DAMASK_EICMD/python/damask/_orientation.py

import inspect
import copy

import numpy as np

from . import Rotation
from . import Crystal
from . import util
from . import tensor


_parameter_doc = \
       """
        family : {'triclinic', 'monoclinic', 'orthorhombic', 'tetragonal', 'hexagonal', 'cubic'}, optional.
            Name of the crystal family.
            Family will be inferred if 'lattice' is given.
        lattice : {'aP', 'mP', 'mS', 'oP', 'oS', 'oI', 'oF', 'tP', 'tI', 'hP', 'cP', 'cI', 'cF'}, optional.
            Name of the Bravais lattice in Pearson notation.
        a : float, optional
            Length of lattice parameter 'a'.
        b : float, optional
            Length of lattice parameter 'b'.
        c : float, optional
            Length of lattice parameter 'c'.
        alpha : float, optional
            Angle between b and c lattice basis.
        beta : float, optional
            Angle between c and a lattice basis.
        gamma : float, optional
            Angle between a and b lattice basis.
        degrees : bool, optional
            Angles are given in degrees. Defaults to False.

       """


class Orientation(Rotation,Crystal):
    """
    Representation of crystallographic orientation as combination of rotation and either crystal family or Bravais lattice.

    The crystal family is one of:

    - triclinic
    - monoclinic
    - orthorhombic
    - tetragonal
    - hexagonal
    - cubic

    and enables symmetry-related operations such as
    "equivalent", "reduced", "disorientation", "IPF_color", or "to_SST".

    The Bravais lattice is given in the Pearson notation:

    - triclinic
       - aP : primitive

    - monoclinic
       - mP : primitive
       - mS : base-centered

    - orthorhombic
       - oP : primitive
       - oS : base-centered
       - oI : body-centered
       - oF : face-centered

    - tetragonal
       - tP : primitive
       - tI : body-centered

    - hexagonal
       - hP : primitive

    - cubic
       - cP : primitive
       - cI : body-centered
       - cF : face-centered

    and inherits the corresponding crystal family.
    Specifying a Bravais lattice, compared to just the crystal family,
    extends the functionality of Orientation objects to include operations such as
    "Schmid", "related", or "to_pole" that require a lattice type and its parameters.

    Examples
    --------
    An array of 3 x 5 random orientations reduced to the fundamental zone of tetragonal symmetry:

    >>> import damask
    >>> o=damask.Orientation.from_random(shape=(3,5),family='tetragonal').reduced

    """

    @util.extend_docstring(_parameter_doc)
    def __init__(self,
                 rotation = np.array([1.0,0.0,0.0,0.0]), *,
                 family = None,
                 lattice = None,
                 a = None,b = None,c = None,
                 alpha = None,beta = None,gamma = None,
                 degrees = False):
        """
        New orientation.

        Parameters
        ----------
        rotation : list, numpy.ndarray, Rotation, optional
            Unit quaternion in positive real hemisphere.
            Use .from_quaternion to perform a sanity check.
            Defaults to no rotation.

        """
        Rotation.__init__(self,rotation)
        Crystal.__init__(self,family=family, lattice=lattice,
                              a=a,b=b,c=c, alpha=alpha,beta=beta,gamma=gamma, degrees=degrees)


    def __repr__(self):
        """Represent."""
        return '\n'.join([Crystal.__repr__(self),
                          Rotation.__repr__(self)])


    def __copy__(self,rotation=None):
        """Create deep copy."""
        dup = copy.deepcopy(self)
        if rotation is not None:
            dup.quaternion = Rotation(rotation).quaternion
        return dup

    copy = __copy__


    def __eq__(self,other):
        """
        Equal to other.

        Parameters
        ----------
        other : Orientation
            Orientation to check for equality.

        """
        matching_type = self.family == other.family and \
                        self.lattice == other.lattice and \
                        self.parameters == other.parameters
        return np.logical_and(matching_type,super(self.__class__,self.reduced).__eq__(other.reduced))

    def __ne__(self,other):
        """
        Not equal to other.

        Parameters
        ----------
        other : Orientation
            Orientation to check for equality.

        """
        return np.logical_not(self==other)


    def isclose(self,other,rtol=1e-5,atol=1e-8,equal_nan=True):
        """
        Report where values are approximately equal to corresponding ones of other Orientation.

        Parameters
        ----------
        other : Orientation
            Orientation to compare against.
        rtol : float, optional
            Relative tolerance of equality.
        atol : float, optional
            Absolute tolerance of equality.
        equal_nan : bool, optional
            Consider matching NaN values as equal. Defaults to True.

        Returns
        -------
        mask : numpy.ndarray bool
            Mask indicating where corresponding orientations are close.

        """
        matching_type = self.family == other.family and \
                        self.lattice == other.lattice and \
                        self.parameters == other.parameters
        return np.logical_and(matching_type,super(self.__class__,self.reduced).isclose(other.reduced))



    def allclose(self,other,rtol=1e-5,atol=1e-8,equal_nan=True):
        """
        Test whether all values are approximately equal to corresponding ones of other Orientation.

        Parameters
        ----------
        other : Orientation
            Orientation to compare against.
        rtol : float, optional
            Relative tolerance of equality.
        atol : float, optional
            Absolute tolerance of equality.
        equal_nan : bool, optional
            Consider matching NaN values as equal. Defaults to True.

        Returns
        -------
        answer : bool
            Whether all values are close between both orientations.

        """
        return np.all(self.isclose(other,rtol,atol,equal_nan))


    def __mul__(self,other):
        """
        Compose this orientation with other.

        Parameters
        ----------
        other : Rotation or Orientation
            Object for composition.

        Returns
        -------
        composition : Orientation
            Compound rotation self*other, i.e. first other then self rotation.

        """
        if isinstance(other,Orientation) or isinstance(other,Rotation):
            return self.copy(rotation=Rotation.__mul__(self,Rotation(other.quaternion)))
        else:
            raise TypeError('use "O@b", i.e. matmul, to apply Orientation "O" to object "b"')


    @staticmethod
    def _split_kwargs(kwargs,target):
        """
        Separate keyword arguments in 'kwargs' targeted at 'target' from general keyword arguments of Orientation objects.

        Parameters
        ----------
        kwargs : dictionary
            Contains all **kwargs.
        target: method
            Function to scan for kwarg signature.

        Returns
        -------
        rot_kwargs: dictionary
            Valid keyword arguments of 'target' function of Rotation class.
        ori_kwargs: dictionary
            Valid keyword arguments of Orientation object.

        """
        kws = ()
        for t in (target,Orientation.__init__):
            kws += ({key: kwargs[key] for key in set(inspect.signature(t).parameters) & set(kwargs)},)

        invalid_keys = set(kwargs)-(set(kws[0])|set(kws[1]))
        if invalid_keys:
            raise TypeError(f"{inspect.stack()[1][3]}() got an unexpected keyword argument '{invalid_keys.pop()}'")

        return kws


    @classmethod
    @util.extended_docstring(Rotation.from_random,_parameter_doc)
    def from_random(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_random)
        return cls(rotation=Rotation.from_random(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_quaternion,_parameter_doc)
    def from_quaternion(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_quaternion)
        return cls(rotation=Rotation.from_quaternion(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_Euler_angles,_parameter_doc)
    def from_Euler_angles(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_Euler_angles)
        return cls(rotation=Rotation.from_Euler_angles(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_axis_angle,_parameter_doc)
    def from_axis_angle(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_axis_angle)
        return cls(rotation=Rotation.from_axis_angle(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_basis,_parameter_doc)
    def from_basis(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_basis)
        return cls(rotation=Rotation.from_basis(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_matrix,_parameter_doc)
    def from_matrix(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_matrix)
        return cls(rotation=Rotation.from_matrix(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_Rodrigues_vector,_parameter_doc)
    def from_Rodrigues_vector(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_Rodrigues_vector)
        return cls(rotation=Rotation.from_Rodrigues_vector(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_homochoric,_parameter_doc)
    def from_homochoric(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_homochoric)
        return cls(rotation=Rotation.from_homochoric(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_cubochoric,_parameter_doc)
    def from_cubochoric(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_cubochoric)
        return cls(rotation=Rotation.from_cubochoric(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_spherical_component,_parameter_doc)
    def from_spherical_component(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_spherical_component)
        return cls(rotation=Rotation.from_spherical_component(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extended_docstring(Rotation.from_fiber_component,_parameter_doc)
    def from_fiber_component(cls,**kwargs):
        kwargs_rot,kwargs_ori = Orientation._split_kwargs(kwargs,Rotation.from_fiber_component)
        return cls(rotation=Rotation.from_fiber_component(**kwargs_rot),**kwargs_ori)


    @classmethod
    @util.extend_docstring(_parameter_doc)
    def from_directions(cls,uvw,hkl,**kwargs):
        """
        Initialize orientation object from two crystallographic directions.

        Parameters
        ----------
        uvw : list, numpy.ndarray of shape (...,3)
            lattice direction aligned with lab frame x-direction.
        hkl : list, numpy.ndarray of shape (...,3)
            lattice plane normal aligned with lab frame z-direction.

        """
        o = cls(**kwargs)
        x = o.to_frame(uvw=uvw)
        z = o.to_frame(hkl=hkl)
        om = np.stack([x,np.cross(z,x),z],axis=-2)
        return o.copy(rotation=Rotation.from_matrix(tensor.transpose(om/np.linalg.norm(om,axis=-1,keepdims=True))))


    @property
    def equivalent(self):
        """
        Orientations that are symmetrically equivalent.

        One dimension (length corresponds to number of symmetrically equivalent orientations)
        is added to the left of the Rotation array.

        """
        sym_ops = self.symmetry_operations
        o = sym_ops.broadcast_to(sym_ops.shape+self.shape,mode='right')
        return self.copy(rotation=o*Rotation(self.quaternion).broadcast_to(o.shape,mode='left'))


    @property
    def reduced(self):
        """Select symmetrically equivalent orientation that falls into fundamental zone according to symmetry."""
        eq   = self.equivalent
        ok   = eq.in_FZ
        ok  &= np.cumsum(ok,axis=0) == 1
        loc  = np.where(ok)
        sort = 0 if len(loc) == 1 else np.lexsort(loc[:0:-1])
        return eq[ok][sort].reshape(self.shape)


    @property
    def in_FZ(self):
        """
        Check whether orientation falls into fundamental zone of own symmetry.

        Returns
        -------
        in : numpy.ndarray of bool, quaternion.shape
            Whether Rodrigues-Frank vector falls into fundamental zone.

        Notes
        -----
        Fundamental zones in Rodrigues space are point-symmetric around origin.

        References
        ----------
        A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991
        https://doi.org/10.1107/S0108767391006864

        """
        rho_abs = np.abs(self.as_Rodrigues_vector(compact=True))*(1.-1.e-9)

        with np.errstate(invalid='ignore'):
            # using '*'/prod for 'and'
            if   self.family == 'cubic':
                return (np.prod(np.sqrt(2)-1. >= rho_abs,axis=-1) *
                                   (1. >= np.sum(rho_abs,axis=-1))).astype(bool)
            elif self.family == 'hexagonal':
                return (np.prod(1.  >= rho_abs,axis=-1) *
                                (2. >= np.sqrt(3)*rho_abs[...,0] + rho_abs[...,1]) *
                                (2. >= np.sqrt(3)*rho_abs[...,1] + rho_abs[...,0]) *
                                (2. >= np.sqrt(3)                + rho_abs[...,2])).astype(bool)
            elif self.family == 'tetragonal':
                return (np.prod(1.  >= rho_abs[...,:2],axis=-1) *
                        (np.sqrt(2) >= rho_abs[...,0] + rho_abs[...,1]) *
                        (np.sqrt(2) >= rho_abs[...,2] + 1.)).astype(bool)
            elif self.family == 'orthorhombic':
                return (np.prod(1. >= rho_abs,axis=-1)).astype(bool)
            elif self.family == 'monoclinic':
                return (1. >= rho_abs[...,1]).astype(bool)
            else:
                return np.all(np.isfinite(rho_abs),axis=-1)


    @property
    def in_disorientation_FZ(self):
        """
        Check whether orientation falls into fundamental zone of disorientations.

        Returns
        -------
        in : numpy.ndarray of bool, quaternion.shape
            Whether Rodrigues-Frank vector falls into disorientation FZ.

        References
        ----------
        A. Heinz and P. Neumann, Acta Crystallographica Section A 47:780-789, 1991
        https://doi.org/10.1107/S0108767391006864

        """
        rho = self.as_Rodrigues_vector(compact=True)*(1.0-1.0e-9)

        with np.errstate(invalid='ignore'):
            if   self.family == 'cubic':
                return ((rho[...,0] >= rho[...,1]) &
                        (rho[...,1] >= rho[...,2]) &
                        (rho[...,2] >= 0)).astype(bool)
            elif self.family == 'hexagonal':
                return ((rho[...,0] >= rho[...,1]*np.sqrt(3)) &
                        (rho[...,1] >= 0) &
                        (rho[...,2] >= 0)).astype(bool)
            elif self.family == 'tetragonal':
                return ((rho[...,0] >= rho[...,1]) &
                        (rho[...,1] >= 0) &
                        (rho[...,2] >= 0)).astype(bool)
            elif self.family == 'orthorhombic':
                return ((rho[...,0] >= 0) &
                        (rho[...,1] >= 0) &
                        (rho[...,2] >= 0)).astype(bool)
            elif self.family == 'monoclinic':
                return ((rho[...,1] >= 0) &
                        (rho[...,2] >= 0)).astype(bool)
            else:
                return np.ones_like(rho[...,0],dtype=bool)


    def disorientation(self,other,return_operators=False):
        """
        Calculate disorientation between myself and given other orientation.

        Parameters
        ----------
        other : Orientation
            Orientation to calculate disorientation for.
            Shape of other blends with shape of own rotation array.
            For example, shapes of (2,3) for own rotations and (3,2) for other's result in (2,3,2) disorientations.
        return_operators : bool, optional
            Return index pair of symmetrically equivalent orientations that result in disorientation axis falling into FZ.
            Defaults to False.

        Returns
        -------
        disorientation : Orientation
            Disorientation between self and other.
        operators : numpy.ndarray int of shape (...,2), conditional
            Index of symmetrically equivalent orientation that rotated vector to the SST.

        Notes
        -----
        Currently requires same crystal family for both orientations.
        For extension to cases with differing symmetry see A. Heinz and P. Neumann 1991 and 10.1107/S0021889808016373.

        Examples
        --------
        Disorientation between two specific orientations of hexagonal symmetry:

        >>> import damask
        >>> a = damask.Orientation.from_Euler_angles(phi=[123,32,21],degrees=True,family='hexagonal')
        >>> b = damask.Orientation.from_Euler_angles(phi=[104,11,87],degrees=True,family='hexagonal')
        >>> a.disorientation(b)
        Crystal family hexagonal
        Quaternion: (real=0.976, imag=<+0.189, +0.018, +0.103>)
        Matrix:
        [[ 0.97831006  0.20710935  0.00389135]
         [-0.19363288  0.90765544  0.37238141]
         [ 0.07359167 -0.36505797  0.92807163]]
        Bunge Eulers / deg: (11.40, 21.86, 0.60)

        Plot a sample from the Mackenzie distribution.

        >>> import matplotlib.pyplot as plt
        >>> import damask
        >>> N = 10000
        >>> a = damask.Orientation.from_random(shape=N,family='cubic')
        >>> b = damask.Orientation.from_random(shape=N,family='cubic')
        >>> d = a.disorientation(b).as_axis_angle(degrees=True,pair=True)[1]
        >>> plt.hist(d,25)
        >>> plt.show()

        """
        if self.family != other.family:
            raise NotImplementedError('disorientation between different crystal families')

        blend = util.shapeblender(self.shape,other.shape)
        s =  self.equivalent
        o = other.equivalent

        s_ = s.reshape((s.shape[0],1)+ self.shape).broadcast_to((s.shape[0],o.shape[0])+blend,mode='right')
        o_ = o.reshape((1,o.shape[0])+other.shape).broadcast_to((s.shape[0],o.shape[0])+blend,mode='right')
        r_ = s_.misorientation(o_)
        _r = ~r_

        forward = r_.in_FZ & r_.in_disorientation_FZ
        reverse = _r.in_FZ & _r.in_disorientation_FZ
        ok  = forward | reverse
        ok &= (np.cumsum(ok.reshape((-1,)+ok.shape[2:]),axis=0) == 1).reshape(ok.shape)
        r = np.where(np.any(forward[...,np.newaxis],axis=(0,1),keepdims=True),
                     r_.quaternion,
                     _r.quaternion)
        loc  = np.where(ok)
        sort = 0 if len(loc) == 2 else np.lexsort(loc[:1:-1])
        quat = r[ok][sort].reshape(blend+(4,))

        return (
                (self.copy(rotation=quat),
                 (np.vstack(loc[:2]).T)[sort].reshape(blend+(2,)))
                if return_operators else
                self.copy(rotation=quat)
               )


    def average(self,weights=None,return_cloud=False):
        """
        Return orientation average over last dimension.

        Parameters
        ----------
        weights : numpy.ndarray, optional
            Relative weights of orientations.
        return_cloud : bool, optional
            Return the set of symmetrically equivalent orientations that was used in averaging.
            Defaults to False.

        Returns
        -------
        average : Orientation
            Weighted average of original Orientation field.
        cloud : Orientations, conditional
            Set of symmetrically equivalent orientations that were used in averaging.

        References
        ----------
        J.C. Glez and J. Driver, Journal of Applied Crystallography 34:280-288, 2001
        https://doi.org/10.1107/S0021889801003077

        """
        eq = self.equivalent
        m  = eq.misorientation(self[...,0].reshape((1,)+self.shape[:-1]+(1,))
                                          .broadcast_to(eq.shape))\
               .as_axis_angle()[...,3]
        r = Rotation(np.squeeze(np.take_along_axis(eq.quaternion,
                                                   np.argmin(m,axis=0)[np.newaxis,...,np.newaxis],
                                                   axis=0),
                                axis=0))
        return (
                (self.copy(rotation=Rotation(r).average(weights)),
                 self.copy(rotation=Rotation(r)))
                if return_cloud else
                self.copy(rotation=Rotation(r).average(weights))
               )


    def to_SST(self,vector,proper=False,return_operators=False):
        """
        Rotate vector to ensure it falls into (improper or proper) standard stereographic triangle of crystal symmetry.

        Parameters
        ----------
        vector : numpy.ndarray of shape (...,3)
            Lab frame vector to align with crystal frame direction.
            Shape of vector blends with shape of own rotation array.
            For example, a rotation array of shape (3,2) and a (2,4) vector array result in (3,2,4) outputs.
        proper : bool, optional
            Consider only vectors with z >= 0, hence combine two neighboring SSTs.
            Defaults to False.
        return_operators : bool, optional
            Return the symmetrically equivalent orientation that rotated vector to SST.
            Defaults to False.

        Returns
        -------
        vector_SST : numpy.ndarray of shape (...,3)
            Rotated vector falling into SST.
        operators : numpy.ndarray int of shape (...), conditional
            Index of symmetrically equivalent orientation that rotated vector to SST.

        """
        eq  = self.equivalent
        blend = util.shapeblender(eq.shape,np.array(vector).shape[:-1])
        poles = eq.broadcast_to(blend,mode='right') @ np.broadcast_to(np.array(vector),blend+(3,))
        ok    = self.in_SST(poles,proper=proper)
        ok   &= np.cumsum(ok,axis=0) == 1
        loc   = np.where(ok)
        sort  = 0 if len(loc) == 1 else np.lexsort(loc[:0:-1])
        return (
                (poles[ok][sort].reshape(blend[1:]+(3,)), (np.vstack(loc[:1]).T)[sort].reshape(blend[1:]))
                if return_operators else
                poles[ok][sort].reshape(blend[1:]+(3,))
               )


    def in_SST(self,vector,proper=False):
        """
        Check whether given crystal frame vector falls into standard stereographic triangle of own symmetry.

        Parameters
        ----------
        vector : numpy.ndarray of shape (...,3)
            Vector to check.
        proper : bool, optional
            Consider only vectors with z >= 0, hence combine two neighboring SSTs.
            Defaults to False.

        Returns
        -------
        in : numpy.ndarray, shape (...)
            Whether vector falls into SST.

        """
        if not isinstance(vector,np.ndarray) or vector.shape[-1] != 3:
            raise ValueError('input is not a field of three-dimensional vectors')

        if self.standard_triangle is None:                                                          # direct exit for no symmetry
            return  np.ones_like(vector[...,0],bool)

        if proper:
            components_proper   = np.around(np.einsum('...ji,...i',
                                                      np.broadcast_to(self.standard_triangle['proper'], vector.shape+(3,)),
                                                      vector), 12)
            components_improper = np.around(np.einsum('...ji,...i',
                                                      np.broadcast_to(self.standard_triangle['improper'], vector.shape+(3,)),
                                                      vector), 12)
            return   np.all(components_proper   >= 0.0,axis=-1) \
                   | np.all(components_improper >= 0.0,axis=-1)
        else:
            components = np.around(np.einsum('...ji,...i',
                                             np.broadcast_to(self.standard_triangle['improper'], vector.shape+(3,)),
                                             np.block([vector[...,:2],np.abs(vector[...,2:3])])), 12)

            return np.all(components >= 0.0,axis=-1)


    def IPF_color(self,vector,in_SST=True,proper=False):
        """
        Map vector to RGB color within standard stereographic triangle of own symmetry.

        Parameters
        ----------
        vector : numpy.ndarray of shape (...,3)
            Vector to colorize.
            Shape of vector blends with shape of own rotation array.
            For example, a rotation array of shape (3,2) and a (2,4) vector array result in (3,2,4) outputs.
        in_SST : bool, optional
            Consider symmetrically equivalent orientations such that poles are located in SST.
            Defaults to True.
        proper : bool, optional
            Consider only vectors with z >= 0, hence combine two neighboring SSTs (with mirrored colors).
            Defaults to False.

        Returns
        -------
        rgb : numpy.ndarray of shape (...,3)
           RGB array of IPF colors.

        Examples
        --------
        Inverse pole figure color of the e_3 direction for a crystal in "Cube" orientation with cubic symmetry:

        >>> import damask
        >>> o = damask.Orientation(family='cubic')
        >>> o.IPF_color([0,0,1])
        array([1., 0., 0.])

        """
        if np.array(vector).shape[-1] != 3:
            raise ValueError('input is not a field of three-dimensional vectors')

        vector_ = self.to_SST(vector,proper) if in_SST else \
                  self @ np.broadcast_to(vector,self.shape+(3,))

        if self.standard_triangle is None:                                                          # direct exit for no symmetry
            return np.zeros_like(vector_)

        if proper:
            components_proper   = np.around(np.einsum('...ji,...i',
                                                      np.broadcast_to(self.standard_triangle['proper'], vector_.shape+(3,)),
                                                      vector_), 12)
            components_improper = np.around(np.einsum('...ji,...i',
                                                      np.broadcast_to(self.standard_triangle['improper'], vector_.shape+(3,)),
                                                      vector_), 12)
            in_SST = np.all(components_proper   >= 0.0,axis=-1) \
                   | np.all(components_improper >= 0.0,axis=-1)
            components = np.where((in_SST & np.all(components_proper   >= 0.0,axis=-1))[...,np.newaxis],
                                  components_proper,components_improper)
        else:
            components = np.around(np.einsum('...ji,...i',
                                             np.broadcast_to(self .standard_triangle['improper'], vector_.shape+(3,)),
                                             np.block([vector_[...,:2],np.abs(vector_[...,2:3])])), 12)

            in_SST = np.all(components >= 0.0,axis=-1)

        with np.errstate(invalid='ignore',divide='ignore'):
            rgb = (components/np.linalg.norm(components,axis=-1,keepdims=True))**0.5                # smoothen color ramps
            rgb = np.clip(rgb,0.,1.)                                                                # clip intensity
            rgb /= np.max(rgb,axis=-1,keepdims=True)                                                # normalize to (HS)V = 1
        rgb[np.broadcast_to(~in_SST[...,np.newaxis],rgb.shape)] = 0.0

        return rgb


    @property
    def symmetry_operations(self):
        """Symmetry operations as Rotations."""
        _symmetry_operations = {
            'cubic':         [
                              [ 1.0,            0.0,            0.0,            0.0            ],
                              [ 0.0,            1.0,            0.0,            0.0            ],
                              [ 0.0,            0.0,            1.0,            0.0            ],
                              [ 0.0,            0.0,            0.0,            1.0            ],
                              [ 0.0,            0.0,            0.5*np.sqrt(2), 0.5*np.sqrt(2) ],
                              [ 0.0,            0.0,            0.5*np.sqrt(2),-0.5*np.sqrt(2) ],
                              [ 0.0,            0.5*np.sqrt(2), 0.0,            0.5*np.sqrt(2) ],
                              [ 0.0,            0.5*np.sqrt(2), 0.0,           -0.5*np.sqrt(2) ],
                              [ 0.0,            0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0            ],
                              [ 0.0,           -0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0            ],
                              [ 0.5,            0.5,            0.5,            0.5            ],
                              [-0.5,            0.5,            0.5,            0.5            ],
                              [-0.5,            0.5,            0.5,           -0.5            ],
                              [-0.5,            0.5,           -0.5,            0.5            ],
                              [-0.5,           -0.5,            0.5,            0.5            ],
                              [-0.5,           -0.5,            0.5,           -0.5            ],
                              [-0.5,           -0.5,           -0.5,            0.5            ],
                              [-0.5,            0.5,           -0.5,           -0.5            ],
                              [-0.5*np.sqrt(2), 0.0,            0.0,            0.5*np.sqrt(2) ],
                              [ 0.5*np.sqrt(2), 0.0,            0.0,            0.5*np.sqrt(2) ],
                              [-0.5*np.sqrt(2), 0.0,            0.5*np.sqrt(2), 0.0            ],
                              [-0.5*np.sqrt(2), 0.0,           -0.5*np.sqrt(2), 0.0            ],
                              [-0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0,            0.0            ],
                              [-0.5*np.sqrt(2),-0.5*np.sqrt(2), 0.0,            0.0            ],
                            ],
            'hexagonal':    [
                              [ 1.0,            0.0,            0.0,            0.0            ],
                              [-0.5*np.sqrt(3), 0.0,            0.0,           -0.5            ],
                              [ 0.5,            0.0,            0.0,            0.5*np.sqrt(3) ],
                              [ 0.0,            0.0,            0.0,            1.0            ],
                              [-0.5,            0.0,            0.0,            0.5*np.sqrt(3) ],
                              [-0.5*np.sqrt(3), 0.0,            0.0,            0.5            ],
                              [ 0.0,            1.0,            0.0,            0.0            ],
                              [ 0.0,           -0.5*np.sqrt(3), 0.5,            0.0            ],
                              [ 0.0,            0.5,           -0.5*np.sqrt(3), 0.0            ],
                              [ 0.0,            0.0,            1.0,            0.0            ],
                              [ 0.0,           -0.5,           -0.5*np.sqrt(3), 0.0            ],
                              [ 0.0,            0.5*np.sqrt(3), 0.5,            0.0            ],
                            ],
            'tetragonal':   [
                              [ 1.0,            0.0,            0.0,            0.0            ],
                              [ 0.0,            1.0,            0.0,            0.0            ],
                              [ 0.0,            0.0,            1.0,            0.0            ],
                              [ 0.0,            0.0,            0.0,            1.0            ],
                              [ 0.0,            0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0            ],
                              [ 0.0,           -0.5*np.sqrt(2), 0.5*np.sqrt(2), 0.0            ],
                              [ 0.5*np.sqrt(2), 0.0,            0.0,            0.5*np.sqrt(2) ],
                              [-0.5*np.sqrt(2), 0.0,            0.0,            0.5*np.sqrt(2) ],
                            ],
            'orthorhombic': [
                              [ 1.0,0.0,0.0,0.0 ],
                              [ 0.0,1.0,0.0,0.0 ],
                              [ 0.0,0.0,1.0,0.0 ],
                              [ 0.0,0.0,0.0,1.0 ],
                            ],
            'monoclinic':   [
                              [ 1.0,0.0,0.0,0.0 ],
                              [ 0.0,0.0,1.0,0.0 ],
                            ],
            'triclinic':    [
                              [ 1.0,0.0,0.0,0.0 ],
                            ]}
        return Rotation.from_quaternion(_symmetry_operations[self.family],accept_homomorph=True)


####################################################################################################
    # functions that require lattice, not just family

    def to_pole(self,*,uvw=None,hkl=None,with_symmetry=False):
        """
        Calculate lab frame vector along lattice direction [uvw] or plane normal (hkl).

        Parameters
        ----------
        uvw|hkl : numpy.ndarray of shape (...,3)
            Miller indices of crystallographic direction or plane normal.
            Shape of vector blends with shape of own rotation array.
            For example, a rotation array of shape (3,2) and a (2,4) vector array result in (3,2,4) outputs.
        with_symmetry : bool, optional
            Calculate all N symmetrically equivalent vectors.

        Returns
        -------
        vector : numpy.ndarray of shape (...,3) or (...,N,3)
            Lab frame vector (or vectors if with_symmetry) along [uvw] direction or (hkl) plane normal.

        """
        v = self.to_frame(uvw=uvw,hkl=hkl)
        blend = util.shapeblender(self.shape,v.shape[:-1])
        if with_symmetry:
            sym_ops = self.symmetry_operations
            shape = v.shape[:-1]+sym_ops.shape
            blend += sym_ops.shape
            v = sym_ops.broadcast_to(shape) \
              @ np.broadcast_to(v.reshape(util.shapeshifter(v.shape,shape+(3,))),shape+(3,))
        return ~(self.broadcast_to(blend)) \
               @ np.broadcast_to(v,blend+(3,))


    def Schmid(self,*,N_slip=None,N_twin=None):
        u"""
        Calculate Schmid matrix P = d ⨂ n in the lab frame for selected deformation systems.

        Parameters
        ----------
        N_slip|N_twin : iterable of int
            Number of deformation systems per family of the deformation system.
            Use '*' to select all.

        Returns
        -------
        P : numpy.ndarray of shape (N,...,3,3)
            Schmid matrix for each of the N deformation systems.

        Examples
        --------
        Schmid matrix (in lab frame) of first octahedral slip system of a face-centered
        cubic crystal in "Goss" orientation.

        >>> import damask
        >>> import numpy as np
        >>> np.set_printoptions(3,suppress=True,floatmode='fixed')
        >>> O = damask.Orientation.from_Euler_angles(phi=[0,45,0],degrees=True,lattice='cF')
        >>> O.Schmid(N_slip=[1])
        array([[ 0.000,  0.000,  0.000],
               [ 0.577, -0.000,  0.816],
               [ 0.000,  0.000,  0.000]])

        """
        if (N_slip is not None) ^ (N_twin is None):
            raise KeyError('Specify either "N_slip" or "N_twin"')

        kinematics,active = (self.kinematics('slip'),N_slip) if N_twin is None else \
                            (self.kinematics('twin'),N_twin)
        if active == '*': active = [len(a) for a in kinematics['direction']]

        d = self.to_frame(uvw=np.vstack([kinematics['direction'][i][:n] for i,n in enumerate(active)]))
        p = self.to_frame(hkl=np.vstack([kinematics['plane'][i][:n] for i,n in enumerate(active)]))
        P = np.einsum('...i,...j',d/np.linalg.norm(d,axis=1,keepdims=True),
                                  p/np.linalg.norm(p,axis=1,keepdims=True))

        shape = P.shape[0:1]+self.shape+(3,3)
        return ~self.broadcast_to(shape[:-2]) \
               @ np.broadcast_to(P.reshape(util.shapeshifter(P.shape,shape)),shape)


    def related(self,model):
        """
        Orientations derived from the given relationship.

        One dimension (length according to number of related orientations)
        is added to the left of the Rotation array.

        """
        lattice,o = self.relation_operations(model)
        target = Crystal(lattice=lattice)
        o = o.broadcast_to(o.shape+self.shape,mode='right')
        return Orientation(rotation=o*Rotation(self.quaternion).broadcast_to(o.shape,mode='left'),
                          lattice=lattice,
                          b = self.b if target.ratio['b'] is None else self.a*target.ratio['b'],
                          c = self.c if target.ratio['c'] is None else self.a*target.ratio['c'],
                          alpha = None if 'alpha' in target.immutable else self.alpha,
                          beta  = None if 'beta'  in target.immutable else self.beta,
                          gamma = None if 'gamma' in target.immutable else self.gamma,
                         )