DAMASK_EICMD/lib/damask/orientation.py

# -*- coding: UTF-8 no BOM -*-

###################################################
# NOTE: everything here needs to be a np array #
###################################################

import math,os
import numpy as np

# ******************************************************************************************
class Rodrigues:

    def __init__(self, vector = np.zeros(3)):
      self.vector = vector

    def asQuaternion(self):
      norm = np.linalg.norm(self.vector)
      halfAngle = np.arctan(norm)
      return Quaternion(np.cos(halfAngle),np.sin(halfAngle)*self.vector/norm)

    def asAngleAxis(self):
      norm = np.linalg.norm(self.vector)
      halfAngle = np.arctan(norm)
      return (2.0*halfAngle,self.vector/norm)



# ******************************************************************************************
class Quaternion:
    """
    Orientation represented as unit quaternion.
    
    All methods and naming conventions based on http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions.

    w is the real part, (x, y, z) are the imaginary parts.
    Representation of rotation is in ACTIVE form!
    (Derived directly or through angleAxis, Euler angles, or active matrix)
    Vector "a" (defined in coordinate system "A") is actively rotated to new coordinates "b".
    b = Q * a
    b = np.dot(Q.asMatrix(),a)
    """

    def __init__(self,
                 quatArray = [1.0,0.0,0.0,0.0]):
      """Initializes to identity if not given"""
      self.w, \
      self.x, \
      self.y, \
      self.z = quatArray
      self.homomorph()

    def __iter__(self):
      """Components"""
      return iter([self.w,self.x,self.y,self.z])

    def __copy__(self):
      """Create copy"""
      Q = Quaternion([self.w,self.x,self.y,self.z])
      return Q

    copy = __copy__

    def __repr__(self):
      """Readbable string"""
      return 'Quaternion(real=%+.6f, imag=<%+.6f, %+.6f, %+.6f>)' % \
          (self.w, self.x, self.y, self.z)

    def __pow__(self, exponent):
      """Power"""
      omega = math.acos(self.w)
      vRescale = math.sin(exponent*omega)/math.sin(omega)
      Q = Quaternion()
      Q.w = math.cos(exponent*omega)
      Q.x = self.x * vRescale
      Q.y = self.y * vRescale
      Q.z = self.z * vRescale
      return Q

    def __ipow__(self, exponent):
      """In-place power"""
      omega    = math.acos(self.w)
      vRescale = math.sin(exponent*omega)/math.sin(omega)
      self.w  = np.cos(exponent*omega)
      self.x *= vRescale
      self.y *= vRescale
      self.z *= vRescale
      return self

    def __mul__(self, other):
      """Multiplication"""
      try:                                                          # quaternion
          Aw = self.w
          Ax = self.x
          Ay = self.y
          Az = self.z
          Bw = other.w
          Bx = other.x
          By = other.y
          Bz = other.z
          Q = Quaternion()
          Q.w = - Ax * Bx - Ay * By - Az * Bz + Aw * Bw
          Q.x = + Ax * Bw + Ay * Bz - Az * By + Aw * Bx
          Q.y = - Ax * Bz + Ay * Bw + Az * Bx + Aw * By
          Q.z = + Ax * By - Ay * Bx + Az * Bw + Aw * Bz
          return Q
      except: pass
      try:                                                         # vector (perform active rotation, i.e. q*v*q.conjugated)
          w = self.w
          x = self.x
          y = self.y
          z = self.z
          Vx = other[0]
          Vy = other[1]
          Vz = other[2]

          return np.array([\
             w * w * Vx + 2 * y * w * Vz - 2 * z * w * Vy + \
             x * x * Vx + 2 * y * x * Vy + 2 * z * x * Vz - \
             z * z * Vx - y * y * Vx,
             2 * x * y * Vx + y * y * Vy + 2 * z * y * Vz + \
             2 * w * z * Vx - z * z * Vy + w * w * Vy - \
             2 * x * w * Vz - x * x * Vy,
             2 * x * z * Vx + 2 * y * z * Vy + \
             z * z * Vz - 2 * w * y * Vx - y * y * Vz + \
             2 * w * x * Vy - x * x * Vz + w * w * Vz ])
      except: pass
      try:                                                        # scalar
          Q = self.copy()
          Q.w *= other
          Q.x *= other
          Q.y *= other
          Q.z *= other
          return Q
      except:
          return self.copy()

    def __imul__(self, other):
      """In-place multiplication"""
      try:                                                        # Quaternion
          Aw = self.w
          Ax = self.x
          Ay = self.y
          Az = self.z
          Bw = other.w
          Bx = other.x
          By = other.y
          Bz = other.z
          self.w = - Ax * Bx - Ay * By - Az * Bz + Aw * Bw
          self.x = + Ax * Bw + Ay * Bz - Az * By + Aw * Bx
          self.y = - Ax * Bz + Ay * Bw + Az * Bx + Aw * By
          self.z = + Ax * By - Ay * Bx + Az * Bw + Aw * Bz
      except: pass
      return self

    def __div__(self, other):
      """Division"""
      if isinstance(other, (int,float)):
        w = self.w / other
        x = self.x / other
        y = self.y / other
        z = self.z / other
        return self.__class__([w,x,y,z])
      else:
          return NotImplemented

    def __idiv__(self, other):
      """In-place division"""
      if isinstance(other, (int,float)):
          self.w /= other
          self.x /= other
          self.y /= other
          self.z /= other
      return self

    def __add__(self, other):
      """Addition"""
      if isinstance(other, Quaternion):
        w = self.w + other.w
        x = self.x + other.x
        y = self.y + other.y
        z = self.z + other.z
        return self.__class__([w,x,y,z])
      else:
          return NotImplemented

    def __iadd__(self, other):
      """In-place addition"""
      if isinstance(other, Quaternion):
          self.w += other.w
          self.x += other.x
          self.y += other.y
          self.z += other.z
      return self

    def __sub__(self, other):
      """Subtraction"""
      if isinstance(other, Quaternion):
          Q = self.copy()
          Q.w -= other.w
          Q.x -= other.x
          Q.y -= other.y
          Q.z -= other.z
          return Q
      else:
          return self.copy()

    def __isub__(self, other):
      """In-place subtraction"""
      if isinstance(other, Quaternion):
          self.w -= other.w
          self.x -= other.x
          self.y -= other.y
          self.z -= other.z
      return self

    def __neg__(self):
      """Additive inverse"""
      self.w = -self.w
      self.x = -self.x
      self.y = -self.y
      self.z = -self.z
      return self

    def __abs__(self):
      """Norm"""
      return math.sqrt(self.w ** 2 + \
                       self.x ** 2 + \
                       self.y ** 2 + \
                       self.z ** 2)

    magnitude = __abs__

    def __eq__(self,other):
      """Equal at e-8 precision"""
      return (abs(self.w-other.w) < 1e-8 and \
              abs(self.x-other.x) < 1e-8 and \
              abs(self.y-other.y) < 1e-8 and \
              abs(self.z-other.z) < 1e-8) \
              or \
             (abs(-self.w-other.w) < 1e-8 and \
              abs(-self.x-other.x) < 1e-8 and \
              abs(-self.y-other.y) < 1e-8 and \
              abs(-self.z-other.z) < 1e-8)

    def __ne__(self,other):
      """Not equal at e-8 precision"""
      return not self.__eq__(self,other)

    def __cmp__(self,other):
      """Linear ordering"""
      return (self.Rodrigues()>other.Rodrigues()) - (self.Rodrigues()<other.Rodrigues())

    def magnitude_squared(self):
      return self.w ** 2 + \
             self.x ** 2 + \
             self.y ** 2 + \
             self.z ** 2

    def identity(self):
      self.w = 1.
      self.x = 0.
      self.y = 0.
      self.z = 0.
      return self

    def normalize(self):
      d = self.magnitude()
      if d > 0.0:
          self /= d
      return self

    def conjugate(self):
      self.x = -self.x
      self.y = -self.y
      self.z = -self.z
      return self

    def inverse(self):
      d = self.magnitude()
      if d > 0.0:
        self.conjugate()
        self /= d
      return self

    def homomorph(self):
      if self.w < 0.0:
        self.w = -self.w
        self.x = -self.x
        self.y = -self.y
        self.z = -self.z
      return self

    def normalized(self):
      return self.copy().normalize()

    def conjugated(self):
      return self.copy().conjugate()

    def inversed(self):
      return self.copy().inverse()

    def homomorphed(self):
      return self.copy().homomorph()

    def asList(self):
      return [i for i in self]

    def asM(self):                                                # to find Averaging Quaternions (see F. Landis Markley et al.)
      return np.outer([i for i in self],[i for i in self])

    def asMatrix(self):
      return np.array(
        [[1.0-2.0*(self.y*self.y+self.z*self.z),     2.0*(self.x*self.y-self.z*self.w),     2.0*(self.x*self.z+self.y*self.w)],
         [    2.0*(self.x*self.y+self.z*self.w), 1.0-2.0*(self.x*self.x+self.z*self.z),     2.0*(self.y*self.z-self.x*self.w)],
         [    2.0*(self.x*self.z-self.y*self.w),     2.0*(self.x*self.w+self.y*self.z), 1.0-2.0*(self.x*self.x+self.y*self.y)]])

    def asAngleAxis(self,
                    degrees = False):
      if self.w > 1:
          self.normalize()

      s = math.sqrt(1. - self.w**2)
      x = 2*self.w**2 - 1.
      y = 2*self.w * s

      angle = math.atan2(y,x)
      if angle < 0.0:
        angle *= -1.
        s     *= -1.

      return (np.degrees(angle) if degrees else angle,
              np.array([1.0, 0.0, 0.0] if np.abs(angle) < 1e-6 else [self.x / s, self.y / s, self.z / s]))

    def asRodrigues(self):
      return np.inf*np.ones(3) if self.w == 0.0 else np.array([self.x, self.y, self.z])/self.w

    def asEulers(self,
                 type = "bunge",
                 degrees = False,
                 standardRange = False):
      """
      Orientation as Bunge-Euler angles.

      Conversion of ACTIVE rotation to Euler angles taken from:
      Melcher, A.; Unser, A.; Reichhardt, M.; Nestler, B.; Poetschke, M.; Selzer, M.
      Conversion of EBSD data by a quaternion based algorithm to be used for grain structure simulations
      Technische Mechanik 30 (2010) pp 401--413.
      """
      angles = [0.0,0.0,0.0]

      if type.lower() == 'bunge' or type.lower() == 'zxz':
        if   abs(self.x) < 1e-4 and abs(self.y) < 1e-4:
          x = self.w**2 - self.z**2
          y = 2.*self.w*self.z
          angles[0] = math.atan2(y,x)
        elif abs(self.w) < 1e-4 and abs(self.z) < 1e-4:
          x = self.x**2 - self.y**2
          y = 2.*self.x*self.y
          angles[0] = math.atan2(y,x)
          angles[1] = math.pi
        else:
          chi = math.sqrt((self.w**2 + self.z**2)*(self.x**2 + self.y**2))

          x = (self.w * self.x - self.y * self.z)/2./chi
          y = (self.w * self.y + self.x * self.z)/2./chi
          angles[0] = math.atan2(y,x)

          x = self.w**2 + self.z**2 - (self.x**2 + self.y**2)
          y = 2.*chi
          angles[1] = math.atan2(y,x)

          x = (self.w * self.x + self.y * self.z)/2./chi
          y = (self.z * self.x - self.y * self.w)/2./chi
          angles[2] = math.atan2(y,x)

        if standardRange:
          angles[0] %= 2*math.pi
          if angles[1] < 0.0:
            angles[1] += math.pi
            angles[2] *= -1.0
          angles[2] %= 2*math.pi

      return np.degrees(angles) if degrees else angles


#    # Static constructors
    @classmethod
    def fromIdentity(cls):
      return cls()


    @classmethod
    def fromRandom(cls,randomSeed = None):
      if randomSeed is None:
        randomSeed = int(os.urandom(4).encode('hex'), 16)
      np.random.seed(randomSeed)
      r = np.random.random(3)
      w = math.cos(2.0*math.pi*r[0])*math.sqrt(r[2])
      x = math.sin(2.0*math.pi*r[1])*math.sqrt(1.0-r[2])
      y = math.cos(2.0*math.pi*r[1])*math.sqrt(1.0-r[2])
      z = math.sin(2.0*math.pi*r[0])*math.sqrt(r[2])
      return cls([w,x,y,z])


    @classmethod
    def fromRodrigues(cls, rodrigues):
      if not isinstance(rodrigues, np.ndarray): rodrigues = np.array(rodrigues)
      halfangle = math.atan(np.linalg.norm(rodrigues))
      c = math.cos(halfangle)
      w = c
      x,y,z = c*rodrigues
      return cls([w,x,y,z])


    @classmethod
    def fromAngleAxis(cls,
                      angle,
                      axis,
                      degrees = False):
      if not isinstance(axis, np.ndarray): axis = np.array(axis,dtype='d')
      axis = axis.astype(float)/np.linalg.norm(axis)
      angle = np.radians(angle) if degrees else angle
      s = math.sin(0.5 * angle)
      w = math.cos(0.5 * angle)
      x = axis[0] * s
      y = axis[1] * s
      z = axis[2] * s
      return cls([w,x,y,z])


    @classmethod
    def fromEulers(cls,
                   eulers,
                   type = 'Bunge',
                   degrees = False):
      if not isinstance(eulers, np.ndarray): eulers = np.array(eulers,dtype='d')
      eulers = np.radians(eulers) if degrees else eulers

      c = np.cos(0.5 * eulers)
      s = np.sin(0.5 * eulers)

      if type.lower() == 'bunge' or type.lower() == 'zxz':
        w =   c[0] * c[1] * c[2] - s[0] * c[1] * s[2]
        x =   c[0] * s[1] * c[2] + s[0] * s[1] * s[2]
        y = - c[0] * s[1] * s[2] + s[0] * s[1] * c[2]
        z =   c[0] * c[1] * s[2] + s[0] * c[1] * c[2]
      else:
        w = c[0] * c[1] * c[2] - s[0] * s[1] * s[2]
        x = s[0] * s[1] * c[2] + c[0] * c[1] * s[2]
        y = s[0] * c[1] * c[2] + c[0] * s[1] * s[2]
        z = c[0] * s[1] * c[2] - s[0] * c[1] * s[2]
      return cls([w,x,y,z])


# Modified Method to calculate Quaternion from Orientation Matrix,
# Source: http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/

    @classmethod
    def fromMatrix(cls, m):
      if m.shape != (3,3) and np.prod(m.shape) == 9:
        m = m.reshape(3,3)

      tr = np.trace(m)
      if tr > 1e-8:
        s = math.sqrt(tr + 1.0)*2.0

        return cls(
          [ s*0.25,
            (m[2,1] - m[1,2])/s,
            (m[0,2] - m[2,0])/s,
            (m[1,0] - m[0,1])/s,
          ])

      elif m[0,0] > m[1,1] and m[0,0] > m[2,2]:
        t = m[0,0] - m[1,1] - m[2,2] + 1.0
        s = 2.0*math.sqrt(t)

        return cls(
          [ (m[2,1] - m[1,2])/s,
            s*0.25,
            (m[0,1] + m[1,0])/s,
            (m[2,0] + m[0,2])/s,
          ])

      elif m[1,1] > m[2,2]:
        t = -m[0,0] + m[1,1] - m[2,2] + 1.0
        s = 2.0*math.sqrt(t)

        return cls(
          [ (m[0,2] - m[2,0])/s,
            (m[0,1] + m[1,0])/s,
            s*0.25,
            (m[1,2] + m[2,1])/s,
          ])

      else:
        t = -m[0,0] - m[1,1] + m[2,2] + 1.0
        s = 2.0*math.sqrt(t)

        return cls(
          [ (m[1,0] - m[0,1])/s,
            (m[2,0] + m[0,2])/s,
            (m[1,2] + m[2,1])/s,
            s*0.25,
          ])


    @classmethod
    def new_interpolate(cls, q1, q2, t):
        """
        Interpolation
 
        See http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20070017872_2007014421.pdf
        for (another?) way to interpolate quaternions.
        """
        assert isinstance(q1, Quaternion) and isinstance(q2, Quaternion)
        Q = cls()

        costheta = q1.w * q2.w + q1.x * q2.x + q1.y * q2.y + q1.z * q2.z
        if costheta < 0.:
            costheta = -costheta
            q1 = q1.conjugated()
        elif costheta > 1:
            costheta = 1

        theta = math.acos(costheta)
        if abs(theta) < 0.01:
            Q.w = q2.w
            Q.x = q2.x
            Q.y = q2.y
            Q.z = q2.z
            return Q

        sintheta = math.sqrt(1.0 - costheta * costheta)
        if abs(sintheta) < 0.01:
            Q.w = (q1.w + q2.w) * 0.5
            Q.x = (q1.x + q2.x) * 0.5
            Q.y = (q1.y + q2.y) * 0.5
            Q.z = (q1.z + q2.z) * 0.5
            return Q

        ratio1 = math.sin((1 - t) * theta) / sintheta
        ratio2 = math.sin(t * theta) / sintheta

        Q.w = q1.w * ratio1 + q2.w * ratio2
        Q.x = q1.x * ratio1 + q2.x * ratio2
        Q.y = q1.y * ratio1 + q2.y * ratio2
        Q.z = q1.z * ratio1 + q2.z * ratio2
        return Q


# ******************************************************************************************
class Symmetry:

  lattices = [None,'orthorhombic','tetragonal','hexagonal','cubic',]

  def __init__(self, symmetry = None):
    """Lattice with given symmetry, defaults to None"""
    if isinstance(symmetry, str) and symmetry.lower() in Symmetry.lattices:
      self.lattice = symmetry.lower()
    else:
      self.lattice = None


  def __copy__(self):
    """Copy"""
    return self.__class__(self.lattice)

  copy = __copy__


  def __repr__(self):
    """Readbable string"""
    return '%s' % (self.lattice)


  def __eq__(self, other):
    """Equal"""
    return self.lattice == other.lattice

  def __neq__(self, other):
    """Not equal"""
    return not self.__eq__(other)

  def __cmp__(self,other):
    """Linear ordering"""
    myOrder = Symmetry.lattices.index(self.lattice)
    otherOrder = Symmetry.lattices.index(other.lattice)
    return (myOrder > otherOrder) - (myOrder < otherOrder)

  def symmetryQuats(self,who = []):
    """List of symmetry operations as quaternions."""
    if self.lattice == 'cubic':
      symQuats =  [
                    [ 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*math.sqrt(2), 0.5*math.sqrt(2) ],
                    [ 0.0,              0.0,              0.5*math.sqrt(2),-0.5*math.sqrt(2) ],
                    [ 0.0,              0.5*math.sqrt(2), 0.0,              0.5*math.sqrt(2) ],
                    [ 0.0,              0.5*math.sqrt(2), 0.0,             -0.5*math.sqrt(2) ],
                    [ 0.0,              0.5*math.sqrt(2),-0.5*math.sqrt(2), 0.0              ],
                    [ 0.0,             -0.5*math.sqrt(2),-0.5*math.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*math.sqrt(2), 0.0,              0.0,              0.5*math.sqrt(2) ],
                    [ 0.5*math.sqrt(2), 0.0,              0.0,              0.5*math.sqrt(2) ],
                    [-0.5*math.sqrt(2), 0.0,              0.5*math.sqrt(2), 0.0              ],
                    [-0.5*math.sqrt(2), 0.0,             -0.5*math.sqrt(2), 0.0              ],
                    [-0.5*math.sqrt(2), 0.5*math.sqrt(2), 0.0,              0.0              ],
                    [-0.5*math.sqrt(2),-0.5*math.sqrt(2), 0.0,              0.0              ],
                  ]
    elif self.lattice == 'hexagonal':
      symQuats =  [
                    [ 1.0,0.0,0.0,0.0 ],
                    [-0.5*math.sqrt(3), 0.0, 0.0,-0.5 ],
                    [ 0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
                    [ 0.0,0.0,0.0,1.0 ],
                    [-0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
                    [-0.5*math.sqrt(3), 0.0, 0.0, 0.5 ],
                    [ 0.0,1.0,0.0,0.0 ],
                    [ 0.0,-0.5*math.sqrt(3), 0.5, 0.0 ],
                    [ 0.0, 0.5,-0.5*math.sqrt(3), 0.0 ],
                    [ 0.0,0.0,1.0,0.0 ],
                    [ 0.0,-0.5,-0.5*math.sqrt(3), 0.0 ],
                    [ 0.0, 0.5*math.sqrt(3), 0.5, 0.0 ],
                  ]
    elif self.lattice == 'tetragonal':
      symQuats =  [
                    [ 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*math.sqrt(2), 0.5*math.sqrt(2), 0.0 ],
                    [ 0.0,-0.5*math.sqrt(2), 0.5*math.sqrt(2), 0.0 ],
                    [ 0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
                    [-0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
                  ]
    elif self.lattice == 'orthorhombic':
      symQuats =  [
                    [ 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 ],
                  ]
    else:
      symQuats =  [
                    [ 1.0,0.0,0.0,0.0 ],
                  ]

    return list(map(Quaternion,
               np.array(symQuats)[np.atleast_1d(np.array(who)) if who != [] else range(len(symQuats))]))
    
    
  def equivalentQuaternions(self,
                            quaternion,
                            who = []):
    """List of symmetrically equivalent quaternions based on own symmetry."""
    return [quaternion*q for q in self.symmetryQuats(who)]


  def inFZ(self,R):
    """Check whether given Rodrigues vector falls into fundamental zone of own symmetry."""
    if isinstance(R, Quaternion): R = R.asRodrigues()                                               # translate accidentially passed quaternion
# fundamental zone in Rodrigues space is point symmetric around origin
    R = abs(R)                                                                                      
    if self.lattice == 'cubic':
      return     math.sqrt(2.0)-1.0 >= R[0] \
             and math.sqrt(2.0)-1.0 >= R[1] \
             and math.sqrt(2.0)-1.0 >= R[2] \
             and 1.0 >= R[0] + R[1] + R[2]
    elif self.lattice == 'hexagonal':
      return     1.0 >= R[0] and 1.0 >= R[1] and 1.0 >= R[2] \
             and 2.0 >= math.sqrt(3)*R[0] + R[1] \
             and 2.0 >= math.sqrt(3)*R[1] + R[0] \
             and 2.0 >= math.sqrt(3) + R[2]
    elif self.lattice == 'tetragonal':
      return     1.0 >= R[0] and 1.0 >= R[1] \
             and math.sqrt(2.0) >= R[0] + R[1] \
             and math.sqrt(2.0) >= R[2] + 1.0
    elif self.lattice == 'orthorhombic':
      return     1.0 >= R[0] and 1.0 >= R[1] and 1.0 >= R[2]
    else:
      return True


  def inDisorientationSST(self,R):
    """
    Check whether given Rodrigues vector (of misorientation) falls into standard stereographic triangle of own symmetry.

    Determination of disorientations follow the work of A. Heinz and P. Neumann:
    Representation of Orientation and Disorientation Data for Cubic, Hexagonal, Tetragonal and Orthorhombic Crystals
    Acta Cryst. (1991). A47, 780-789
    """
    if isinstance(R, Quaternion): R = R.asRodrigues()       # translate accidentially passed quaternion

    epsilon = 0.0
    if self.lattice == 'cubic':
      return R[0] >= R[1]+epsilon                and R[1] >= R[2]+epsilon    and R[2] >= epsilon

    elif self.lattice == 'hexagonal':
      return R[0] >= math.sqrt(3)*(R[1]-epsilon) and R[1] >= epsilon         and R[2] >= epsilon

    elif self.lattice == 'tetragonal':
      return R[0] >= R[1]-epsilon                and R[1] >= epsilon         and R[2] >= epsilon

    elif self.lattice == 'orthorhombic':
      return R[0] >= epsilon                     and R[1] >= epsilon         and R[2] >= epsilon

    else:
      return True


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

    proper considers only vectors with z >= 0, hence uses two neighboring SSTs.
    Return inverse pole figure color if requested.
    """
#     basis = {'cubic' :        np.linalg.inv(np.array([[0.,0.,1.],                                 # direction of red
#                                                       [1.,0.,1.]/np.sqrt(2.),                     # direction of green
#                                                       [1.,1.,1.]/np.sqrt(3.)]).transpose()),      # direction of blue
#              'hexagonal' :    np.linalg.inv(np.array([[0.,0.,1.],                                 # direction of red
#                                                       [1.,0.,0.],                                 # direction of green
#                                                       [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).transpose()),      # direction of blue
#              'tetragonal' :   np.linalg.inv(np.array([[0.,0.,1.],                                 # direction of red
#                                                       [1.,0.,0.],                                 # direction of green
#                                                       [1.,1.,0.]/np.sqrt(2.)]).transpose()),      # direction of blue
#              'orthorhombic' : np.linalg.inv(np.array([[0.,0.,1.],                                 # direction of red
#                                                       [1.,0.,0.],                                 # direction of green
#                                                       [0.,1.,0.]]).transpose()),                  # direction of blue
#             }

    if self.lattice == 'cubic':
      basis = {'improper':np.array([ [-1.            ,  0.            ,  1. ],
                                     [ np.sqrt(2.)   , -np.sqrt(2.)   ,  0. ],
                                     [ 0.            ,  np.sqrt(3.)   ,  0. ] ]),
                 'proper':np.array([ [ 0.            , -1.            ,  1. ],
                                     [-np.sqrt(2.)   , np.sqrt(2.)    ,  0. ],
                                     [ np.sqrt(3.)   ,  0.            ,  0. ] ]),
              }
    elif self.lattice == 'hexagonal':
      basis = {'improper':np.array([ [ 0.            ,  0.            ,  1. ],
                                     [ 1.            , -np.sqrt(3.)   ,  0. ],
                                     [ 0.            ,  2.            ,  0. ] ]),
               'proper':np.array([   [ 0.            ,  0.            ,  1. ],
                                     [-1.            ,  np.sqrt(3.)   ,  0. ],
                                     [ np.sqrt(3.)   , -1.            ,  0. ] ]),
              }
    elif self.lattice == 'tetragonal':
      basis = {'improper':np.array([ [ 0.            ,  0.            ,  1. ],
                                     [ 1.            , -1.            ,  0. ],
                                     [ 0.            ,  np.sqrt(2.)   ,  0. ] ]),
               'proper':np.array([   [ 0.            ,  0.            ,  1. ],
                                     [-1.            ,  1.            ,  0. ],
                                     [ np.sqrt(2.)   ,  0.            ,  0. ] ]),
              }
    elif self.lattice == 'orthorhombic':
      basis = {'improper':np.array([ [ 0., 0., 1.],
                                     [ 1., 0., 0.],
                                     [ 0., 1., 0.] ]),
                 'proper':np.array([ [ 0., 0., 1.],
                                     [-1., 0., 0.],
                                     [ 0., 1., 0.] ]),
              }
    else:                                                                                           # direct exit for unspecified symmetry
      if color:
        return (True,np.zeros(3,'d'))
      else:
        return True

    v = np.array(vector,dtype = float)
    if proper:                                                                                      # check both improper ...
      theComponents = np.dot(basis['improper'],v)
      inSST = np.all(theComponents >= 0.0)
      if not inSST:                                                                                 # ... and proper SST
        theComponents = np.dot(basis['proper'],v)
        inSST = np.all(theComponents >= 0.0)
    else:      
      v[2] = abs(v[2])                                                                              # z component projects identical 
      theComponents = np.dot(basis['improper'],v)                                                   # for positive and negative values
      inSST = np.all(theComponents >= 0.0)

    if color:                                                                                       # have to return color array
      if inSST:
        rgb = np.power(theComponents/np.linalg.norm(theComponents),0.5)                             # smoothen color ramps
        rgb = np.minimum(np.ones(3,'d'),rgb)                                                        # limit to maximum intensity
        rgb /= max(rgb)                                                                             # normalize to (HS)V = 1
      else:
        rgb = np.zeros(3,'d')
      return (inSST,rgb)
    else:
      return inSST

# code derived from https://github.com/ezag/pyeuclid
# suggested reading: http://web.mit.edu/2.998/www/QuaternionReport1.pdf



# ******************************************************************************************
class Orientation:

  __slots__ = ['quaternion','symmetry']

  def __init__(self,
               quaternion = Quaternion.fromIdentity(),
               Rodrigues  = None,
               angleAxis  = None,
               matrix     = None,
               Eulers     = None,
               random     = False,                                                                  # integer to have a fixed seed or True for real random
               symmetry   = None,
               degrees    = False,
              ):
    if random:                                                                                      # produce random orientation
      if isinstance(random, bool ):
        self.quaternion = Quaternion.fromRandom()
      else:
        self.quaternion = Quaternion.fromRandom(randomSeed=random)
    elif isinstance(Eulers, np.ndarray) and Eulers.shape == (3,):                                   # based on given Euler angles
      self.quaternion = Quaternion.fromEulers(Eulers,type='bunge',degrees=degrees)
    elif isinstance(matrix, np.ndarray) :                                                           # based on given rotation matrix
      self.quaternion = Quaternion.fromMatrix(matrix)
    elif isinstance(angleAxis, np.ndarray) and angleAxis.shape == (4,):                             # based on given angle and rotation axis
      self.quaternion = Quaternion.fromAngleAxis(angleAxis[0],angleAxis[1:4],degrees=degrees)
    elif isinstance(Rodrigues, np.ndarray) and Rodrigues.shape == (3,):                             # based on given Rodrigues vector
      self.quaternion = Quaternion.fromRodrigues(Rodrigues)
    elif isinstance(quaternion, Quaternion):                                                        # based on given quaternion
      self.quaternion = quaternion.homomorphed()
    elif isinstance(quaternion, np.ndarray) and quaternion.shape == (4,):                           # based on given quaternion-like array
      self.quaternion = Quaternion(quaternion).homomorphed()

    self.symmetry = Symmetry(symmetry)

  def __copy__(self):
    """Copy"""
    return self.__class__(quaternion=self.quaternion,symmetry=self.symmetry.lattice)

  copy = __copy__


  def __repr__(self):
    """Value as all implemented representations"""
    return 'Symmetry: %s\n' % (self.symmetry) + \
           'Quaternion: %s\n' % (self.quaternion) + \
           'Matrix:\n%s\n' % ( '\n'.join(['\t'.join(map(str,self.asMatrix()[i,:])) for i in range(3)]) ) + \
           'Bunge Eulers / deg: %s' % ('\t'.join(map(str,self.asEulers('bunge',degrees=True))) )

  def asQuaternion(self):
    return self.quaternion.asList()

  def asEulers(self,
               type = 'bunge',
               degrees = False,
               standardRange = False):
    return self.quaternion.asEulers(type, degrees, standardRange)
  eulers = property(asEulers)

  def asRodrigues(self):
    return self.quaternion.asRodrigues()
  rodrigues = property(asRodrigues)

  def asAngleAxis(self,
                  degrees = False):
    return self.quaternion.asAngleAxis(degrees)
  angleAxis = property(asAngleAxis)

  def asMatrix(self):
    return self.quaternion.asMatrix()
  matrix = property(asMatrix)

  def inFZ(self):
    return self.symmetry.inFZ(self.quaternion.asRodrigues())
  infz = property(inFZ)

  def equivalentQuaternions(self,
                            who = []):
    return self.symmetry.equivalentQuaternions(self.quaternion,who)

  def equivalentOrientations(self,
                             who = []):
    return [Orientation(quaternion = q, symmetry = self.symmetry.lattice) for q in self.equivalentQuaternions(who)]

  def reduced(self):
    """Transform orientation to fall into fundamental zone according to symmetry"""
    for me in self.symmetry.equivalentQuaternions(self.quaternion):
      if self.symmetry.inFZ(me.asRodrigues()): break

    return Orientation(quaternion=me,symmetry=self.symmetry.lattice)


  def disorientation(self,
                     other,
                     SST = True):
    """
    Disorientation between myself and given other orientation.

    Rotation axis falls into SST if SST == True.
    (Currently requires same symmetry for both orientations.
     Look into A. Heinz and P. Neumann 1991 for cases with differing sym.)
    """
    if self.symmetry != other.symmetry: raise TypeError('disorientation between different symmetry classes not supported yet.')

    misQ = self.quaternion.conjugated()*other.quaternion
    mySymQs    =  self.symmetry.symmetryQuats() if SST else self.symmetry.symmetryQuats()[:1]       # take all or only first sym operation
    otherSymQs = other.symmetry.symmetryQuats()
    
    for i,sA in enumerate(mySymQs):
      for j,sB in enumerate(otherSymQs):
        theQ = sA.conjugated()*misQ*sB
        for k in range(2):
          theQ.conjugate()
          breaker = self.symmetry.inFZ(theQ) \
                    and (not SST or other.symmetry.inDisorientationSST(theQ))
          if breaker: break
        if breaker: break
      if breaker: break

# disorientation, own sym, other sym, self-->other: True, self<--other: False
    return (Orientation(quaternion = theQ,symmetry = self.symmetry.lattice),
            i,j,k == 1)                                                                             


  def inversePole(self,
                  axis,
                  proper = False,
                  SST = True):
    """Axis rotated according to orientation (using crystal symmetry to ensure location falls into SST)"""
    if SST:                                                                                         # pole requested to be within SST
      for i,q in enumerate(self.symmetry.equivalentQuaternions(self.quaternion)):                   # test all symmetric equivalent quaternions
        pole = q.conjugated()*axis                                                                  # align crystal direction to axis
        if self.symmetry.inSST(pole,proper): break                                                  # found SST version
    else:
      pole = self.quaternion.conjugated()*axis                                                      # align crystal direction to axis

    return (pole,i if SST else 0)

  def IPFcolor(self,axis):
    """TSL color of inverse pole figure for given axis"""
    color = np.zeros(3,'d')

    for q in self.symmetry.equivalentQuaternions(self.quaternion):
      pole = q.conjugated()*axis                                                                    # align crystal direction to axis
      inSST,color = self.symmetry.inSST(pole,color=True)
      if inSST: break

    return color

  @classmethod
  def average(cls,
              orientations,
              multiplicity = []):
    """
    Average orientation

    ref: F. Landis Markley, Yang Cheng, John Lucas Crassidis, and Yaakov Oshman.
         Averaging Quaternions,
         Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4 (2007), pp. 1193-1197.
         doi: 10.2514/1.28949
    usage:
         a = Orientation(Eulers=np.radians([10, 10, 0]), symmetry='hexagonal')
         b = Orientation(Eulers=np.radians([20, 0, 0]),  symmetry='hexagonal')
         avg = Orientation.average([a,b])
    """
    if not all(isinstance(item, Orientation) for item in orientations):
      raise TypeError("Only instances of Orientation can be averaged.")

    N = len(orientations)
    if multiplicity == [] or not multiplicity:
      multiplicity = np.ones(N,dtype='i')

    reference = orientations[0]                                                                     # take first as reference
    for i,(o,n) in enumerate(zip(orientations,multiplicity)):
      closest = o.equivalentOrientations(reference.disorientation(o,SST = False)[2])[0]             # select sym orientation with lowest misorientation
      M = closest.quaternion.asM() * n if i == 0 else M + closest.quaternion.asM() * n              # noqa add (multiples) of this orientation to average noqa
    eig, vec = np.linalg.eig(M/N)

    return Orientation(quaternion = Quaternion(quatArray = np.real(vec.T[eig.argmax()])),
                       symmetry = reference.symmetry.lattice)


  def related(self,
              relationModel,
              direction,
              targetSymmetry = 'cubic'):
    """
    Orientation relationship

    positive number: fcc --> bcc
    negative number: bcc --> fcc
    """
    if relationModel not in ['KS','GT','GTdash','NW','Pitsch','Bain']:  return None
    if int(direction) == 0:  return None

    # KS from S. Morito et al./Journal of Alloys and Compounds 5775 (2013) S587-S592
    # for KS rotation matrices also check  K. Kitahara et al./Acta Materialia 54 (2006) 1279-1288
    # GT from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
    # GT' from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
    # NW from H. Kitahara et al./Materials Characterization 54 (2005) 378-386
    # Pitsch from Y. He et al./Acta Materialia 53 (2005) 1179-1190
    # Bain from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81

    variant  = int(abs(direction))-1
    (me,other)  = (0,1) if direction > 0 else (1,0)

    planes = {'KS': \
                    np.array([[[  1,  1,  1],[  0,  1,  1]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[  1,  1, -1],[  0,  1,  1]],
                              [[  1,  1, -1],[  0,  1,  1]],
                              [[  1,  1, -1],[  0,  1,  1]],
                              [[  1,  1, -1],[  0,  1,  1]],
                              [[  1,  1, -1],[  0,  1,  1]],
                              [[  1,  1, -1],[  0,  1,  1]]]),
              'GT': \
                    np.array([[[  1,  1,  1],[  1,  0,  1]],
                              [[  1,  1,  1],[  1,  1,  0]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[ -1, -1,  1],[ -1,  0,  1]],
                              [[ -1, -1,  1],[ -1, -1,  0]],
                              [[ -1, -1,  1],[  0, -1,  1]],
                              [[ -1,  1,  1],[ -1,  0,  1]],
                              [[ -1,  1,  1],[ -1,  1,  0]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  1,  0,  1]],
                              [[  1, -1,  1],[  1, -1,  0]],
                              [[  1, -1,  1],[  0, -1,  1]],
                              [[  1,  1,  1],[  1,  1,  0]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[  1,  1,  1],[  1,  0,  1]],
                              [[ -1, -1,  1],[ -1, -1,  0]],
                              [[ -1, -1,  1],[  0, -1,  1]],
                              [[ -1, -1,  1],[ -1,  0,  1]],
                              [[ -1,  1,  1],[ -1,  1,  0]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[ -1,  0,  1]],
                              [[  1, -1,  1],[  1, -1,  0]],
                              [[  1, -1,  1],[  0, -1,  1]],
                              [[  1, -1,  1],[  1,  0,  1]]]),
              'GTdash': \
                    np.array([[[  7, 17, 17],[ 12,  5, 17]],
                              [[ 17,  7, 17],[ 17, 12,  5]],
                              [[ 17, 17,  7],[  5, 17, 12]],
                              [[ -7,-17, 17],[-12, -5, 17]],
                              [[-17, -7, 17],[-17,-12,  5]],
                              [[-17,-17,  7],[ -5,-17, 12]],
                              [[  7,-17,-17],[ 12, -5,-17]],
                              [[ 17, -7,-17],[ 17,-12, -5]],
                              [[ 17,-17, -7],[  5,-17,-12]],
                              [[ -7, 17,-17],[-12,  5,-17]],
                              [[-17,  7,-17],[-17, 12, -5]],
                              [[-17, 17, -7],[ -5, 17,-12]],
                              [[  7, 17, 17],[ 12, 17,  5]],
                              [[ 17,  7, 17],[  5, 12, 17]],
                              [[ 17, 17,  7],[ 17,  5, 12]],
                              [[ -7,-17, 17],[-12,-17,  5]],
                              [[-17, -7, 17],[ -5,-12, 17]],
                              [[-17,-17,  7],[-17, -5, 12]],
                              [[  7,-17,-17],[ 12,-17, -5]],
                              [[ 17, -7,-17],[ 5, -12,-17]],
                              [[ 17,-17,  7],[ 17, -5,-12]],
                              [[ -7, 17,-17],[-12, 17, -5]],
                              [[-17,  7,-17],[ -5, 12,-17]],
                              [[-17, 17, -7],[-17,  5,-12]]]),
              'NW': \
                    np.array([[[  1,  1,  1],[  0,  1,  1]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[  1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[ -1,  1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[  1, -1,  1],[  0,  1,  1]],
                              [[ -1, -1,  1],[  0,  1,  1]],
                              [[ -1, -1,  1],[  0,  1,  1]],
                              [[ -1, -1,  1],[  0,  1,  1]]]),
              'Pitsch': \
                    np.array([[[  0,  1,  0],[ -1,  0,  1]],
                              [[  0,  0,  1],[  1, -1,  0]],
                              [[  1,  0,  0],[  0,  1, -1]],
                              [[  1,  0,  0],[  0, -1, -1]],
                              [[  0,  1,  0],[ -1,  0, -1]],
                              [[  0,  0,  1],[ -1, -1,  0]],
                              [[  0,  1,  0],[ -1,  0, -1]],
                              [[  0,  0,  1],[ -1, -1,  0]],
                              [[  1,  0,  0],[  0, -1, -1]],
                              [[  1,  0,  0],[  0, -1,  1]],
                              [[  0,  1,  0],[  1,  0, -1]],
                              [[  0,  0,  1],[ -1,  1,  0]]]),
              'Bain': \
                    np.array([[[  1,  0,  0],[  1,  0,  0]],
                              [[  0,  1,  0],[  0,  1,  0]],
                              [[  0,  0,  1],[  0,  0,  1]]]),
              }

    normals = {'KS': \
                    np.array([[[ -1,  0,  1],[ -1, -1,  1]],
                              [[ -1,  0,  1],[ -1,  1, -1]],
                              [[  0,  1, -1],[ -1, -1,  1]],
                              [[  0,  1, -1],[ -1,  1, -1]],
                              [[  1, -1,  0],[ -1, -1,  1]],
                              [[  1, -1,  0],[ -1,  1, -1]],
                              [[  1,  0, -1],[ -1, -1,  1]],
                              [[  1,  0, -1],[ -1,  1, -1]],
                              [[ -1, -1,  0],[ -1, -1,  1]],
                              [[ -1, -1,  0],[ -1,  1, -1]],
                              [[  0,  1,  1],[ -1, -1,  1]],
                              [[  0,  1,  1],[ -1,  1, -1]],
                              [[  0, -1,  1],[ -1, -1,  1]],
                              [[  0, -1,  1],[ -1,  1, -1]],
                              [[ -1,  0, -1],[ -1, -1,  1]],
                              [[ -1,  0, -1],[ -1,  1, -1]],
                              [[  1,  1,  0],[ -1, -1,  1]],
                              [[  1,  1,  0],[ -1,  1, -1]],
                              [[ -1,  1,  0],[ -1, -1,  1]],
                              [[ -1,  1,  0],[ -1,  1, -1]],
                              [[  0, -1, -1],[ -1, -1,  1]],
                              [[  0, -1, -1],[ -1,  1, -1]],
                              [[  1,  0,  1],[ -1, -1,  1]],
                              [[  1,  0,  1],[ -1,  1, -1]]]),
              'GT': \
                    np.array([[[ -5,-12, 17],[-17, -7, 17]],
                              [[ 17, -5,-12],[ 17,-17, -7]],
                              [[-12, 17, -5],[ -7, 17,-17]],
                              [[  5, 12, 17],[ 17,  7, 17]],
                              [[-17,  5,-12],[-17, 17, -7]],
                              [[ 12,-17, -5],[  7,-17,-17]],
                              [[ -5, 12,-17],[-17,  7,-17]],
                              [[ 17,  5, 12],[ 17, 17,  7]],
                              [[-12,-17,  5],[ -7,-17, 17]],
                              [[  5,-12,-17],[ 17, -7,-17]],
                              [[-17, -5, 12],[-17,-17,  7]],
                              [[ 12, 17,  5],[  7, 17, 17]],
                              [[ -5, 17,-12],[-17, 17, -7]],
                              [[-12, -5, 17],[ -7,-17, 17]],
                              [[ 17,-12, -5],[ 17, -7,-17]],
                              [[  5,-17,-12],[ 17,-17, -7]],
                              [[ 12,  5, 17],[  7, 17, 17]],
                              [[-17, 12, -5],[-17,  7,-17]],
                              [[ -5,-17, 12],[-17,-17,  7]],
                              [[-12,  5,-17],[ -7, 17,-17]],
                              [[ 17, 12,  5],[ 17,  7, 17]],
                              [[  5, 17, 12],[ 17, 17,  7]],
                              [[ 12, -5,-17],[  7,-17,-17]],
                              [[-17,-12,  5],[-17,  7, 17]]]),
              'GTdash': \
                    np.array([[[  0,  1, -1],[  1,  1, -1]],
                              [[ -1,  0,  1],[ -1,  1,  1]],
                              [[  1, -1,  0],[  1, -1,  1]],
                              [[  0, -1, -1],[ -1, -1, -1]],
                              [[  1,  0,  1],[  1, -1,  1]],
                              [[  1, -1,  0],[  1, -1, -1]],
                              [[  0,  1, -1],[ -1,  1, -1]],
                              [[  1,  0,  1],[  1,  1,  1]],
                              [[ -1, -1,  0],[ -1, -1,  1]],
                              [[  0, -1, -1],[  1, -1, -1]],
                              [[ -1,  0,  1],[ -1, -1,  1]],
                              [[ -1, -1,  0],[ -1, -1, -1]],
                              [[  0, -1,  1],[  1, -1,  1]],
                              [[  1,  0, -1],[  1,  1, -1]],
                              [[ -1,  1,  0],[ -1,  1,  1]],
                              [[  0,  1,  1],[ -1,  1,  1]],
                              [[ -1,  0, -1],[ -1, -1, -1]],
                              [[ -1,  1,  0],[ -1,  1, -1]],
                              [[  0, -1,  1],[ -1, -1,  1]],
                              [[ -1,  0, -1],[ -1,  1, -1]],
                              [[  1,  1,  0],[  1,  1,  1]],
                              [[  0,  1,  1],[  1,  1,  1]],
                              [[  1,  0, -1],[  1, -1, -1]],
                              [[  1,  1,  0],[  1,  1, -1]]]),
              'NW': \
                    np.array([[[  2, -1, -1],[  0, -1,  1]],
                              [[ -1,  2, -1],[  0, -1,  1]],
                              [[ -1, -1,  2],[  0, -1,  1]],
                              [[ -2, -1, -1],[  0, -1,  1]],
                              [[  1,  2, -1],[  0, -1,  1]],
                              [[  1, -1,  2],[  0, -1,  1]],
                              [[  2,  1, -1],[  0, -1,  1]],
                              [[ -1, -2, -1],[  0, -1,  1]],
                              [[ -1,  1,  2],[  0, -1,  1]],
                              [[ -1,  2,  1],[  0, -1,  1]],
                              [[ -1,  2,  1],[  0, -1,  1]],
                              [[ -1, -1, -2],[  0, -1,  1]]]),
              'Pitsch': \
                    np.array([[[  1,  0,  1],[  1, -1,  1]],
                              [[  1,  1,  0],[  1,  1, -1]],
                              [[  0,  1,  1],[ -1,  1,  1]],
                              [[  0,  1, -1],[ -1,  1, -1]],
                              [[ -1,  0,  1],[ -1, -1,  1]],
                              [[  1, -1,  0],[  1, -1, -1]],
                              [[  1,  0, -1],[  1, -1, -1]],
                              [[ -1,  1,  0],[ -1,  1, -1]],
                              [[  0, -1,  1],[ -1, -1,  1]],
                              [[  0,  1,  1],[ -1,  1,  1]],
                              [[  1,  0,  1],[  1, -1,  1]],
                              [[  1,  1,  0],[  1,  1, -1]]]),
              'Bain': \
                    np.array([[[  0,  1,  0],[  0,  1,  1]],
                              [[  0,  0,  1],[  1,  0,  1]],
                              [[  1,  0,  0],[  1,  1,  0]]]),
              }
    myPlane   = [float(i) for i in planes[relationModel][variant,me]]                               # map(float, planes[...]) does not work in python 3
    myPlane  /= np.linalg.norm(myPlane)
    myNormal  = [float(i) for i in normals[relationModel][variant,me]]                              # map(float, planes[...]) does not work in python 3
    myNormal /= np.linalg.norm(myNormal)
    myMatrix  = np.array([myNormal,np.cross(myPlane,myNormal),myPlane]).T

    otherPlane   = [float(i) for i in planes[relationModel][variant,other]]                         # map(float, planes[...]) does not work in python 3
    otherPlane  /= np.linalg.norm(otherPlane)
    otherNormal  = [float(i) for i in normals[relationModel][variant,other]]                        # map(float, planes[...]) does not work in python 3
    otherNormal /= np.linalg.norm(otherNormal)
    otherMatrix  = np.array([otherNormal,np.cross(otherPlane,otherNormal),otherPlane]).T

    rot=np.dot(otherMatrix,myMatrix.T)

    return Orientation(matrix=np.dot(rot,self.asMatrix()),symmetry=targetSymmetry)