1049 lines
42 KiB
Python
1049 lines
42 KiB
Python
import numpy as np
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from ._Lambert import ball_to_cube, cube_to_ball
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_P = -1
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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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class Rotation:
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u"""
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Orientation stored with functionality for conversion to different representations.
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References
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----------
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D. Rowenhorst et al., Modelling and Simulation in Materials Science and Engineering 23:083501, 2015
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https://doi.org/10.1088/0965-0393/23/8/083501
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Conventions
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-----------
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Convention 1: Coordinate frames are right-handed.
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Convention 2: A rotation angle ω is taken to be positive for a counterclockwise rotation
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when viewing from the end point of the rotation axis towards the origin.
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Convention 3: Rotations will be interpreted in the passive sense.
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Convention 4: Euler angle triplets are implemented using the Bunge convention,
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with the angular ranges as [0, 2π],[0, π],[0, 2π].
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Convention 5: The rotation angle ω is limited to the interval [0, π].
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Convention 6: the real part of a quaternion is positive, Re(q) > 0
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Convention 7: P = -1 (as default).
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Usage
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-----
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Vector "a" (defined in coordinate system "A") is passively rotated
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resulting in new coordinates "b" when expressed in system "B".
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b = Q * a
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b = np.dot(Q.asMatrix(),a)
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"""
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__slots__ = ['quaternion']
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def __init__(self,quaternion = np.array([1.0,0.0,0.0,0.0])):
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"""
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Initializes to identity unless specified.
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Parameters
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----------
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quaternion : numpy.ndarray, optional
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Unit quaternion that follows the conventions. Use .fromQuaternion to perform a sanity check.
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"""
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self.quaternion = quaternion.copy()
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def __copy__(self):
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"""Copy."""
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return self.__class__(self.quaternion)
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copy = __copy__
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def __repr__(self):
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"""Orientation displayed as unit quaternion, rotation matrix, and Bunge-Euler angles."""
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return '\n'.join([
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'Quaternion: (real={:.3f}, imag=<{:+.3f}, {:+.3f}, {:+.3f}>)'.format(*(self.quaternion)),
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'Matrix:\n{}'.format(self.asMatrix()),
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'Bunge Eulers / deg: ({:3.2f}, {:3.2f}, {:3.2f})'.format(*self.asEulers(degrees=True)),
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])
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def __mul__(self, other):
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"""
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Multiplication.
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Parameters
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----------
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other : numpy.ndarray or Rotation
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Vector, second or fourth order tensor, or rotation object that is rotated.
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Todo
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----
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Document details active/passive)
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considere rotation of (3,3,3,3)-matrix
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"""
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if isinstance(other, Rotation): # rotate a rotation
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self_q = self.quaternion[0]
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self_p = self.quaternion[1:]
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other_q = other.quaternion[0]
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other_p = other.quaternion[1:]
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R = self.__class__(np.append(self_q*other_q - np.dot(self_p,other_p),
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self_q*other_p + other_q*self_p + _P * np.cross(self_p,other_p)))
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return R.standardize()
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elif isinstance(other, (tuple,np.ndarray)):
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if isinstance(other,tuple) or other.shape == (3,): # rotate a single (3)-vector or meshgrid
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A = self.quaternion[0]**2.0 - np.dot(self.quaternion[1:],self.quaternion[1:])
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B = 2.0 * ( self.quaternion[1]*other[0]
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+ self.quaternion[2]*other[1]
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+ self.quaternion[3]*other[2])
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C = 2.0 * _P*self.quaternion[0]
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return np.array([
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A*other[0] + B*self.quaternion[1] + C*(self.quaternion[2]*other[2] - self.quaternion[3]*other[1]),
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A*other[1] + B*self.quaternion[2] + C*(self.quaternion[3]*other[0] - self.quaternion[1]*other[2]),
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A*other[2] + B*self.quaternion[3] + C*(self.quaternion[1]*other[1] - self.quaternion[2]*other[0]),
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])
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elif other.shape == (3,3,): # rotate a single (3x3)-matrix
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return np.dot(self.asMatrix(),np.dot(other,self.asMatrix().T))
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elif other.shape == (3,3,3,3,):
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raise NotImplementedError
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else:
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return NotImplemented
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else:
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return NotImplemented
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def inverse(self):
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"""In-place inverse rotation/backward rotation."""
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self.quaternion[1:] *= -1
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return self
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def inversed(self):
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"""Inverse rotation/backward rotation."""
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return self.copy().inverse()
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def standardize(self):
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"""In-place quaternion representation with positive q."""
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if self.quaternion[0] < 0.0: self.quaternion*=-1
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return self
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def standardized(self):
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"""Quaternion representation with positive q."""
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return self.copy().standardize()
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def misorientation(self,other):
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"""
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Get Misorientation.
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Parameters
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----------
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other : Rotation
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Rotation to which the misorientation is computed.
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"""
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return other*self.inversed()
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def average(self,other):
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"""
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Calculate the average rotation.
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Parameters
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----------
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other : Rotation
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Rotation from which the average is rotated.
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"""
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return Rotation.fromAverage([self,other])
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################################################################################################
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# convert to different orientation representations (numpy arrays)
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def asQuaternion(self):
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"""
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Unit quaternion [q, p_1, p_2, p_3] unless quaternion == True: damask.quaternion object.
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Parameters
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----------
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quaternion : bool, optional
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return quaternion as DAMASK object.
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"""
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return self.quaternion
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def asEulers(self,
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degrees = False):
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"""
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Bunge-Euler angles: (φ_1, ϕ, φ_2).
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Parameters
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----------
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degrees : bool, optional
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return angles in degrees.
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"""
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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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def asAxisAngle(self,
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degrees = False,
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pair = False):
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"""
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Axis angle representation [n_1, n_2, n_3, ω] unless pair == True: ([n_1, n_2, n_3], ω).
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Parameters
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----------
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degrees : bool, optional
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return rotation angle in degrees.
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pair : bool, optional
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return tuple of axis and angle.
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"""
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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],ax[3]) if pair else ax
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def asMatrix(self):
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"""Rotation matrix."""
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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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"""
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Rodrigues-Frank vector representation [n_1, n_2, n_3, tan(ω/2)] unless vector == True: [n_1, n_2, n_3] * tan(ω/2).
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Parameters
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----------
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vector : bool, optional
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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 = 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 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 Rotation.qu2cu(self.quaternion)
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def asM(self):
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"""
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Intermediate representation supporting quaternion averaging.
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References
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----------
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F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
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https://doi.org/10.2514/1.28949
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"""
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return np.outer(self.quaternion,self.quaternion)
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################################################################################################
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# static constructors. The input data needs to follow the convention, options allow to
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# relax these convections
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@staticmethod
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def fromQuaternion(quaternion,
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acceptHomomorph = False,
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P = -1):
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qu = quaternion if isinstance(quaternion,np.ndarray) and quaternion.dtype == np.dtype(float) \
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else np.array(quaternion,dtype=float)
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if P > 0: qu[1:4] *= -1 # convert from P=1 to P=-1
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if qu[0] < 0.0:
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if acceptHomomorph:
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qu *= -1.
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else:
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raise ValueError('Quaternion has negative first component: {}.'.format(qu[0]))
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if not np.isclose(np.linalg.norm(qu), 1.0):
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raise ValueError('Quaternion is not of unit length: {} {} {} {}.'.format(*qu))
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return Rotation(qu)
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@staticmethod
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def fromEulers(eulers,
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degrees = False):
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eu = eulers if isinstance(eulers, np.ndarray) and eulers.dtype == np.dtype(float) \
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else np.array(eulers,dtype=float)
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eu = np.radians(eu) if degrees else eu
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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]: {} {} {}.'.format(*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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degrees = False,
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normalise = False,
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P = -1):
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ax = angleAxis if isinstance(angleAxis, np.ndarray) and angleAxis.dtype == np.dtype(float) \
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else np.array(angleAxis,dtype=float)
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if P > 0: ax[0:3] *= -1 # convert from P=1 to P=-1
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if degrees: ax[ 3] = np.radians(ax[3])
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if normalise: ax[0:3] /= np.linalg.norm(ax[0:3])
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if ax[3] < 0.0 or ax[3] > np.pi:
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raise ValueError('Axis angle rotation angle outside of [0..π]: {}.'.format(ax[3]))
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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: {} {} {}.'.format(*ax[0:3]))
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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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orthonormal = True,
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reciprocal = False,
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):
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om = basis if isinstance(basis, np.ndarray) else np.array(basis).reshape(3,3)
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if reciprocal:
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om = np.linalg.inv(om.T/np.pi) # transform reciprocal basis set
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orthonormal = False # contains stretch
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if not orthonormal:
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(U,S,Vh) = np.linalg.svd(om) # singular value decomposition
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om = np.dot(U,Vh)
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if not np.isclose(np.linalg.det(om),1.0):
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raise ValueError('matrix is not a proper rotation: {}.'.format(om))
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if not np.isclose(np.dot(om[0],om[1]), 0.0) \
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or not np.isclose(np.dot(om[1],om[2]), 0.0) \
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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: {}.'.format(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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):
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return Rotation.fromBasis(om)
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@staticmethod
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def fromRodrigues(rodrigues,
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normalise = False,
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P = -1):
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ro = rodrigues if isinstance(rodrigues, np.ndarray) and rodrigues.dtype == np.dtype(float) \
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else np.array(rodrigues,dtype=float)
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if P > 0: ro[0:3] *= -1 # convert from P=1 to P=-1
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if normalise: ro[0:3] /= np.linalg.norm(ro[0:3])
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if not np.isclose(np.linalg.norm(ro[0:3]), 1.0):
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raise ValueError('Rodrigues rotation axis is not of unit length: {} {} {}.'.format(*ro[0:3]))
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if ro[3] < 0.0:
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raise ValueError('Rodrigues rotation angle not positive: {}.'.format(ro[3]))
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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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P = -1):
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ho = homochoric if isinstance(homochoric, np.ndarray) and homochoric.dtype == np.dtype(float) \
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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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if np.linalg.norm(ho) > (3.*np.pi/4.)**(1./3.)+1e-9:
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raise ValueError('Coordinate outside of the sphere: {} {} {}.'.format(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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P = -1):
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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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if np.abs(np.max(cu))>np.pi**(2./3.) * 0.5+1e-9:
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raise ValueError('Coordinate outside of the cube: {} {} {}.'.format(*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(Rotation.ho2qu(ho))
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@staticmethod
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def fromAverage(rotations,weights = None):
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"""
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Average rotation.
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References
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----------
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F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
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https://doi.org/10.2514/1.28949
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Parameters
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----------
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rotations : list of Rotations
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Rotations to average from
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weights : list of floats, optional
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Weights for each rotation used for averaging
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"""
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if not all(isinstance(item, Rotation) for item in rotations):
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raise TypeError('Only instances of Rotation can be averaged.')
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N = len(rotations)
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if not weights:
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weights = np.ones(N,dtype='i')
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for i,(r,n) in enumerate(zip(rotations,weights)):
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M = r.asM() * n if i == 0 \
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else M + r.asM() * n # noqa add (multiples) of this rotation to average noqa
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eig, vec = np.linalg.eig(M/N)
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return Rotation.fromQuaternion(np.real(vec.T[eig.argmax()]),acceptHomomorph = True)
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@staticmethod
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def fromRandom():
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r = np.random.random(3)
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A = np.sqrt(r[2])
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B = np.sqrt(1.0-r[2])
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return Rotation(np.array([np.cos(2.0*np.pi*r[0])*A,
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np.sin(2.0*np.pi*r[1])*B,
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np.cos(2.0*np.pi*r[1])*B,
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np.sin(2.0*np.pi*r[0])*A])).standardize()
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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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#---------- Quaternion ----------
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@staticmethod
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def qu2om(qu):
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if len(qu.shape) == 1:
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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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else:
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qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2)
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om = np.block([qq + 2.0*qu[...,1:2]**2,
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2.0*(qu[...,2:3]*qu[...,1:2]+qu[...,0:1]*qu[...,3:4]),
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2.0*(qu[...,3:4]*qu[...,1:2]-qu[...,0:1]*qu[...,2:3]),
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2.0*(qu[...,1:2]*qu[...,2:3]-qu[...,0:1]*qu[...,3:4]),
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qq + 2.0*qu[...,2:3]**2,
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2.0*(qu[...,3:4]*qu[...,2:3]+qu[...,0:1]*qu[...,1:2]),
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|
2.0*(qu[...,1:2]*qu[...,3:4]+qu[...,0:1]*qu[...,2:3]),
|
|
2.0*(qu[...,2:3]*qu[...,3:4]-qu[...,0:1]*qu[...,1:2]),
|
|
qq + 2.0*qu[...,3:4]**2,
|
|
]).reshape(qu.shape[:-1]+(3,3))
|
|
return om if _P < 0.0 else np.swapaxes(om,(-1,-2))
|
|
|
|
@staticmethod
|
|
def qu2eu(qu):
|
|
"""Quaternion to Bunge-Euler angles."""
|
|
if len(qu.shape) == 1:
|
|
q03 = qu[0]**2+qu[3]**2
|
|
q12 = qu[1]**2+qu[2]**2
|
|
chi = np.sqrt(q03*q12)
|
|
if np.abs(q12) < 1.e-8:
|
|
eu = np.array([np.arctan2(-_P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0, 0.0])
|
|
elif np.abs(q03) < 1.e-8:
|
|
eu = np.array([np.arctan2( 2.0*qu[1]*qu[2],qu[1]**2-qu[2]**2), np.pi, 0.0])
|
|
else:
|
|
eu = np.array([np.arctan2((-_P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]-qu[2]*qu[3])*chi ),
|
|
np.arctan2( 2.0*chi, q03-q12 ),
|
|
np.arctan2(( _P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]+qu[2]*qu[3])*chi )])
|
|
else:
|
|
q02 = qu[...,0:1]*qu[...,2:3]
|
|
q13 = qu[...,1:2]*qu[...,3:4]
|
|
q01 = qu[...,0:1]*qu[...,1:2]
|
|
q23 = qu[...,2:3]*qu[...,3:4]
|
|
q03_s = qu[...,0:1]**2+qu[...,3:4]**2
|
|
q12_s = qu[...,1:2]**2+qu[...,2:3]**2
|
|
chi = np.sqrt(q03_s*q12_s)
|
|
|
|
eu = np.where(np.abs(q12_s) < 1.0e-8,
|
|
np.block([np.arctan2(-_P*2.0*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2),
|
|
np.zeros(qu.shape[:-1]+(2,))]),
|
|
np.where(np.abs(q03_s) < 1.0e-8,
|
|
np.block([np.arctan2( 2.0*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2),
|
|
np.broadcast_to(np.pi,qu.shape[:-1]+(1,)),
|
|
np.zeros(qu.shape[:-1]+(1,))]),
|
|
np.block([np.arctan2((-_P*q02+q13)*chi, (-_P*q01-q23)*chi),
|
|
np.arctan2( 2.0*chi, q03_s-q12_s ),
|
|
np.arctan2(( _P*q02+q13)*chi, (-_P*q01+q23)*chi)])
|
|
)
|
|
)
|
|
# reduce Euler angles to definition range
|
|
eu[np.abs(eu)<1.e-6] = 0.0
|
|
eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
|
|
return eu
|
|
|
|
@staticmethod
|
|
def qu2ax(qu):
|
|
"""
|
|
Quaternion to axis angle pair.
|
|
|
|
Modified version of the original formulation, should be numerically more stable
|
|
"""
|
|
if len(qu.shape) == 1:
|
|
if np.abs(np.sum(qu[1:4]**2)) < 1.e-6: # set axis to [001] if the angle is 0/360
|
|
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
|
elif qu[0] > 1.e-6:
|
|
s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2)
|
|
omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
|
|
ax = ax = np.array([ qu[1]*s, qu[2]*s, qu[3]*s, omega ])
|
|
else:
|
|
ax = ax = np.array([ qu[1], qu[2], qu[3], np.pi])
|
|
else:
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
s = np.sign(qu[...,0:1])/np.sqrt(qu[...,1:2]**2+qu[...,2:3]**2+qu[...,3:4]**2)
|
|
omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
|
|
ax = np.where(np.broadcast_to(qu[...,0:1] < 1.0e-6,qu.shape),
|
|
np.block([qu[...,1:4],np.broadcast_to(np.pi,qu.shape[:-1]+(1,))]),
|
|
np.block([qu[...,1:4]*s,omega]))
|
|
ax[np.sum(np.abs(qu[...,1:4])**2,axis=-1) < 1.0e-6,] = [0.0, 0.0, 1.0, 0.0]
|
|
return ax
|
|
|
|
@staticmethod
|
|
def qu2ro(qu):
|
|
"""Quaternion to Rodrigues-Frank vector."""
|
|
if len(qu.shape) == 1:
|
|
if iszero(qu[0]):
|
|
ro = np.array([qu[1], qu[2], qu[3], np.inf])
|
|
else:
|
|
s = np.linalg.norm(qu[1:4])
|
|
ro = np.array([0.0,0.0,_P,0.0] if iszero(s) else \
|
|
[ qu[1]/s, qu[2]/s, qu[3]/s, np.tan(np.arccos(np.clip(qu[0],-1.0,1.0)))])
|
|
else:
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
s = np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True)
|
|
ro = np.where(np.broadcast_to(np.abs(qu[...,0:1]) < 1.0e-12,qu.shape),
|
|
np.block([qu[...,1:2], qu[...,2:3], qu[...,3:4], np.broadcast_to(np.inf,qu.shape[:-1]+(1,))]),
|
|
np.block([qu[...,1:2]/s,qu[...,2:3]/s,qu[...,3:4]/s,
|
|
np.tan(np.arccos(np.clip(qu[...,0:1],-1.0,1.0)))
|
|
])
|
|
)
|
|
ro[np.abs(s).squeeze(-1) < 1.0e-12] = [0.0,0.0,_P,0.0]
|
|
return ro
|
|
|
|
@staticmethod
|
|
def qu2ho(qu):
|
|
"""Quaternion to homochoric vector."""
|
|
if len(qu.shape) == 1:
|
|
omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
|
|
if np.abs(omega) < 1.0e-12:
|
|
ho = np.zeros(3)
|
|
else:
|
|
ho = np.array([qu[1], qu[2], qu[3]])
|
|
f = 0.75 * ( omega - np.sin(omega) )
|
|
ho = ho/np.linalg.norm(ho) * f**(1./3.)
|
|
else:
|
|
with np.errstate(invalid='ignore'):
|
|
omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
|
|
ho = np.where(np.abs(omega) < 1.0e-12,
|
|
np.zeros(3),
|
|
qu[...,1:4]/np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) \
|
|
* (0.75*(omega - np.sin(omega)))**(1./3.))
|
|
return ho
|
|
|
|
@staticmethod
|
|
def qu2cu(qu):
|
|
"""Quaternion to cubochoric vector."""
|
|
return Rotation.ho2cu(Rotation.qu2ho(qu))
|
|
|
|
|
|
#---------- Rotation matrix ----------
|
|
@staticmethod
|
|
def om2qu(om):
|
|
"""
|
|
Rotation matrix to quaternion.
|
|
|
|
The original formulation (direct conversion) had (numerical?) issues
|
|
"""
|
|
return Rotation.eu2qu(Rotation.om2eu(om))
|
|
|
|
@staticmethod
|
|
def om2eu(om):
|
|
"""Rotation matrix to Bunge-Euler angles."""
|
|
if len(om.shape) == 2:
|
|
if not np.isclose(np.abs(om[2,2]),1.0,1.e-4):
|
|
zeta = 1.0/np.sqrt(1.0-om[2,2]**2)
|
|
eu = np.array([np.arctan2(om[2,0]*zeta,-om[2,1]*zeta),
|
|
np.arccos(om[2,2]),
|
|
np.arctan2(om[0,2]*zeta, om[1,2]*zeta)])
|
|
else:
|
|
eu = np.array([np.arctan2( om[0,1],om[0,0]), np.pi*0.5*(1-om[2,2]),0.0]) # following the paper, not the reference implementation
|
|
else:
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
zeta = 1.0/np.sqrt(1.0-om[...,2,2:3]**2)
|
|
eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.0,1e-4),
|
|
np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]),
|
|
np.pi*0.5*(1-om[...,2,2:3]),
|
|
np.zeros(om.shape[:-2]+(1,)),
|
|
]),
|
|
np.block([np.arctan2(om[...,2,0:1]*zeta,-om[...,2,1:2]*zeta),
|
|
np.arccos(om[...,2,2:3]),
|
|
np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta)
|
|
])
|
|
)
|
|
eu[np.abs(eu)<1.e-6] = 0.0
|
|
eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
|
|
return eu
|
|
|
|
|
|
@staticmethod
|
|
def om2ax(om):
|
|
"""Rotation matrix to axis angle pair."""
|
|
if len(om.shape) == 2:
|
|
ax=np.empty(4)
|
|
|
|
# first get the rotation angle
|
|
t = 0.5*(om.trace() -1.0)
|
|
ax[3] = np.arccos(np.clip(t,-1.0,1.0))
|
|
if np.abs(ax[3])<1.e-6:
|
|
ax = np.array([ 0.0, 0.0, 1.0, 0.0])
|
|
else:
|
|
w,vr = np.linalg.eig(om)
|
|
# next, find the eigenvalue (1,0j)
|
|
i = np.where(np.isclose(w,1.0+0.0j))[0][0]
|
|
ax[0:3] = np.real(vr[0:3,i])
|
|
diagDelta = -_P*np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]])
|
|
diagDelta[np.abs(diagDelta)<1.e-6] = 1.0
|
|
ax[0:3] = np.where(np.abs(diagDelta)<0, ax[0:3],np.abs(ax[0:3])*np.sign(diagDelta))
|
|
else:
|
|
diag_delta = -_P*np.block([om[...,1,2:3]-om[...,2,1:2],
|
|
om[...,2,0:1]-om[...,0,2:3],
|
|
om[...,0,1:2]-om[...,1,0:1]
|
|
])
|
|
diag_delta[np.abs(diag_delta)<1.e-6] = 1.0
|
|
t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.0).reshape(om.shape[:-2]+(1,))
|
|
w,vr = np.linalg.eig(om)
|
|
# mask duplicated real eigenvalues
|
|
w[np.isclose(w[...,0],1.0+0.0j),1:] = 0.
|
|
w[np.isclose(w[...,1],1.0+0.0j),2:] = 0.
|
|
vr = np.swapaxes(vr,-1,-2)
|
|
ax = np.where(np.abs(diag_delta)<0,
|
|
np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,)),
|
|
np.abs(np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,))) \
|
|
*np.sign(diag_delta))
|
|
ax = np.block([ax,np.arccos(np.clip(t,-1.0,1.0))])
|
|
ax[np.abs(ax[...,3])<1.e-6] = [ 0.0, 0.0, 1.0, 0.0]
|
|
return ax
|
|
|
|
|
|
@staticmethod
|
|
def om2ro(om):
|
|
"""Rotation matrix to Rodrigues-Frank vector."""
|
|
return Rotation.eu2ro(Rotation.om2eu(om))
|
|
|
|
@staticmethod
|
|
def om2ho(om):
|
|
"""Rotation matrix to homochoric vector."""
|
|
return Rotation.ax2ho(Rotation.om2ax(om))
|
|
|
|
@staticmethod
|
|
def om2cu(om):
|
|
"""Rotation matrix to cubochoric vector."""
|
|
return Rotation.ho2cu(Rotation.om2ho(om))
|
|
|
|
|
|
#---------- Bunge-Euler angles ----------
|
|
@staticmethod
|
|
def eu2qu(eu):
|
|
"""Bunge-Euler angles to quaternion."""
|
|
if len(eu.shape) == 1:
|
|
ee = 0.5*eu
|
|
cPhi = np.cos(ee[1])
|
|
sPhi = np.sin(ee[1])
|
|
qu = np.array([ cPhi*np.cos(ee[0]+ee[2]),
|
|
-_P*sPhi*np.cos(ee[0]-ee[2]),
|
|
-_P*sPhi*np.sin(ee[0]-ee[2]),
|
|
-_P*cPhi*np.sin(ee[0]+ee[2]) ])
|
|
if qu[0] < 0.0: qu*=-1
|
|
else:
|
|
ee = 0.5*eu
|
|
cPhi = np.cos(ee[...,1:2])
|
|
sPhi = np.sin(ee[...,1:2])
|
|
qu = np.block([ cPhi*np.cos(ee[...,0:1]+ee[...,2:3]),
|
|
-_P*sPhi*np.cos(ee[...,0:1]-ee[...,2:3]),
|
|
-_P*sPhi*np.sin(ee[...,0:1]-ee[...,2:3]),
|
|
-_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])])
|
|
qu[qu[...,0]<0.0]*=-1
|
|
return qu
|
|
|
|
|
|
@staticmethod
|
|
def eu2om(eu):
|
|
"""Bunge-Euler angles to rotation matrix."""
|
|
if len(eu.shape) == 1:
|
|
c = np.cos(eu)
|
|
s = np.sin(eu)
|
|
|
|
om = np.array([[+c[0]*c[2]-s[0]*s[2]*c[1], +s[0]*c[2]+c[0]*s[2]*c[1], +s[2]*s[1]],
|
|
[-c[0]*s[2]-s[0]*c[2]*c[1], -s[0]*s[2]+c[0]*c[2]*c[1], +c[2]*s[1]],
|
|
[+s[0]*s[1], -c[0]*s[1], +c[1] ]])
|
|
else:
|
|
c = np.cos(eu)
|
|
s = np.sin(eu)
|
|
om = np.block([+c[...,0:1]*c[...,2:3]-s[...,0:1]*s[...,2:3]*c[...,1:2],
|
|
+s[...,0:1]*c[...,2:3]+c[...,0:1]*s[...,2:3]*c[...,1:2],
|
|
+s[...,2:3]*s[...,1:2],
|
|
-c[...,0:1]*s[...,2:3]-s[...,0:1]*c[...,2:3]*c[...,1:2],
|
|
-s[...,0:1]*s[...,2:3]+c[...,0:1]*c[...,2:3]*c[...,1:2],
|
|
+c[...,2:3]*s[...,1:2],
|
|
+s[...,0:1]*s[...,1:2],
|
|
-c[...,0:1]*s[...,1:2],
|
|
+c[...,1:2]
|
|
]).reshape(eu.shape[:-1]+(3,3))
|
|
om[np.abs(om)<1.e-12] = 0.0
|
|
return om
|
|
|
|
@staticmethod
|
|
def eu2ax(eu):
|
|
"""Bunge-Euler angles to axis angle pair."""
|
|
if len(eu.shape) == 1:
|
|
t = np.tan(eu[1]*0.5)
|
|
sigma = 0.5*(eu[0]+eu[2])
|
|
delta = 0.5*(eu[0]-eu[2])
|
|
tau = np.linalg.norm([t,np.sin(sigma)])
|
|
alpha = np.pi if iszero(np.cos(sigma)) else \
|
|
2.0*np.arctan(tau/np.cos(sigma))
|
|
|
|
if np.abs(alpha)<1.e-6:
|
|
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
|
else:
|
|
ax = -_P/tau * np.array([ t*np.cos(delta), t*np.sin(delta), np.sin(sigma) ]) # passive axis angle pair so a minus sign in front
|
|
ax = np.append(ax,alpha)
|
|
if alpha < 0.0: ax *= -1.0 # ensure alpha is positive
|
|
else:
|
|
t = np.tan(eu[...,1:2]*0.5)
|
|
sigma = 0.5*(eu[...,0:1]+eu[...,2:3])
|
|
delta = 0.5*(eu[...,0:1]-eu[...,2:3])
|
|
tau = np.linalg.norm(np.block([t,np.sin(sigma)]),axis=-1,keepdims=True)
|
|
alpha = np.where(np.abs(np.cos(sigma))<1.e-12,np.pi,2.0*np.arctan(tau/np.cos(sigma)))
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
ax = np.where(np.broadcast_to(np.abs(alpha)<1.0e-12,eu.shape[:-1]+(4,)),
|
|
[0.0,0.0,1.0,0.0],
|
|
np.block([-_P/tau*t*np.cos(delta),
|
|
-_P/tau*t*np.sin(delta),
|
|
-_P/tau* np.sin(sigma),
|
|
alpha
|
|
]))
|
|
ax[(alpha<0.0).squeeze()] *=-1
|
|
return ax
|
|
|
|
@staticmethod
|
|
def eu2ro(eu):
|
|
"""Bunge-Euler angles to Rodrigues-Frank vector."""
|
|
if len(eu.shape) == 1:
|
|
ro = Rotation.eu2ax(eu) # convert to axis angle pair representation
|
|
if ro[3] >= np.pi: # Differs from original implementation. check convention 5
|
|
ro[3] = np.inf
|
|
elif iszero(ro[3]):
|
|
ro = np.array([ 0.0, 0.0, _P, 0.0 ])
|
|
else:
|
|
ro[3] = np.tan(ro[3]*0.5)
|
|
else:
|
|
ax = Rotation.eu2ax(eu)
|
|
ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)])
|
|
ro[ax[...,3]>=np.pi,3] = np.inf
|
|
ro[np.abs(ax[...,3])<1.e-16] = [ 0.0, 0.0, _P, 0.0 ]
|
|
return ro
|
|
|
|
@staticmethod
|
|
def eu2ho(eu):
|
|
"""Bunge-Euler angles to homochoric vector."""
|
|
return Rotation.ax2ho(Rotation.eu2ax(eu))
|
|
|
|
@staticmethod
|
|
def eu2cu(eu):
|
|
"""Bunge-Euler angles to cubochoric vector."""
|
|
return Rotation.ho2cu(Rotation.eu2ho(eu))
|
|
|
|
|
|
#---------- Axis angle pair ----------
|
|
@staticmethod
|
|
def ax2qu(ax):
|
|
"""Axis angle pair to quaternion."""
|
|
if len(ax.shape) == 1:
|
|
if np.abs(ax[3])<1.e-6:
|
|
qu = np.array([ 1.0, 0.0, 0.0, 0.0 ])
|
|
else:
|
|
c = np.cos(ax[3]*0.5)
|
|
s = np.sin(ax[3]*0.5)
|
|
qu = np.array([ c, ax[0]*s, ax[1]*s, ax[2]*s ])
|
|
return qu
|
|
else:
|
|
c = np.cos(ax[...,3:4]*.5)
|
|
s = np.sin(ax[...,3:4]*.5)
|
|
qu = np.where(np.abs(ax[...,3:4])<1.e-6,[1.0, 0.0, 0.0, 0.0],np.block([c, ax[...,:3]*s]))
|
|
return qu
|
|
|
|
@staticmethod
|
|
def ax2om(ax):
|
|
"""Axis angle pair to rotation matrix."""
|
|
if len(ax.shape) == 1:
|
|
c = np.cos(ax[3])
|
|
s = np.sin(ax[3])
|
|
omc = 1.0-c
|
|
om=np.diag(ax[0:3]**2*omc + c)
|
|
|
|
for idx in [[0,1,2],[1,2,0],[2,0,1]]:
|
|
q = omc*ax[idx[0]] * ax[idx[1]]
|
|
om[idx[0],idx[1]] = q + s*ax[idx[2]]
|
|
om[idx[1],idx[0]] = q - s*ax[idx[2]]
|
|
else:
|
|
c = np.cos(ax[...,3:4])
|
|
s = np.sin(ax[...,3:4])
|
|
omc = 1. -c
|
|
om = np.block([c+omc*ax[...,0:1]**2,
|
|
omc*ax[...,0:1]*ax[...,1:2] + s*ax[...,2:3],
|
|
omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2],
|
|
omc*ax[...,0:1]*ax[...,1:2] - s*ax[...,2:3],
|
|
c+omc*ax[...,1:2]**2,
|
|
omc*ax[...,1:2]*ax[...,2:3] + s*ax[...,0:1],
|
|
omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2],
|
|
omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1],
|
|
c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3))
|
|
return om if _P < 0.0 else np.swapaxes(om,(-1,-2))
|
|
|
|
@staticmethod
|
|
def ax2eu(ax):
|
|
"""Rotation matrix to Bunge Euler angles."""
|
|
return Rotation.om2eu(Rotation.ax2om(ax))
|
|
|
|
@staticmethod
|
|
def ax2ro(ax):
|
|
"""Axis angle pair to Rodrigues-Frank vector."""
|
|
if len(ax.shape) == 1:
|
|
if np.abs(ax[3])<1.e-6:
|
|
ro = [ 0.0, 0.0, _P, 0.0 ]
|
|
else:
|
|
ro = [ax[0], ax[1], ax[2]]
|
|
# 180 degree case
|
|
ro += [np.inf] if np.isclose(ax[3],np.pi,atol=1.0e-15,rtol=0.0) else \
|
|
[np.tan(ax[3]*0.5)]
|
|
return np.array(ro)
|
|
else:
|
|
ro = np.block([ax[...,:3],
|
|
np.where(np.isclose(ax[...,3:4],np.pi,atol=1.e-15,rtol=.0),
|
|
np.inf,
|
|
np.tan(ax[...,3:4]*0.5))
|
|
])
|
|
ro[np.abs(ax[...,3])<1.e-6] = [.0,.0,_P,.0]
|
|
return ro
|
|
|
|
@staticmethod
|
|
def ax2ho(ax):
|
|
"""Axis angle pair to homochoric vector."""
|
|
if len(ax.shape) == 1:
|
|
f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0)
|
|
ho = ax[0:3] * f
|
|
return ho
|
|
else:
|
|
f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1.0/3.0)
|
|
ho = ax[...,:3] * f
|
|
return ho
|
|
|
|
@staticmethod
|
|
def ax2cu(ax):
|
|
"""Axis angle pair to cubochoric vector."""
|
|
return Rotation.ho2cu(Rotation.ax2ho(ax))
|
|
|
|
|
|
#---------- Rodrigues-Frank vector ----------
|
|
@staticmethod
|
|
def ro2qu(ro):
|
|
"""Rodrigues-Frank vector to quaternion."""
|
|
return Rotation.ax2qu(Rotation.ro2ax(ro))
|
|
|
|
@staticmethod
|
|
def ro2om(ro):
|
|
"""Rodgrigues-Frank vector to rotation matrix."""
|
|
return Rotation.ax2om(Rotation.ro2ax(ro))
|
|
|
|
@staticmethod
|
|
def ro2eu(ro):
|
|
"""Rodrigues-Frank vector to Bunge-Euler angles."""
|
|
return Rotation.om2eu(Rotation.ro2om(ro))
|
|
|
|
@staticmethod
|
|
def ro2ax(ro):
|
|
"""Rodrigues-Frank vector to axis angle pair."""
|
|
if len(ro.shape) == 1:
|
|
if np.abs(ro[3]) < 1.e-6:
|
|
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
|
elif not np.isfinite(ro[3]):
|
|
ax = np.array([ ro[0], ro[1], ro[2], np.pi ])
|
|
else:
|
|
angle = 2.0*np.arctan(ro[3])
|
|
ta = np.linalg.norm(ro[0:3])
|
|
ax = np.array([ ro[0]*ta, ro[1]*ta, ro[2]*ta, angle ])
|
|
else:
|
|
with np.errstate(invalid='ignore',divide='ignore'):
|
|
ax = np.where(np.isfinite(ro[...,3:4]),
|
|
np.block([ro[...,0:3]*np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True),2.*np.arctan(ro[...,3:4])]),
|
|
np.block([ro[...,0:3],np.broadcast_to(np.pi,ro[...,3:4].shape)]))
|
|
ax[np.abs(ro[...,3]) < 1.e-6] = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
|
return ax
|
|
|
|
@staticmethod
|
|
def ro2ho(ro):
|
|
"""Rodrigues-Frank vector to homochoric vector."""
|
|
if len(ro.shape) == 1:
|
|
if np.sum(ro[0:3]**2.0) < 1.e-6:
|
|
ho = np.zeros(3)
|
|
else:
|
|
f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
|
|
ho = ro[0:3] * (0.75*f)**(1.0/3.0)
|
|
else:
|
|
f = np.where(np.isfinite(ro[...,3:4]),2.0*np.arctan(ro[...,3:4]) -np.sin(2.0*np.arctan(ro[...,3:4])),np.pi)
|
|
ho = np.where(np.broadcast_to(np.sum(ro[...,0:3]**2.0,axis=-1,keepdims=True) < 1.e-6,ro[...,0:3].shape),
|
|
np.zeros(3), ro[...,0:3]* (0.75*f)**(1.0/3.0))
|
|
|
|
return ho
|
|
|
|
@staticmethod
|
|
def ro2cu(ro):
|
|
"""Rodrigues-Frank vector to cubochoric vector."""
|
|
return Rotation.ho2cu(Rotation.ro2ho(ro))
|
|
|
|
|
|
#---------- Homochoric vector----------
|
|
@staticmethod
|
|
def ho2qu(ho):
|
|
"""Homochoric vector to quaternion."""
|
|
return Rotation.ax2qu(Rotation.ho2ax(ho))
|
|
|
|
@staticmethod
|
|
def ho2om(ho):
|
|
"""Homochoric vector to rotation matrix."""
|
|
return Rotation.ax2om(Rotation.ho2ax(ho))
|
|
|
|
@staticmethod
|
|
def ho2eu(ho):
|
|
"""Homochoric vector to Bunge-Euler angles."""
|
|
return Rotation.ax2eu(Rotation.ho2ax(ho))
|
|
|
|
@staticmethod
|
|
def ho2ax(ho):
|
|
"""Homochoric vector to axis angle pair."""
|
|
tfit = np.array([+1.0000000000018852, -0.5000000002194847,
|
|
-0.024999992127593126, -0.003928701544781374,
|
|
-0.0008152701535450438, -0.0002009500426119712,
|
|
-0.00002397986776071756, -0.00008202868926605841,
|
|
+0.00012448715042090092, -0.0001749114214822577,
|
|
+0.0001703481934140054, -0.00012062065004116828,
|
|
+0.000059719705868660826, -0.00001980756723965647,
|
|
+0.000003953714684212874, -0.00000036555001439719544])
|
|
if len(ho.shape) == 1:
|
|
# normalize h and store the magnitude
|
|
hmag_squared = np.sum(ho**2.)
|
|
if iszero(hmag_squared):
|
|
ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
|
|
else:
|
|
hm = hmag_squared
|
|
|
|
# convert the magnitude to the rotation angle
|
|
s = tfit[0] + tfit[1] * hmag_squared
|
|
for i in range(2,16):
|
|
hm *= hmag_squared
|
|
s += tfit[i] * hm
|
|
ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0)))
|
|
else:
|
|
hmag_squared = np.sum(ho**2.,axis=-1,keepdims=True)
|
|
hm = hmag_squared.copy()
|
|
s = tfit[0] + tfit[1] * hmag_squared
|
|
for i in range(2,16):
|
|
hm *= hmag_squared
|
|
s += tfit[i] * hm
|
|
with np.errstate(invalid='ignore'):
|
|
ax = np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-6,ho.shape[:-1]+(4,)),
|
|
[ 0.0, 0.0, 1.0, 0.0 ],
|
|
np.block([ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))]))
|
|
return ax
|
|
|
|
@staticmethod
|
|
def ho2ro(ho):
|
|
"""Axis angle pair to Rodrigues-Frank vector."""
|
|
return Rotation.ax2ro(Rotation.ho2ax(ho))
|
|
|
|
@staticmethod
|
|
def ho2cu(ho):
|
|
"""Homochoric vector to cubochoric vector."""
|
|
if len(ho.shape) == 1:
|
|
return ball_to_cube(ho)
|
|
else:
|
|
raise NotImplementedError
|
|
|
|
|
|
#---------- Cubochoric ----------
|
|
@staticmethod
|
|
def cu2qu(cu):
|
|
"""Cubochoric vector to quaternion."""
|
|
return Rotation.ho2qu(Rotation.cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def cu2om(cu):
|
|
"""Cubochoric vector to rotation matrix."""
|
|
return Rotation.ho2om(Rotation.cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def cu2eu(cu):
|
|
"""Cubochoric vector to Bunge-Euler angles."""
|
|
return Rotation.ho2eu(Rotation.cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def cu2ax(cu):
|
|
"""Cubochoric vector to axis angle pair."""
|
|
return Rotation.ho2ax(Rotation.cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def cu2ro(cu):
|
|
"""Cubochoric vector to Rodrigues-Frank vector."""
|
|
return Rotation.ho2ro(Rotation.cu2ho(cu))
|
|
|
|
@staticmethod
|
|
def cu2ho(cu):
|
|
"""Cubochoric vector to homochoric vector."""
|
|
if len(cu.shape) == 1:
|
|
return cube_to_ball(cu)
|
|
else:
|
|
raise NotImplementedError
|