2,-1,4))
|
||||
|
|
@ -349,8 +771,8 @@ class TestRotation:
|
|||
print(r,c)
|
||||
assert np.allclose(single(r),c) and np.allclose(single(r),vectorized(r))
|
||||
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._ho2ax,rotation_conversion.ho2ax),
|
||||
(Rotation._ho2cu,rotation_conversion.ho2cu)])
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._ho2ax,ho2ax),
|
||||
(Rotation._ho2cu,ho2cu)])
|
||||
def test_homochoric_vectorization(self,default,vectorized,single):
|
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ho = np.array([rot.as_homochoric() for rot in default])
|
||||
vectorized(ho.reshape(ho.shape[0]//2,-1,3))
|
||||
|
|
@ -359,7 +781,7 @@ class TestRotation:
|
|||
print(h,c)
|
||||
assert np.allclose(single(h),c) and np.allclose(single(h),vectorized(h))
|
||||
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._cu2ho,rotation_conversion.cu2ho)])
|
||||
@pytest.mark.parametrize('vectorized, single',[(Rotation._cu2ho,cu2ho)])
|
||||
def test_cubochoric_vectorization(self,default,vectorized,single):
|
||||
cu = np.array([rot.as_cubochoric() for rot in default])
|
||||
vectorized(cu.reshape(cu.shape[0]//2,-1,3))
|
||||
|
|
|
|||
Loading…
Reference in New Issue