no 'dangling' functions
@staticmethod is what we need here
This commit is contained in:
Martin Diehl
parent
1e1cb3f151
commit
c84a6e90c9
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@ -4,6 +4,13 @@ from . import Lambert
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P = -1
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def isone(a):
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return np.isclose(a,1.0,atol=1.0e-7,rtol=0.0)
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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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####################################################################################################
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class Rotation:
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u"""
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@ -183,7 +190,7 @@ class Rotation:
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return angles in degrees.
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"""
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eu = qu2eu(self.quaternion)
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eu = Rotation.qu2eu(self.quaternion)
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if degrees: eu = np.degrees(eu)
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return eu
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@ -201,13 +208,13 @@ class Rotation:
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return tuple of axis and angle.
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"""
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ax = qu2ax(self.quaternion)
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ax = Rotation.qu2ax(self.quaternion)
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if degrees: ax[3] = np.degrees(ax[3])
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return (ax[:3],np.degrees(ax[3])) if pair else ax
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def asMatrix(self):
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"""Rotation matrix."""
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return qu2om(self.quaternion)
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return Rotation.qu2om(self.quaternion)
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def asRodrigues(self,
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vector = False):
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@ -221,16 +228,16 @@ class Rotation:
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return as actual Rodrigues--Frank vector, i.e. rotation axis scaled by tan(ω/2).
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"""
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ro = qu2ro(self.quaternion)
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ro = Rotation.qu2ro(self.quaternion)
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return ro[:3]*ro[3] if vector else ro
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def asHomochoric(self):
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"""Homochoric vector: (h_1, h_2, h_3)."""
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return qu2ho(self.quaternion)
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return Rotation.qu2ho(self.quaternion)
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def asCubochoric(self):
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"""Cubochoric vector: (c_1, c_2, c_3)."""
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return qu2cu(self.quaternion)
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return Rotation.qu2cu(self.quaternion)
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def asM(self):
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"""
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@ -276,7 +283,7 @@ class Rotation:
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if np.any(eu < 0.0) or np.any(eu > 2.0*np.pi) or eu[1] > np.pi:
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raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].\n{} {} {}.'.format(*eu))
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return Rotation(eu2qu(eu))
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return Rotation(Rotation.eu2qu(eu))
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@staticmethod
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def fromAxisAngle(angleAxis,
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@ -294,7 +301,7 @@ class Rotation:
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if not np.isclose(np.linalg.norm(ax[0:3]), 1.0):
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raise ValueError('Axis angle rotation axis is not of unit length.\n{} {} {}'.format(*ax[0:3]))
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return Rotation(ax2qu(ax))
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return Rotation(Rotation.ax2qu(ax))
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@staticmethod
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def fromBasis(basis,
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@ -316,7 +323,7 @@ class Rotation:
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or not np.isclose(np.dot(om[2],om[0]), 0.0):
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raise ValueError('matrix is not orthogonal.\n{}'.format(om))
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return Rotation(om2qu(om))
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return Rotation(Rotation.om2qu(om))
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@staticmethod
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def fromMatrix(om,
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@ -338,7 +345,7 @@ class Rotation:
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if ro[3] < 0.0:
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raise ValueError('Rodriques rotation angle not positive.\n'.format(ro[3]))
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return Rotation(ro2qu(ro))
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return Rotation(Rotation.ro2qu(ro))
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@staticmethod
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def fromHomochoric(homochoric,
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@ -348,7 +355,7 @@ class Rotation:
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else np.array(homochoric,dtype=float)
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if P > 0: ho *= -1 # convert from P=1 to P=-1
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return Rotation(ho2qu(ho))
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return Rotation(Rotation.ho2qu(ho))
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@staticmethod
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def fromCubochoric(cubochoric,
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@ -356,10 +363,10 @@ class Rotation:
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cu = cubochoric if isinstance(cubochoric, np.ndarray) and cubochoric.dtype == np.dtype(float) \
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else np.array(cubochoric,dtype=float)
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ho = cu2ho(cu)
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ho = Rotation.cu2ho(cu)
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if P > 0: ho *= -1 # convert from P=1 to P=-1
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return Rotation(ho2qu(ho))
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return Rotation(Rotation.ho2qu(ho))
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@staticmethod
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@ -437,417 +444,410 @@ class Rotation:
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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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def isone(a):
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return np.isclose(a,1.0,atol=1.0e-7,rtol=0.0)
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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[1,0] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3])
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om[0,1] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3])
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om[2,1] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1])
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om[1,2] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1])
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om[0,2] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2])
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om[2,0] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2])
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return om if P > 0.0 else om.T
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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 iszero(chi):
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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]) if iszero(q12) else \
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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, i.e a lower limit of 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 iszero(qu[1]**2+qu[2]**2+qu[3]**2): # set axis to [001] if the angle is 0/360
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ax = [ 0.0, 0.0, 1.0, 0.0 ]
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elif not iszero(qu[0]):
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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 = [ qu[1]*s, qu[2]*s, qu[3]*s, omega ]
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else:
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ax = [ qu[1], qu[2], qu[3], np.pi]
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return np.array(ax)
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def qu2ro(qu):
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"""Quaternion to Rodriques-Frank vector."""
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if iszero(qu[0]):
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ro = [qu[1], qu[2], qu[3], np.inf]
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else:
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s = np.linalg.norm([qu[1],qu[2],qu[3]])
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ro = [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)))] # avoid numerical difficulties
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return np.array(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)) # avoid numerical difficulties
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if iszero(omega):
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ho = np.array([ 0.0, 0.0, 0.0 ])
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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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def qu2cu(qu):
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"""Quaternion to cubochoric vector."""
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return ho2cu(qu2ho(qu))
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#---------- Rotation matrix ----------
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def om2qu(om):
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"""
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Rotation matrix to quaternion.
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The original formulation (direct conversion) had (numerical?) issues
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"""
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return eu2qu(om2eu(om))
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def om2eu(om):
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"""Rotation matrix to Bunge-Euler angles."""
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if abs(om[2,2]) < 1.0:
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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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# reduce Euler angles to definition range, i.e a lower limit of 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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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 iszero(ax[3]):
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ax = [ 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 = 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(iszero(diagDelta), ax[0:3],np.abs(ax[0:3])*np.sign(-P*diagDelta))
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return np.array(ax)
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def om2ro(om):
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"""Rotation matrix to Rodriques-Frank vector."""
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return eu2ro(om2eu(om))
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def om2ho(om):
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"""Rotation matrix to homochoric vector."""
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return ax2ho(om2ax(om))
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def om2cu(om):
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"""Rotation matrix to cubochoric vector."""
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return ho2cu(om2ho(om))
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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.where(iszero(om))] = 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 iszero(alpha):
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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 Rodriques-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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def eu2ho(eu):
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"""Bunge-Euler angles to homochoric vector."""
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return ax2ho(eu2ax(eu))
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def eu2cu(eu):
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"""Bunge-Euler angles to cubochoric vector."""
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return ho2cu(eu2ho(eu))
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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 iszero(ax[3]):
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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 ax2eu(ax):
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"""Rotation matrix to Bunge Euler angles."""
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return om2eu(ax2om(ax))
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def ax2ro(ax):
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"""Axis angle pair to Rodriques-Frank vector."""
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if iszero(ax[3]):
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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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return np.array(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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def ax2cu(ax):
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"""Axis angle pair to cubochoric vector."""
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return ho2cu(ax2ho(ax))
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#---------- Rodrigues-Frank vector ----------
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def ro2qu(ro):
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"""Rodriques-Frank vector to quaternion."""
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return ax2qu(ro2ax(ro))
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def ro2om(ro):
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"""Rodgrigues-Frank vector to rotation matrix."""
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return ax2om(ro2ax(ro))
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def ro2eu(ro):
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"""Rodriques-Frank vector to Bunge-Euler angles."""
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return om2eu(ro2om(ro))
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def ro2ax(ro):
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"""Rodriques-Frank vector to axis angle pair."""
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ta = ro[3]
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if iszero(ta):
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ax = [ 0.0, 0.0, 1.0, 0.0 ]
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elif not np.isfinite(ta):
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ax = [ ro[0], ro[1], ro[2], np.pi ]
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else:
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angle = 2.0*np.arctan(ta)
|
||||
ta = 1.0/np.linalg.norm(ro[0:3])
|
||||
ax = [ ro[0]/ta, ro[1]/ta, ro[2]/ta, angle ]
|
||||
|
||||
return np.array(ax)
|
||||
|
||||
|
||||
def ro2ho(ro):
|
||||
"""Rodriques-Frank vector to homochoric vector."""
|
||||
if iszero(np.sum(ro[0:3]**2.0)):
|
||||
ho = [ 0.0, 0.0, 0.0 ]
|
||||
else:
|
||||
f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
|
||||
ho = ro[0:3] * (0.75*f)**(1.0/3.0)
|
||||
|
||||
return np.array(ho)
|
||||
|
||||
|
||||
def ro2cu(ro):
|
||||
"""Rodriques-Frank vector to cubochoric vector."""
|
||||
return ho2cu(ro2ho(ro))
|
||||
|
||||
|
||||
#---------- Homochoric vector----------
|
||||
|
||||
def ho2qu(ho):
|
||||
"""Homochoric vector to quaternion."""
|
||||
return ax2qu(ho2ax(ho))
|
||||
|
||||
|
||||
def ho2om(ho):
|
||||
"""Homochoric vector to rotation matrix."""
|
||||
return ax2om(ho2ax(ho))
|
||||
|
||||
|
||||
def ho2eu(ho):
|
||||
"""Homochoric vector to Bunge-Euler angles."""
|
||||
return ax2eu(ho2ax(ho))
|
||||
|
||||
|
||||
def ho2ax(ho):
|
||||
"""Homochoric vector to axis angle pair."""
|
||||
tfit = np.array([+1.0000000000018852, -0.5000000002194847,
|
||||
-0.024999992127593126, -0.003928701544781374,
|
||||
-0.0008152701535450438, -0.0002009500426119712,
|
||||
-0.00002397986776071756, -0.00008202868926605841,
|
||||
+0.00012448715042090092, -0.0001749114214822577,
|
||||
+0.0001703481934140054, -0.00012062065004116828,
|
||||
+0.000059719705868660826, -0.00001980756723965647,
|
||||
+0.000003953714684212874, -0.00000036555001439719544])
|
||||
# normalize h and store the magnitude
|
||||
hmag_squared = np.sum(ho**2.)
|
||||
if iszero(hmag_squared):
|
||||
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
||||
else:
|
||||
hm = hmag_squared
|
||||
|
||||
# convert the magnitude to the rotation angle
|
||||
s = tfit[0] + tfit[1] * hmag_squared
|
||||
for i in range(2,16):
|
||||
hm *= hmag_squared
|
||||
s += tfit[i] * hm
|
||||
ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0)))
|
||||
return ax
|
||||
|
||||
|
||||
def ho2ro(ho):
|
||||
"""Axis angle pair to Rodriques-Frank vector."""
|
||||
return ax2ro(ho2ax(ho))
|
||||
|
||||
|
||||
def ho2cu(ho):
|
||||
"""Homochoric vector to cubochoric vector."""
|
||||
return Lambert.BallToCube(ho)
|
||||
|
||||
|
||||
#---------- Cubochoric ----------
|
||||
|
||||
def cu2qu(cu):
|
||||
"""Cubochoric vector to quaternion."""
|
||||
return ho2qu(cu2ho(cu))
|
||||
|
||||
|
||||
def cu2om(cu):
|
||||
"""Cubochoric vector to rotation matrix."""
|
||||
return ho2om(cu2ho(cu))
|
||||
|
||||
|
||||
def cu2eu(cu):
|
||||
"""Cubochoric vector to Bunge-Euler angles."""
|
||||
return ho2eu(cu2ho(cu))
|
||||
|
||||
|
||||
def cu2ax(cu):
|
||||
"""Cubochoric vector to axis angle pair."""
|
||||
return ho2ax(cu2ho(cu))
|
||||
|
||||
|
||||
def cu2ro(cu):
|
||||
"""Cubochoric vector to Rodriques-Frank vector."""
|
||||
return ho2ro(cu2ho(cu))
|
||||
|
||||
|
||||
def cu2ho(cu):
|
||||
"""Cubochoric vector to homochoric vector."""
|
||||
return Lambert.CubeToBall(cu)
|
||||
#---------- Quaternion ----------
|
||||
@staticmethod
|
||||
def qu2om(qu):
|
||||
"""Quaternion to rotation matrix."""
|
||||
qq = qu[0]**2-(qu[1]**2 + qu[2]**2 + qu[3]**2)
|
||||
om = np.diag(qq + 2.0*np.array([qu[1],qu[2],qu[3]])**2)
|
||||
|
||||
om[1,0] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3])
|
||||
om[0,1] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3])
|
||||
om[2,1] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1])
|
||||
om[1,2] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1])
|
||||
om[0,2] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2])
|
||||
om[2,0] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2])
|
||||
return om if P > 0.0 else om.T
|
||||
|
||||
@staticmethod
|
||||
def qu2eu(qu):
|
||||
"""Quaternion to Bunge-Euler angles."""
|
||||
q03 = qu[0]**2+qu[3]**2
|
||||
q12 = qu[1]**2+qu[2]**2
|
||||
chi = np.sqrt(q03*q12)
|
||||
|
||||
if iszero(chi):
|
||||
eu = np.array([np.arctan2(-P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0, 0.0]) if iszero(q12) else \
|
||||
np.array([np.arctan2(2.0*qu[1]*qu[2],qu[1]**2-qu[2]**2), np.pi, 0.0])
|
||||
else:
|
||||
eu = np.array([np.arctan2((-P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-P*qu[0]*qu[1]-qu[2]*qu[3])*chi ),
|
||||
np.arctan2( 2.0*chi, q03-q12 ),
|
||||
np.arctan2(( P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-P*qu[0]*qu[1]+qu[2]*qu[3])*chi )])
|
||||
|
||||
# reduce Euler angles to definition range, i.e a lower limit of 0.0
|
||||
eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
|
||||
return eu
|
||||
|
||||
@staticmethod
|
||||
def qu2ax(qu):
|
||||
"""
|
||||
Quaternion to axis angle pair.
|
||||
|
||||
Modified version of the original formulation, should be numerically more stable
|
||||
"""
|
||||
if iszero(qu[1]**2+qu[2]**2+qu[3]**2): # set axis to [001] if the angle is 0/360
|
||||
ax = [ 0.0, 0.0, 1.0, 0.0 ]
|
||||
elif not iszero(qu[0]):
|
||||
s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2)
|
||||
omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
|
||||
ax = [ qu[1]*s, qu[2]*s, qu[3]*s, omega ]
|
||||
else:
|
||||
ax = [ qu[1], qu[2], qu[3], np.pi]
|
||||
|
||||
return np.array(ax)
|
||||
|
||||
@staticmethod
|
||||
def qu2ro(qu):
|
||||
"""Quaternion to Rodriques-Frank vector."""
|
||||
if iszero(qu[0]):
|
||||
ro = [qu[1], qu[2], qu[3], np.inf]
|
||||
else:
|
||||
s = np.linalg.norm([qu[1],qu[2],qu[3]])
|
||||
ro = [0.0,0.0,P,0.0] if iszero(s) else \
|
||||
[ qu[1]/s, qu[2]/s, qu[3]/s, np.tan(np.arccos(np.clip(qu[0],-1.0,1.0)))] # avoid numerical difficulties
|
||||
|
||||
return np.array(ro)
|
||||
|
||||
@staticmethod
|
||||
def qu2ho(qu):
|
||||
"""Quaternion to homochoric vector."""
|
||||
omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0)) # avoid numerical difficulties
|
||||
|
||||
if iszero(omega):
|
||||
ho = np.array([ 0.0, 0.0, 0.0 ])
|
||||
else:
|
||||
ho = np.array([qu[1], qu[2], qu[3]])
|
||||
f = 0.75 * ( omega - np.sin(omega) )
|
||||
ho = ho/np.linalg.norm(ho) * f**(1./3.)
|
||||
|
||||
return ho
|
||||
|
||||
@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 abs(om[2,2]) < 1.0:
|
||||
zeta = 1.0/np.sqrt(1.0-om[2,2]**2)
|
||||
eu = np.array([np.arctan2(om[2,0]*zeta,-om[2,1]*zeta),
|
||||
np.arccos(om[2,2]),
|
||||
np.arctan2(om[0,2]*zeta, om[1,2]*zeta)])
|
||||
else:
|
||||
eu = np.array([np.arctan2( om[0,1],om[0,0]), np.pi*0.5*(1-om[2,2]),0.0]) # following the paper, not the reference implementation
|
||||
|
||||
# reduce Euler angles to definition range, i.e a lower limit of 0.0
|
||||
eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
|
||||
return eu
|
||||
|
||||
@staticmethod
|
||||
def om2ax(om):
|
||||
"""Rotation matrix to axis angle pair."""
|
||||
ax=np.empty(4)
|
||||
|
||||
# first get the rotation angle
|
||||
t = 0.5*(om.trace() -1.0)
|
||||
ax[3] = np.arccos(np.clip(t,-1.0,1.0))
|
||||
|
||||
if iszero(ax[3]):
|
||||
ax = [ 0.0, 0.0, 1.0, 0.0]
|
||||
else:
|
||||
w,vr = np.linalg.eig(om)
|
||||
# next, find the eigenvalue (1,0j)
|
||||
i = np.where(np.isclose(w,1.0+0.0j))[0][0]
|
||||
ax[0:3] = np.real(vr[0:3,i])
|
||||
diagDelta = np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]])
|
||||
ax[0:3] = np.where(iszero(diagDelta), ax[0:3],np.abs(ax[0:3])*np.sign(-P*diagDelta))
|
||||
|
||||
return np.array(ax)
|
||||
|
||||
@staticmethod
|
||||
def om2ro(om):
|
||||
"""Rotation matrix to Rodriques-Frank vector."""
|
||||
return Rotation.eu2ro(Rotation.om2eu(om))
|
||||
|
||||
@staticmethod
|
||||
def om2ho(om):
|
||||
"""Rotation matrix to homochoric vector."""
|
||||
return Rotation.ax2ho(Rotation.om2ax(om))
|
||||
|
||||
@staticmethod
|
||||
def om2cu(om):
|
||||
"""Rotation matrix to cubochoric vector."""
|
||||
return Rotation.ho2cu(Rotation.om2ho(om))
|
||||
|
||||
|
||||
#---------- Bunge-Euler angles ----------
|
||||
@staticmethod
|
||||
def eu2qu(eu):
|
||||
"""Bunge-Euler angles to quaternion."""
|
||||
ee = 0.5*eu
|
||||
cPhi = np.cos(ee[1])
|
||||
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
|
||||
|
||||
@staticmethod
|
||||
def eu2om(eu):
|
||||
"""Bunge-Euler angles to rotation matrix."""
|
||||
c = np.cos(eu)
|
||||
s = np.sin(eu)
|
||||
|
||||
om = np.array([[+c[0]*c[2]-s[0]*s[2]*c[1], +s[0]*c[2]+c[0]*s[2]*c[1], +s[2]*s[1]],
|
||||
[-c[0]*s[2]-s[0]*c[2]*c[1], -s[0]*s[2]+c[0]*c[2]*c[1], +c[2]*s[1]],
|
||||
[+s[0]*s[1], -c[0]*s[1], +c[1] ]])
|
||||
|
||||
om[np.where(iszero(om))] = 0.0
|
||||
return om
|
||||
|
||||
@staticmethod
|
||||
def eu2ax(eu):
|
||||
"""Bunge-Euler angles to axis angle pair."""
|
||||
t = np.tan(eu[1]*0.5)
|
||||
sigma = 0.5*(eu[0]+eu[2])
|
||||
delta = 0.5*(eu[0]-eu[2])
|
||||
tau = np.linalg.norm([t,np.sin(sigma)])
|
||||
alpha = np.pi if iszero(np.cos(sigma)) else \
|
||||
2.0*np.arctan(tau/np.cos(sigma))
|
||||
|
||||
if iszero(alpha):
|
||||
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
||||
else:
|
||||
ax = -P/tau * np.array([ t*np.cos(delta), t*np.sin(delta), np.sin(sigma) ]) # passive axis angle pair so a minus sign in front
|
||||
ax = np.append(ax,alpha)
|
||||
if alpha < 0.0: ax *= -1.0 # ensure alpha is positive
|
||||
|
||||
return ax
|
||||
|
||||
@staticmethod
|
||||
def eu2ro(eu):
|
||||
"""Bunge-Euler angles to Rodriques-Frank vector."""
|
||||
ro = eu2ax(eu) # convert to axis angle pair representation
|
||||
if ro[3] >= np.pi: # Differs from original implementation. check convention 5
|
||||
ro[3] = np.inf
|
||||
elif iszero(ro[3]):
|
||||
ro = np.array([ 0.0, 0.0, P, 0.0 ])
|
||||
else:
|
||||
ro[3] = np.tan(ro[3]*0.5)
|
||||
|
||||
return ro
|
||||
|
||||
@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 iszero(ax[3]):
|
||||
qu = np.array([ 1.0, 0.0, 0.0, 0.0 ])
|
||||
else:
|
||||
c = np.cos(ax[3]*0.5)
|
||||
s = np.sin(ax[3]*0.5)
|
||||
qu = np.array([ c, ax[0]*s, ax[1]*s, ax[2]*s ])
|
||||
|
||||
return qu
|
||||
|
||||
@staticmethod
|
||||
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
|
||||
|
||||
@staticmethod
|
||||
def ax2eu(ax):
|
||||
"""Rotation matrix to Bunge Euler angles."""
|
||||
return Rotation.om2eu(Rotation.ax2om(ax))
|
||||
|
||||
@staticmethod
|
||||
def ax2ro(ax):
|
||||
"""Axis angle pair to Rodriques-Frank vector."""
|
||||
if iszero(ax[3]):
|
||||
ro = [ 0.0, 0.0, P, 0.0 ]
|
||||
else:
|
||||
ro = [ax[0], ax[1], ax[2]]
|
||||
# 180 degree case
|
||||
ro += [np.inf] if np.isclose(ax[3],np.pi,atol=1.0e-15,rtol=0.0) else \
|
||||
[np.tan(ax[3]*0.5)]
|
||||
|
||||
return np.array(ro)
|
||||
|
||||
@staticmethod
|
||||
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
|
||||
|
||||
@staticmethod
|
||||
def ax2cu(ax):
|
||||
"""Axis angle pair to cubochoric vector."""
|
||||
return Rotation.ho2cu(Rotation.ax2ho(ax))
|
||||
|
||||
|
||||
#---------- Rodrigues-Frank vector ----------
|
||||
@staticmethod
|
||||
def ro2qu(ro):
|
||||
"""Rodriques-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):
|
||||
"""Rodriques-Frank vector to Bunge-Euler angles."""
|
||||
return Rotation.om2eu(Rotation.ro2om(ro))
|
||||
|
||||
@staticmethod
|
||||
def ro2ax(ro):
|
||||
"""Rodriques-Frank vector to axis angle pair."""
|
||||
ta = ro[3]
|
||||
|
||||
if iszero(ta):
|
||||
ax = [ 0.0, 0.0, 1.0, 0.0 ]
|
||||
elif not np.isfinite(ta):
|
||||
ax = [ ro[0], ro[1], ro[2], np.pi ]
|
||||
else:
|
||||
angle = 2.0*np.arctan(ta)
|
||||
ta = 1.0/np.linalg.norm(ro[0:3])
|
||||
ax = [ ro[0]/ta, ro[1]/ta, ro[2]/ta, angle ]
|
||||
|
||||
return np.array(ax)
|
||||
|
||||
@staticmethod
|
||||
def ro2ho(ro):
|
||||
"""Rodriques-Frank vector to homochoric vector."""
|
||||
if iszero(np.sum(ro[0:3]**2.0)):
|
||||
ho = [ 0.0, 0.0, 0.0 ]
|
||||
else:
|
||||
f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
|
||||
ho = ro[0:3] * (0.75*f)**(1.0/3.0)
|
||||
|
||||
return np.array(ho)
|
||||
|
||||
@staticmethod
|
||||
def ro2cu(ro):
|
||||
"""Rodriques-Frank vector to cubochoric vector."""
|
||||
return ho2cu(ro2ho(ro))
|
||||
|
||||
|
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#---------- Homochoric vector----------
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@staticmethod
|
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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])
|
||||
# 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
|
||||
|
||||
@staticmethod
|
||||
def ho2ro(ho):
|
||||
"""Axis angle pair to Rodriques-Frank vector."""
|
||||
return Rotation.ax2ro(Rotation.ho2ax(ho))
|
||||
|
||||
@staticmethod
|
||||
def ho2cu(ho):
|
||||
"""Homochoric vector to cubochoric vector."""
|
||||
return Lambert.BallToCube(ho)
|
||||
|
||||
|
||||
#---------- 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 Rodriques-Frank vector."""
|
||||
return Rotation.ho2ro(Rotation.cu2ho(cu))
|
||||
|
||||
@staticmethod
|
||||
def cu2ho(cu):
|
||||
"""Cubochoric vector to homochoric vector."""
|
||||
return Lambert.CubeToBall(cu)
|
||||
|
|
|
|||
Loading…
Reference in New Issue