838 lines
30 KiB
Python
838 lines
30 KiB
Python
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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],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 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.\n{}'.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.\n{} {} {} {}'.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].\n{} {} {}.'.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].\n'.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.\n{} {} {}'.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.\n{}'.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.\n{}'.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.\n{} {} {}'.format(*ro[0:3]))
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if ro[3] < 0.0:
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raise ValueError('Rodrigues rotation angle not positive.\n'.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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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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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
|
||
|
|
def fromAverage(rotations,
|
||
|
|
weights = []):
|
||
|
|
"""
|
||
|
|
Average rotation.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
F. Landis Markley et al., Journal of Guidance, Control, and Dynamics 30(4):1193-1197, 2007
|
||
|
|
https://doi.org/10.2514/1.28949
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
rotations : list of Rotations
|
||
|
|
Rotations to average from
|
||
|
|
weights : list of floats, optional
|
||
|
|
Weights for each rotation used for averaging
|
||
|
|
|
||
|
|
"""
|
||
|
|
if not all(isinstance(item, Rotation) for item in rotations):
|
||
|
|
raise TypeError("Only instances of Rotation can be averaged.")
|
||
|
|
|
||
|
|
N = len(rotations)
|
||
|
|
if weights == [] or not weights:
|
||
|
|
weights = np.ones(N,dtype='i')
|
||
|
|
|
||
|
|
for i,(r,n) in enumerate(zip(rotations,weights)):
|
||
|
|
M = r.asM() * n if i == 0 \
|
||
|
|
else M + r.asM() * n # noqa add (multiples) of this rotation to average noqa
|
||
|
|
eig, vec = np.linalg.eig(M/N)
|
||
|
|
|
||
|
|
return Rotation.fromQuaternion(np.real(vec.T[eig.argmax()]),acceptHomomorph = True)
|
||
|
|
|
||
|
|
|
||
|
|
@staticmethod
|
||
|
|
def fromRandom():
|
||
|
|
r = np.random.random(3)
|
||
|
|
A = np.sqrt(r[2])
|
||
|
|
B = np.sqrt(1.0-r[2])
|
||
|
|
return Rotation(np.array([np.cos(2.0*np.pi*r[0])*A,
|
||
|
|
np.sin(2.0*np.pi*r[1])*B,
|
||
|
|
np.cos(2.0*np.pi*r[1])*B,
|
||
|
|
np.sin(2.0*np.pi*r[0])*A])).standardize()
|
||
|
|
|
||
|
|
|
||
|
|
####################################################################################################
|
||
|
|
# Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations
|
||
|
|
####################################################################################################
|
||
|
|
# Copyright (c) 2017-2019, Martin Diehl/Max-Planck-Institut für Eisenforschung GmbH
|
||
|
|
# Copyright (c) 2013-2014, Marc De Graef/Carnegie Mellon University
|
||
|
|
# All rights reserved.
|
||
|
|
#
|
||
|
|
# Redistribution and use in source and binary forms, with or without modification, are
|
||
|
|
# permitted provided that the following conditions are met:
|
||
|
|
#
|
||
|
|
# - Redistributions of source code must retain the above copyright notice, this list
|
||
|
|
# of conditions and the following disclaimer.
|
||
|
|
# - Redistributions in binary form must reproduce the above copyright notice, this
|
||
|
|
# list of conditions and the following disclaimer in the documentation and/or
|
||
|
|
# other materials provided with the distribution.
|
||
|
|
# - Neither the names of Marc De Graef, Carnegie Mellon University nor the names
|
||
|
|
# of its contributors may be used to endorse or promote products derived from
|
||
|
|
# this software without specific prior written permission.
|
||
|
|
#
|
||
|
|
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
|
||
|
|
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
||
|
|
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
||
|
|
# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
|
||
|
|
# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
||
|
|
# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
|
||
|
|
# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
|
||
|
|
# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
|
||
|
|
# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
|
||
|
|
# USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
||
|
|
####################################################################################################
|
||
|
|
#---------- Quaternion ----------
|
||
|
|
@staticmethod
|
||
|
|
def qu2om(qu):
|
||
|
|
"""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 Rodrigues-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)))]
|
||
|
|
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))
|
||
|
|
|
||
|
|
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 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."""
|
||
|
|
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 Rodrigues-Frank vector."""
|
||
|
|
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)
|
||
|
|
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 Rodrigues-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):
|
||
|
|
"""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."""
|
||
|
|
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):
|
||
|
|
"""Rodrigues-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):
|
||
|
|
"""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])
|
||
|
|
# 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 Rodrigues-Frank vector."""
|
||
|
|
return Rotation.ax2ro(Rotation.ho2ax(ho))
|
||
|
|
|
||
|
|
@staticmethod
|
||
|
|
def ho2cu(ho):
|
||
|
|
"""Homochoric vector to cubochoric vector."""
|
||
|
|
return ball_to_cube(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 Rodrigues-Frank vector."""
|
||
|
|
return Rotation.ho2ro(Rotation.cu2ho(cu))
|
||
|
|
|
||
|
|
@staticmethod
|
||
|
|
def cu2ho(cu):
|
||
|
|
"""Cubochoric vector to homochoric vector."""
|
||
|
|
return cube_to_ball(cu)
|