diff --git a/processing/post/addOrientations.py b/processing/post/addOrientations.py
index e1752b0a4..706a959ad 100755
--- a/processing/post/addOrientations.py
+++ b/processing/post/addOrientations.py
@@ -2,6 +2,7 @@
 
 import os
 import sys
+from io import StringIO
 from optparse import OptionParser
 
 import numpy as np
@@ -24,14 +25,8 @@ Additional (globally fixed) rotations of the lab frame and/or crystal frame can
 
 """, version = scriptID)
 
-representations = {
-                  'quaternion': ['qu',4],
-                  'rodrigues':  ['ro',4],
-                  'Rodrigues':  ['Ro',3],
-                  'eulers':     ['eu',3],
-                  'matrix':     ['om',9],
-                  'angleaxis':  ['ax',4],
-                  }
+representations = ['quaternion', 'rodrigues', 'eulers', 'matrix', 'axisangle']
+
 
 parser.add_option('-o',
                   '--output',
@@ -93,10 +88,10 @@ parser.set_defaults(output = [],
                    )
 
 (options, filenames) = parser.parse_args()
+if filenames == []: filenames = [None]
 
-#options.output = list(map(lambda x: x.lower(), options.output))
 if options.output == [] or (not set(options.output).issubset(set(representations))):
-  parser.error('output must be chosen from {}.'.format(', '.join(representations)))
+    parser.error('output must be chosen from {}.'.format(', '.join(representations)))
 
 input = [options.eulers     is not None,
          options.rodrigues  is not None,
@@ -109,95 +104,47 @@ input = [options.eulers     is not None,
 
 if np.sum(input) != 1: parser.error('needs exactly one input format.')
 
-(label,dim,inputtype) = [(options.eulers,representations['eulers'][1],'eulers'),
-                         (options.rodrigues,representations['rodrigues'][1],'rodrigues'),
-                         ([options.x,options.y,options.z],[3,3,3],'frame'),
-                         (options.matrix,representations['matrix'][1],'matrix'),
-                         (options.quaternion,representations['quaternion'][1],'quaternion'),
-                        ][np.where(input)[0][0]]                                                    # select input label that was requested
-
-r = damask.Rotation.fromAxisAngle(np.array(options.crystalrotation),options.degrees,normalise=True)
-R = damask.Rotation.fromAxisAngle(np.array(options.labrotation),options.degrees,normalise=True)
-
-
-# --- loop over input files ------------------------------------------------------------------------
-
-if filenames == []: filenames = [None]
+r = damask.Rotation.from_axis_angle(np.array(options.crystalrotation),options.degrees,normalise=True)
+R = damask.Rotation.from_axis_angle(np.array(options.labrotation),options.degrees,normalise=True)
 
 for name in filenames:
-  try: 
-    table = damask.ASCIItable(name = name)
-  except IOError:
-    continue
-  damask.util.report(scriptName,name)
+    damask.util.report(scriptName,name)
 
-# ------------------------------------------ read header ------------------------------------------
+    table = damask.Table.from_ASCII(StringIO(''.join(sys.stdin.read())) if name is None else name)
 
-  table.head_read()
+    if   options.eulers is not None:
+        label = options.eulers
+        print(np.max(table.get(options.eulers),axis=0))
+        o = damask.Rotation.from_Eulers(table.get(options.eulers), options.degrees)
+    elif options.rodrigues is not None:
+        label = options.rodrigues
+        o = damask.Rotation.from_Rodrigues(table.get(options.rodrigues))
+    elif options.matrix is not None:
+        label = options.matrix
+        o = damask.Rotation.from_matrix(table.get(options.matrix).reshape(-1,3,3))
+    elif options.x is not None:
+        label = '<{},{},{}>'.format(options.x,options.y,options.z)
+        M = np.block([table.get(options.x),table.get(options.y),table.get(options.z)]).reshape(-1,3,3)
+        o = damask.Rotation.from_matrix(M/np.linalg.norm(M,axis=0))
+    elif options.quaternion is not None:
+        label = options.quaternion
+        o = damask.Rotation.from_quaternion(table.get(options.quaternion))
 
-# ------------------------------------------ sanity checks -----------------------------------------
+    o = r.broadcast_to(o.shape) @ o @ R.broadcast_to(o.shape)
 
-  errors  = []
-  remarks = []
-  
-  if not np.all(table.label_dimension(label) == dim):  errors.append('input {} does not have dimension {}.'.format(label,dim))
-  else:  column = table.label_index(label)
+    #if options.lattice is not None:
+    #  o = damask.Orientation(rotation = o,lattice = options.lattice).reduced().rotation
 
-  if remarks != []: damask.util.croak(remarks)
-  if errors  != []:
-    damask.util.croak(errors)
-    table.close(dismiss = True)
-    continue
 
-# ------------------------------------------ assemble header ---------------------------------------
+    if 'rodrigues' in options.output:
+        table.add('ro({})'.format(label),o.as_Rodrigues(),                scriptID+' '+' '.join(sys.argv[1:]))
+    if 'eulers' in options.output:
+        table.add('eu({})'.format(label),o.as_Eulers(options.degrees),    scriptID+' '+' '.join(sys.argv[1:]))
+    if 'quaternion' in options.output:
+        table.add('qu({})'.format(label),o.as_quaternion(),               scriptID+' '+' '.join(sys.argv[1:]))
+    if 'matrix' in options.output:
+        table.add('om({})'.format(label),o.as_matrix(),                   scriptID+' '+' '.join(sys.argv[1:]))
+    if 'axisangle' in options.output:
+        table.add('om({})'.format(label),o.as_axisangle(options.degrees), scriptID+' '+' '.join(sys.argv[1:]))
 
-  table.info_append(scriptID + '\t' + ' '.join(sys.argv[1:]))
-  for output in options.output:
-    if output in representations:
-      table.labels_append(['{}_{}({})'.format(i+1,representations[output][0],label) \
-                           for i in range(representations[output][1])])
-  table.head_write()
-
-# ------------------------------------------ process data ------------------------------------------
-
-  outputAlive = True
-  while outputAlive and table.data_read():                                                          # read next data line of ASCII table
-    if   inputtype == 'eulers':
-      d = representations['eulers'][1]
-      o = damask.Rotation.fromEulers(list(map(float,table.data[column:column+d])),options.degrees)
-      
-    elif inputtype == 'rodrigues':
-      d = representations['rodrigues'][1]
-      o = damask.Rotation.fromRodrigues(list(map(float,table.data[column:column+d])))
-      
-    elif inputtype == 'matrix':
-      d = representations['matrix'][1]
-      o = damask.Rotation.fromMatrix(np.array(list(map(float,table.data[column:column+d]))).reshape(3,3))
-
-    elif inputtype == 'frame':
-      M = np.array(list(map(float,table.data[column[0]:column[0]+3] + \
-                                  table.data[column[1]:column[1]+3] + \
-                                  table.data[column[2]:column[2]+3]))).reshape(3,3).T
-      o = damask.Rotation.fromMatrix(M/np.linalg.norm(M,axis=0))
-      
-    elif inputtype == 'quaternion':
-      d = representations['quaternion'][1]
-      o = damask.Rotation.fromQuaternion(list(map(float,table.data[column:column+d])))
-
-    o = r*o*R                                                                                       # apply additional lab and crystal frame rotations
-
-    if options.lattice is not None:
-      o = damask.Orientation(rotation = o,lattice = options.lattice).reduced().rotation
-
-    for output in options.output:
-      if   output == 'quaternion': table.data_append(o.asQuaternion())
-      elif output == 'rodrigues':  table.data_append(o.asRodrigues())
-      elif output == 'Rodrigues':  table.data_append(o.asRodrigues(vector=True))
-      elif output == 'eulers':     table.data_append(o.asEulers(degrees=options.degrees))
-      elif output == 'matrix':     table.data_append(o.asMatrix())
-      elif output == 'angleaxis':  table.data_append(o.asAxisAngle(degrees=options.degrees))
-    outputAlive = table.data_write()                                                                # output processed line
-
-# ------------------------------------------ output finalization -----------------------------------  
-
-  table.close()                                                                                     # close ASCII tables
+    table.to_ASCII(sys.stdout if name is None else name)
diff --git a/processing/post/rotateData.py b/processing/post/rotateData.py
index d270eb79e..ee172c64d 100755
--- a/processing/post/rotateData.py
+++ b/processing/post/rotateData.py
@@ -37,27 +37,26 @@ parser.add_option('--degrees',
 parser.set_defaults(rotation = (1.,1.,1.,0),                                                        # no rotation about (1,1,1)
                     degrees = False,
                    )
-                    
+
 (options,filenames) = parser.parse_args()
 if filenames == []: filenames = [None]
 
 if options.data is None:
-  parser.error('no data column specified.')
+    parser.error('no data column specified.')
 
-r = damask.Rotation.fromAxisAngle(options.rotation,options.degrees,normalise=True)
+r = damask.Rotation.from_axis_angle(options.rotation,options.degrees,normalise=True)
 
 for name in filenames:
     damask.util.report(scriptName,name)
-    
+
     table = damask.Table.from_ASCII(StringIO(''.join(sys.stdin.read())) if name is None else name)
 
     for data in options.data:
         d = table.get(data)
         if table.shapes[data] == (9,): d=d.reshape(-1,3,3)
-        for i,l in enumerate(d):
-            d[i] = r*l
+        d = r.broadcast_to(d.shape[0:1]) @ d
         if table.shapes[data] == (9,): d=d.reshape(-1,9)
-        
+
         table.set(data,d,scriptID+' '+' '.join(sys.argv[1:]))
 
     table.to_ASCII(sys.stdout if name is None else name)
diff --git a/processing/pre/geom_addPrimitive.py b/processing/pre/geom_addPrimitive.py
index f33ba27b1..c04fcf565 100755
--- a/processing/pre/geom_addPrimitive.py
+++ b/processing/pre/geom_addPrimitive.py
@@ -84,9 +84,9 @@ if [options.angleaxis,options.quaternion].count(None) == 0:
   parser.error('more than one rotation specified.')
 
 if options.angleaxis is not None:
-  rotation = damask.Rotation.fromAxisAngle(np.array(options.angleaxis),options.degrees,normalise=True)
+  rotation = damask.Rotation.from_axis_angle(np.array(options.angleaxis),options.degrees,normalise=True)
 elif options.quaternion is not None:
-  rotation = damask.Rotation.fromQuaternion(options.quaternion)
+  rotation = damask.Rotation.from_quaternion(options.quaternion)
 else:
   rotation = damask.Rotation()
 
diff --git a/processing/pre/geom_fromDREAM3D.py b/processing/pre/geom_fromDREAM3D.py
index 7d5b1442d..c48698e53 100755
--- a/processing/pre/geom_fromDREAM3D.py
+++ b/processing/pre/geom_fromDREAM3D.py
@@ -97,7 +97,7 @@ for name in filenames:
     dataset = os.path.join(group_pointwise,options.quaternion)
     try:
       quats = np.reshape(inFile[dataset][...],(np.product(grid),4))
-      rot   = [damask.Rotation.fromQuaternion(q,True,P=+1) for q in quats]
+      rot   = [damask.Rotation.from_quaternion(q,True,P=+1) for q in quats]
     except KeyError:
       errors.append('Pointwise orientation (quaternion) data ({}) not readable'.format(dataset))
 
@@ -123,7 +123,7 @@ for name in filenames:
 
     dataset = os.path.join(group_average,options.quaternion)
     try:
-      rot = [damask.Rotation.fromQuaternion(q,True,P=+1) for q in inFile[dataset][...][1:]]         # skip first entry (unindexed)
+      rot = [damask.Rotation.from_quaternion(q,True,P=+1) for q in inFile[dataset][...][1:]]        # skip first entry (unindexed)
     except KeyError:
       errors.append('Average orientation data ({}) not readable'.format(dataset))
 
@@ -140,7 +140,7 @@ for name in filenames:
   config_header = ['<texture>']
   for i in range(np.nanmax(microstructure)):
     config_header += ['[{}{}]'.format(label,i+1),
-                      '(gauss)\tphi1 {:.2f}\tPhi {:.2f}\tphi2 {:.2f}'.format(*rot[i].asEulers(degrees = True)),
+                      '(gauss)\tphi1 {:.2f}\tPhi {:.2f}\tphi2 {:.2f}'.format(*rot[i].as_Eulers(degrees = True)),
                      ]
   config_header += ['<microstructure>']
   for i in range(np.nanmax(microstructure)):
diff --git a/processing/pre/geom_fromTable.py b/processing/pre/geom_fromTable.py
index f6c9ba38e..be5b3a23c 100755
--- a/processing/pre/geom_fromTable.py
+++ b/processing/pre/geom_fromTable.py
@@ -89,7 +89,7 @@ for name in filenames:
         for i,data in enumerate(unique):
             ori = damask.Rotation(data[0:4])
             config_header += ['[Grain{}]'.format(i+1),
-                              '(gauss)\tphi1 {:.2f}\tPhi {:.2f}\tphi2 {:.2f}'.format(*ori.asEulers(degrees = True)),
+                              '(gauss)\tphi1 {:.2f}\tPhi {:.2f}\tphi2 {:.2f}'.format(*ori.as_Eulers(degrees = True)),
                              ]
             if options.axes is not None: config_header += ['axes\t{} {} {}'.format(*options.axes)]
 
diff --git a/processing/pre/geom_rotate.py b/processing/pre/geom_rotate.py
index 977e00b65..dc2e8e2a3 100755
--- a/processing/pre/geom_rotate.py
+++ b/processing/pre/geom_rotate.py
@@ -59,15 +59,15 @@ if [options.rotation,options.eulers,options.matrix,options.quaternion].count(Non
   parser.error('no rotation specified.')
 
 if options.quaternion is not None:
-  rot = damask.Rotation.fromQuaternion(np.array(options.quaternion))                            # we might need P=+1 here, too...
+  rot = damask.Rotation.from_quaternion(np.array(options.quaternion))                               # we might need P=+1 here, too...
 if options.rotation is not None:
-  rot = damask.Rotation.fromAxisAngle(np.array(options.rotation),degrees=options.degrees,normalise=True,P=+1)
+  rot = damask.Rotation.from_axis_angle(np.array(options.rotation),degrees=options.degrees,normalise=True,P=+1)
 if options.matrix is not None:
-  rot = damask.Rotation.fromMatrix(np.array(options.Matrix))
+  rot = damask.Rotation.from_matrix(np.array(options.Matrix))
 if options.eulers is not None:
-  rot = damask.Rotation.fromEulers(np.array(options.eulers),degrees=options.degrees)
+  rot = damask.Rotation.from_Eulers(np.array(options.eulers),degrees=options.degrees)
 
-eulers = rot.asEulers(degrees=True)
+eulers = rot.as_Eulers(degrees=True)
 
 
 if filenames == []: filenames = [None]
diff --git a/python/damask/_lattice.py b/python/damask/_lattice.py
index f33cc66f7..0f6cffd04 100644
--- a/python/damask/_lattice.py
+++ b/python/damask/_lattice.py
@@ -635,6 +635,6 @@ class Lattice:
             otherDir    = miller[otherDir_id]/   np.linalg.norm(miller[otherDir_id])
             otherMatrix = np.array([otherDir,np.cross(otherPlane,otherDir),otherPlane])
 
-            r['rotations'].append(Rotation.fromMatrix(np.dot(otherMatrix.T,myMatrix)))
+            r['rotations'].append(Rotation.from_matrix(np.dot(otherMatrix.T,myMatrix)))
 
         return r
diff --git a/python/damask/_orientation.py b/python/damask/_orientation.py
index a558ec85d..e4dbc7d7c 100644
--- a/python/damask/_orientation.py
+++ b/python/damask/_orientation.py
@@ -37,7 +37,7 @@ class Orientation:
         if isinstance(rotation, Rotation):
             self.rotation = rotation
         else:
-            self.rotation = Rotation.fromQuaternion(rotation)                                       # assume quaternion
+            self.rotation = Rotation.from_quaternion(rotation)                                      # assume quaternion
 
         if self.rotation.quaternion.shape != (4,):
             raise NotImplementedError('Support for multiple rotations missing')
@@ -68,8 +68,8 @@ class Orientation:
                 r = b*aInv
                 for k in range(2):
                     r.inverse()
-                    breaker = self.lattice.symmetry.inFZ(r.asRodrigues(vector=True)) \
-                              and (not SST or other.lattice.symmetry.inDisorientationSST(r.asRodrigues(vector=True)))
+                    breaker = self.lattice.symmetry.inFZ(r.as_Rodrigues(vector=True)) \
+                              and (not SST or other.lattice.symmetry.inDisorientationSST(r.as_Rodrigues(vector=True)))
                     if breaker: break
                 if breaker: break
             if breaker: break
@@ -78,7 +78,7 @@ class Orientation:
                                                                                                     # ... own sym, other sym,
                                                                                                     # self-->other: True, self<--other: False
     def inFZ(self):
-        return self.lattice.symmetry.inFZ(self.rotation.asRodrigues(vector=True))
+        return self.lattice.symmetry.inFZ(self.rotation.as_Rodrigues(vector=True))
 
 
     def equivalentOrientations(self,members=[]):
@@ -100,7 +100,7 @@ class Orientation:
     def reduced(self):
         """Transform orientation to fall into fundamental zone according to symmetry."""
         for me in self.equivalentOrientations():
-            if self.lattice.symmetry.inFZ(me.rotation.asRodrigues(vector=True)): break
+            if self.lattice.symmetry.inFZ(me.rotation.as_Rodrigues(vector=True)): break
 
         return self.__class__(me.rotation,self.lattice)
 
diff --git a/python/damask/_rotation.py b/python/damask/_rotation.py
index 162a118b9..7ab235adc 100644
--- a/python/damask/_rotation.py
+++ b/python/damask/_rotation.py
@@ -9,10 +9,6 @@ _sc   = np.pi**(1./6.)/6.**(1./6.)
 _beta = np.pi**(5./6.)/6.**(1./6.)/2.
 _R1   = (3.*np.pi/4.)**(1./3.)
 
-def iszero(a):
-    return np.isclose(a,0.0,atol=1.0e-12,rtol=0.0)
-
-
 class Rotation:
     u"""
     Orientation stored with functionality for conversion to different representations.
@@ -34,7 +30,7 @@ class Rotation:
     Rotate vector "a" (defined in coordinate system "A") to
     coordinates "b" expressed in system "B":
 
-    - b = Q * a
+    - b = Q @ a
     - b = np.dot(Q.asMatrix(),a)
 
     References
@@ -53,11 +49,17 @@ class Rotation:
         Parameters
         ----------
         quaternion : numpy.ndarray, optional
-            Unit quaternion that follows the conventions. Use .fromQuaternion to perform a sanity check.
+            Unit quaternion that follows the conventions. Use .from_quaternion to perform a sanity check.
 
         """
         self.quaternion = quaternion.copy()
 
+
+    @property
+    def shape(self):
+        return self.quaternion.shape[:-1]
+
+
     def __copy__(self):
         """Copy."""
         return self.__class__(self.quaternion)
@@ -71,14 +73,14 @@ class Rotation:
             raise NotImplementedError('Support for multiple rotations missing')
         return '\n'.join([
                'Quaternion: (real={:.3f}, imag=<{:+.3f}, {:+.3f}, {:+.3f}>)'.format(*(self.quaternion)),
-               'Matrix:\n{}'.format(self.asMatrix()),
-               'Bunge Eulers / deg: ({:3.2f}, {:3.2f}, {:3.2f})'.format(*self.asEulers(degrees=True)),
+               'Matrix:\n{}'.format(self.as_matrix()),
+               'Bunge Eulers / deg: ({:3.2f}, {:3.2f}, {:3.2f})'.format(*self.as_Eulers(degrees=True)),
                 ])
 
 
-    def __mul__(self, other):
+    def __matmul__(self, other):
         """
-        Multiplication.
+        Rotation of vector, second or fourth order tensor, or rotation object.
 
         Parameters
         ----------
@@ -87,42 +89,40 @@ class Rotation:
 
         Todo
         ----
-        Document details active/passive)
-        considere rotation of (3,3,3,3)-matrix
+        Check rotation of 4th order tensor
 
         """
-        if self.quaternion.shape != (4,):
-            raise NotImplementedError('Support for multiple rotations missing')
-        if isinstance(other, Rotation):                                                             # rotate a rotation
-            self_q  = self.quaternion[0]
-            self_p  = self.quaternion[1:]
-            other_q = other.quaternion[0]
-            other_p = other.quaternion[1:]
-            R = self.__class__(np.append(self_q*other_q - np.dot(self_p,other_p),
-                                         self_q*other_p + other_q*self_p + _P * np.cross(self_p,other_p)))
-            return R.standardize()
-        elif isinstance(other, (tuple,np.ndarray)):
-            if isinstance(other,tuple) or other.shape == (3,):                                      # rotate a single (3)-vector or meshgrid
-                A = self.quaternion[0]**2.0 - np.dot(self.quaternion[1:],self.quaternion[1:])
-                B = 2.0 * (  self.quaternion[1]*other[0]
-                           + self.quaternion[2]*other[1]
-                           + self.quaternion[3]*other[2])
-                C = 2.0 * _P*self.quaternion[0]
+        if isinstance(other, Rotation):
+            q_m = self.quaternion[...,0:1]
+            p_m = self.quaternion[...,1:]
+            q_o = other.quaternion[...,0:1]
+            p_o = other.quaternion[...,1:]
+            q = (q_m*q_o - np.einsum('...i,...i',p_m,p_o).reshape(self.shape+(1,)))
+            p = q_m*p_o + q_o*p_m + _P * np.cross(p_m,p_o)
+            return self.__class__(np.block([q,p])).standardize()
 
-                return np.array([
-                  A*other[0] + B*self.quaternion[1] + C*(self.quaternion[2]*other[2] - self.quaternion[3]*other[1]),
-                  A*other[1] + B*self.quaternion[2] + C*(self.quaternion[3]*other[0] - self.quaternion[1]*other[2]),
-                  A*other[2] + B*self.quaternion[3] + C*(self.quaternion[1]*other[1] - self.quaternion[2]*other[0]),
-                  ])
-            elif other.shape == (3,3,):                                                             # rotate a single (3x3)-matrix
-                return np.dot(self.asMatrix(),np.dot(other,self.asMatrix().T))
-            elif other.shape == (3,3,3,3,):
-                raise NotImplementedError('Support for rotation of 4th order tensors missing')
+        elif isinstance(other,np.ndarray):
+            if self.shape + (3,) == other.shape:
+                q_m = self.quaternion[...,0]
+                p_m = self.quaternion[...,1:]
+                A = q_m**2.0 - np.einsum('...i,...i',p_m,p_m)
+                B = 2.0 * np.einsum('...i,...i',p_m,other)
+                C = 2.0 * _P * q_m
+                return np.block([(A * other[...,i]).reshape(self.shape+(1,)) +
+                                 (B * p_m[...,i]).reshape(self.shape+(1,)) +
+                                 (C * (  p_m[...,(i+1)%3]*other[...,(i+2)%3]\
+                                       - p_m[...,(i+2)%3]*other[...,(i+1)%3])).reshape(self.shape+(1,))
+                                 for i in [0,1,2]])
+            if self.shape + (3,3) == other.shape:
+                R = self.as_matrix()
+                return np.einsum('...im,...jn,...mn',R,R,other)
+            if self.shape + (3,3,3,3) == other.shape:
+                R = self.as_matrix()
+                return np.einsum('...im,...jn,...ko,...lp,...mnop',R,R,R,R,other)
             else:
-                return NotImplemented
+                raise ValueError('Can only rotate vectors, 2nd order tensors, and 4th order tensors')
         else:
-            return NotImplemented
-
+            raise TypeError('Cannot rotate {}'.format(type(other)))
 
     def inverse(self):
         """In-place inverse rotation/backward rotation."""
@@ -157,6 +157,17 @@ class Rotation:
         return other*self.inversed()
 
 
+    def broadcast_to(self,shape):
+        if self.shape == ():
+            q = np.broadcast_to(self.quaternion,shape+(4,))
+        else:
+            q = np.block([np.broadcast_to(self.quaternion[...,0:1],shape+(1,)),
+                          np.broadcast_to(self.quaternion[...,1:2],shape+(1,)),
+                          np.broadcast_to(self.quaternion[...,2:3],shape+(1,)),
+                          np.broadcast_to(self.quaternion[...,3:4],shape+(1,))])
+        return self.__class__(q)
+
+
     def average(self,other):
         """
         Calculate the average rotation.
@@ -198,7 +209,7 @@ class Rotation:
             return angles in degrees.
 
         """
-        eu = Rotation.qu2eu(self.quaternion)
+        eu = Rotation._qu2eu(self.quaternion)
         if degrees: eu = np.degrees(eu)
         return eu
 
@@ -216,13 +227,13 @@ class Rotation:
             return tuple of axis and angle.
 
         """
-        ax = Rotation.qu2ax(self.quaternion)
+        ax = Rotation._qu2ax(self.quaternion)
         if degrees: ax[...,3] = np.degrees(ax[...,3])
         return (ax[...,:3],ax[...,3]) if pair else ax
 
     def as_matrix(self):
         """Rotation matrix."""
-        return Rotation.qu2om(self.quaternion)
+        return Rotation._qu2om(self.quaternion)
 
     def as_Rodrigues(self,
                      vector = False):
@@ -235,16 +246,16 @@ class Rotation:
             return as actual Rodrigues--Frank vector, i.e. rotation axis scaled by tan(ω/2).
 
         """
-        ro = Rotation.qu2ro(self.quaternion)
+        ro = Rotation._qu2ro(self.quaternion)
         return ro[...,:3]*ro[...,3] if vector else ro
 
     def as_homochoric(self):
         """Homochoric vector: (h_1, h_2, h_3)."""
-        return Rotation.qu2ho(self.quaternion)
+        return Rotation._qu2ho(self.quaternion)
 
     def as_cubochoric(self):
         """Cubochoric vector: (c_1, c_2, c_3)."""
-        return Rotation.qu2cu(self.quaternion)
+        return Rotation._qu2cu(self.quaternion)
 
     def M(self): # ToDo not sure about the name: as_M or M? we do not have a from_M
         """
@@ -258,30 +269,24 @@ class Rotation:
         """
         return np.einsum('...i,...j',self.quaternion,self.quaternion)
 
-    # for compatibility (old names do not follow convention)
-    asM          = M
-    asQuaternion = as_quaternion
-    asEulers     = as_Eulers
-    asAxisAngle  = as_axis_angle
-    asMatrix     = as_matrix
-    asRodrigues  = as_Rodrigues
-    asHomochoric = as_homochoric
-    asCubochoric = as_cubochoric
 
     ################################################################################################
     # Static constructors. The input data needs to follow the conventions, options allow to
     # relax the conventions.
     @staticmethod
     def from_quaternion(quaternion,
-                        acceptHomomorph = False,
-                        P = -1):
+                        accept_homomorph = False,
+                        P = -1,
+                        acceptHomomorph = None):
 
+        if acceptHomomorph is not None:
+            accept_homomorph = acceptHomomorph
         qu = np.array(quaternion,dtype=float)
         if qu.shape[:-2:-1] != (4,):
             raise ValueError('Invalid shape.')
 
         if P > 0: qu[...,1:4] *= -1                                                                 # convert from P=1 to P=-1
-        if acceptHomomorph:
+        if accept_homomorph:
             qu[qu[...,0] < 0.0] *= -1
         else:
             if np.any(qu[...,0] < 0.0):
@@ -303,7 +308,7 @@ class Rotation:
         if np.any(eu < 0.0) or np.any(eu > 2.0*np.pi) or np.any(eu[...,1] > np.pi):                 # ToDo: No separate check for PHI
             raise ValueError('Euler angles outside of [0..2π],[0..π],[0..2π].')
 
-        return Rotation(Rotation.eu2qu(eu))
+        return Rotation(Rotation._eu2qu(eu))
 
     @staticmethod
     def from_axis_angle(axis_angle,
@@ -323,7 +328,7 @@ class Rotation:
         if not np.all(np.isclose(np.linalg.norm(ax[...,0:3],axis=-1), 1.0)):
             raise ValueError('Axis angle rotation axis is not of unit length.')
 
-        return Rotation(Rotation.ax2qu(ax))
+        return Rotation(Rotation._ax2qu(ax))
 
     @staticmethod
     def from_basis(basis,
@@ -347,7 +352,7 @@ class Rotation:
            or not np.all(np.isclose(np.einsum('...i,...i',om[...,2],om[...,0]), 0.0)):
             raise ValueError('Orientation matrix is not orthogonal.')
 
-        return Rotation(Rotation.om2qu(om))
+        return Rotation(Rotation._om2qu(om))
 
     @staticmethod
     def from_matrix(om):
@@ -370,7 +375,7 @@ class Rotation:
         if not np.all(np.isclose(np.linalg.norm(ro[...,0:3],axis=-1), 1.0)):
             raise ValueError('Rodrigues vector rotation axis is not of unit length.')
 
-        return Rotation(Rotation.ro2qu(ro))
+        return Rotation(Rotation._ro2qu(ro))
 
     @staticmethod
     def from_homochoric(homochoric,
@@ -382,10 +387,10 @@ class Rotation:
 
         if P > 0: ho *= -1                                                                          # convert from P=1 to P=-1
 
-        if np.any(np.linalg.norm(ho,axis=-1) > (3.*np.pi/4.)**(1./3.)+1e-9):
+        if np.any(np.linalg.norm(ho,axis=-1) >_R1+1e-9):
             raise ValueError('Homochoric coordinate outside of the sphere.')
 
-        return Rotation(Rotation.ho2qu(ho))
+        return Rotation(Rotation._ho2qu(ho))
 
     @staticmethod
     def from_cubochoric(cubochoric,
@@ -398,10 +403,10 @@ class Rotation:
         if np.abs(np.max(cu))>np.pi**(2./3.) * 0.5+1e-9:
             raise ValueError('Cubochoric coordinate outside of the cube: {} {} {}.'.format(*cu))
 
-        ho = Rotation.cu2ho(cu)
+        ho = Rotation._cu2ho(cu)
         if P > 0: ho *= -1                                                                          # convert from P=1 to P=-1
 
-        return Rotation(Rotation.ho2qu(ho))
+        return Rotation(Rotation._ho2qu(ho))
 
 
     @staticmethod
@@ -430,11 +435,11 @@ class Rotation:
             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
+            M =          r.M() * n if i == 0 \
+                else M + r.M() * 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)
+        return Rotation.from_quaternion(np.real(vec.T[eig.argmax()]),accept_homomorph = True)
 
     @staticmethod
     def from_random(shape=None):
@@ -456,15 +461,11 @@ class Rotation:
 
 
     # for compatibility (old names do not follow convention)
+    asM            = M
     fromQuaternion = from_quaternion
     fromEulers     = from_Eulers
-    fromAxisAngle  = from_axis_angle
-    fromBasis      = from_basis
-    fromMatrix     = from_matrix
-    fromRodrigues  = from_Rodrigues
-    fromHomochoric = from_homochoric
-    fromCubochoric = from_cubochoric
-    fromRandom     = from_random
+    asAxisAngle    = as_axis_angle
+    __mul__        = __matmul__
 
 ####################################################################################################
 # Code below available according to the following conditions on https://github.com/MarDiehl/3Drotations
@@ -498,519 +499,379 @@ class Rotation:
 ####################################################################################################
     #---------- Quaternion ----------
     @staticmethod
-    def qu2om(qu):
-        if len(qu.shape) == 1:
-            """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[0,1] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3])
-            om[1,0] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3])
-            om[1,2] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1])
-            om[2,1] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1])
-            om[2,0] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2])
-            om[0,2] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2])
-        else:
-            qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2)
-            om = np.block([qq + 2.0*qu[...,1:2]**2,
-                           2.0*(qu[...,2:3]*qu[...,1:2]+qu[...,0:1]*qu[...,3:4]),
-                           2.0*(qu[...,3:4]*qu[...,1:2]-qu[...,0:1]*qu[...,2:3]),
-                           2.0*(qu[...,1:2]*qu[...,2:3]-qu[...,0:1]*qu[...,3:4]),
-                           qq + 2.0*qu[...,2:3]**2,
-                           2.0*(qu[...,3:4]*qu[...,2:3]+qu[...,0:1]*qu[...,1:2]),
-                           2.0*(qu[...,1:2]*qu[...,3:4]+qu[...,0:1]*qu[...,2:3]),
-                           2.0*(qu[...,2:3]*qu[...,3:4]-qu[...,0:1]*qu[...,1:2]),
-                           qq + 2.0*qu[...,3:4]**2,
-                          ]).reshape(qu.shape[:-1]+(3,3))
-        return om if _P < 0.0 else np.swapaxes(om,(-1,-2))
+    def _qu2om(qu):
+        qq = qu[...,0:1]**2-(qu[...,1:2]**2 + qu[...,2:3]**2 + qu[...,3:4]**2)
+        om = np.block([qq + 2.0*qu[...,1:2]**2,
+                       2.0*(qu[...,2:3]*qu[...,1:2]-_P*qu[...,0:1]*qu[...,3:4]),
+                       2.0*(qu[...,3:4]*qu[...,1:2]+_P*qu[...,0:1]*qu[...,2:3]),
+                       2.0*(qu[...,1:2]*qu[...,2:3]+_P*qu[...,0:1]*qu[...,3:4]),
+                       qq + 2.0*qu[...,2:3]**2,
+                       2.0*(qu[...,3:4]*qu[...,2:3]-_P*qu[...,0:1]*qu[...,1:2]),
+                       2.0*(qu[...,1:2]*qu[...,3:4]-_P*qu[...,0:1]*qu[...,2:3]),
+                       2.0*(qu[...,2:3]*qu[...,3:4]+_P*qu[...,0:1]*qu[...,1:2]),
+                       qq + 2.0*qu[...,3:4]**2,
+                      ]).reshape(qu.shape[:-1]+(3,3))
+        return om
 
     @staticmethod
-    def qu2eu(qu):
+    def _qu2eu(qu):
         """Quaternion to Bunge-Euler angles."""
-        if len(qu.shape) == 1:
-            q03 = qu[0]**2+qu[3]**2
-            q12 = qu[1]**2+qu[2]**2
-            chi = np.sqrt(q03*q12)
-            if   np.abs(q12) < 1.e-8:
-                eu = np.array([np.arctan2(-_P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0,   0.0])
-            elif np.abs(q03) < 1.e-8:
-                eu = np.array([np.arctan2(   2.0*qu[1]*qu[2],qu[1]**2-qu[2]**2), np.pi, 0.0])
-            else:
-                eu = np.array([np.arctan2((-_P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]-qu[2]*qu[3])*chi ),
-                               np.arctan2( 2.0*chi, q03-q12 ),
-                               np.arctan2(( _P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]+qu[2]*qu[3])*chi )])
-        else:
-            q02   = qu[...,0:1]*qu[...,2:3]
-            q13   = qu[...,1:2]*qu[...,3:4]
-            q01   = qu[...,0:1]*qu[...,1:2]
-            q23   = qu[...,2:3]*qu[...,3:4]
-            q03_s = qu[...,0:1]**2+qu[...,3:4]**2
-            q12_s = qu[...,1:2]**2+qu[...,2:3]**2
-            chi = np.sqrt(q03_s*q12_s)
+        q02   = qu[...,0:1]*qu[...,2:3]
+        q13   = qu[...,1:2]*qu[...,3:4]
+        q01   = qu[...,0:1]*qu[...,1:2]
+        q23   = qu[...,2:3]*qu[...,3:4]
+        q03_s = qu[...,0:1]**2+qu[...,3:4]**2
+        q12_s = qu[...,1:2]**2+qu[...,2:3]**2
+        chi = np.sqrt(q03_s*q12_s)
 
-            eu = np.where(np.abs(q12_s) < 1.0e-8,
-                    np.block([np.arctan2(-_P*2.0*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2),
-                              np.zeros(qu.shape[:-1]+(2,))]),
-                          np.where(np.abs(q03_s) < 1.0e-8,
-                              np.block([np.arctan2(   2.0*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2),
-                                        np.broadcast_to(np.pi,qu.shape[:-1]+(1,)),
-                                        np.zeros(qu.shape[:-1]+(1,))]),
-                              np.block([np.arctan2((-_P*q02+q13)*chi, (-_P*q01-q23)*chi),
-                                        np.arctan2( 2.0*chi,          q03_s-q12_s    ),
-                                        np.arctan2(( _P*q02+q13)*chi, (-_P*q01+q23)*chi)])
-                                  )
-                         )
+        eu = np.where(np.abs(q12_s) < 1.0e-8,
+                np.block([np.arctan2(-_P*2.0*qu[...,0:1]*qu[...,3:4],qu[...,0:1]**2-qu[...,3:4]**2),
+                          np.zeros(qu.shape[:-1]+(2,))]),
+                      np.where(np.abs(q03_s) < 1.0e-8,
+                          np.block([np.arctan2(   2.0*qu[...,1:2]*qu[...,2:3],qu[...,1:2]**2-qu[...,2:3]**2),
+                                    np.broadcast_to(np.pi,qu.shape[:-1]+(1,)),
+                                    np.zeros(qu.shape[:-1]+(1,))]),
+                          np.block([np.arctan2((-_P*q02+q13)*chi, (-_P*q01-q23)*chi),
+                                    np.arctan2( 2.0*chi,          q03_s-q12_s    ),
+                                    np.arctan2(( _P*q02+q13)*chi, (-_P*q01+q23)*chi)])
+                              )
+                     )
         # reduce Euler angles to definition range
         eu[np.abs(eu)<1.e-6] = 0.0
         eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
         return eu
 
     @staticmethod
-    def qu2ax(qu):
+    def _qu2ax(qu):
         """
         Quaternion to axis angle pair.
 
         Modified version of the original formulation, should be numerically more stable
         """
-        if len(qu.shape) == 1:
-            if np.abs(np.sum(qu[1:4]**2)) < 1.e-6:                                                  # set axis to [001] if the angle is 0/360
-                ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
-            elif qu[0] > 1.e-6:
-                s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2)
-                omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
-                ax = ax = np.array([ qu[1]*s, qu[2]*s, qu[3]*s, omega ])
-            else:
-                ax = ax = np.array([ qu[1], qu[2], qu[3], np.pi])
-        else:
-            with np.errstate(invalid='ignore',divide='ignore'):
-                s = np.sign(qu[...,0:1])/np.sqrt(qu[...,1:2]**2+qu[...,2:3]**2+qu[...,3:4]**2)
-                omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
-                ax = np.where(np.broadcast_to(qu[...,0:1] < 1.0e-6,qu.shape),
-                              np.block([qu[...,1:4],np.broadcast_to(np.pi,qu.shape[:-1]+(1,))]),
-                              np.block([qu[...,1:4]*s,omega]))
-            ax[np.sum(np.abs(qu[...,1:4])**2,axis=-1) < 1.0e-6,] = [0.0, 0.0, 1.0, 0.0]
+        with np.errstate(invalid='ignore',divide='ignore'):
+            s = np.sign(qu[...,0:1])/np.sqrt(qu[...,1:2]**2+qu[...,2:3]**2+qu[...,3:4]**2)
+            omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
+            ax = np.where(np.broadcast_to(qu[...,0:1] < 1.0e-8,qu.shape),
+                          np.block([qu[...,1:4],np.broadcast_to(np.pi,qu.shape[:-1]+(1,))]),
+                          np.block([qu[...,1:4]*s,omega]))
+        ax[np.isclose(qu[...,0],1.,rtol=0.0)] = [0.0, 0.0, 1.0, 0.0]
         return ax
 
     @staticmethod
-    def qu2ro(qu):
+    def _qu2ro(qu):
         """Quaternion to Rodrigues-Frank vector."""
-        if len(qu.shape) == 1:
-            if iszero(qu[0]):
-                ro = np.array([qu[1], qu[2], qu[3], np.inf])
-            else:
-                s = np.linalg.norm(qu[1:4])
-                ro = np.array([0.0,0.0,_P,0.0] if iszero(s) else \
-                              [ qu[1]/s,  qu[2]/s,  qu[3]/s, np.tan(np.arccos(np.clip(qu[0],-1.0,1.0)))])
-        else:
-            with np.errstate(invalid='ignore',divide='ignore'):
-                s  = np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True)
-                ro = np.where(np.broadcast_to(np.abs(qu[...,0:1]) < 1.0e-12,qu.shape),
-                              np.block([qu[...,1:2], qu[...,2:3], qu[...,3:4], np.broadcast_to(np.inf,qu.shape[:-1]+(1,))]),
-                              np.block([qu[...,1:2]/s,qu[...,2:3]/s,qu[...,3:4]/s,
-                                        np.tan(np.arccos(np.clip(qu[...,0:1],-1.0,1.0)))
-                                       ])
-                           )
-            ro[np.abs(s).squeeze(-1) < 1.0e-12] = [0.0,0.0,_P,0.0]
+        with np.errstate(invalid='ignore',divide='ignore'):
+            s  = np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True)
+            ro = np.where(np.broadcast_to(np.abs(qu[...,0:1]) < 1.0e-12,qu.shape),
+                          np.block([qu[...,1:2], qu[...,2:3], qu[...,3:4], np.broadcast_to(np.inf,qu.shape[:-1]+(1,))]),
+                          np.block([qu[...,1:2]/s,qu[...,2:3]/s,qu[...,3:4]/s,
+                                    np.tan(np.arccos(np.clip(qu[...,0:1],-1.0,1.0)))
+                                   ])
+                       )
+        ro[np.abs(s).squeeze(-1) < 1.0e-12] = [0.0,0.0,_P,0.0]
         return ro
 
     @staticmethod
-    def qu2ho(qu):
+    def _qu2ho(qu):
         """Quaternion to homochoric vector."""
-        if len(qu.shape) == 1:
-            omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
-            if np.abs(omega) < 1.0e-12:
-                ho = np.zeros(3)
-            else:
-                ho = np.array([qu[1], qu[2], qu[3]])
-                f  = 0.75 * ( omega - np.sin(omega) )
-                ho = ho/np.linalg.norm(ho) * f**(1./3.)
-        else:
-            with np.errstate(invalid='ignore'):
-                omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
-                ho = np.where(np.abs(omega) < 1.0e-12,
-                              np.zeros(3),
-                              qu[...,1:4]/np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) \
-                              * (0.75*(omega - np.sin(omega)))**(1./3.))
+        with np.errstate(invalid='ignore'):
+            omega = 2.0 * np.arccos(np.clip(qu[...,0:1],-1.0,1.0))
+            ho = np.where(np.abs(omega) < 1.0e-12,
+                          np.zeros(3),
+                          qu[...,1:4]/np.linalg.norm(qu[...,1:4],axis=-1,keepdims=True) \
+                          * (0.75*(omega - np.sin(omega)))**(1./3.))
         return ho
 
     @staticmethod
-    def qu2cu(qu):
+    def _qu2cu(qu):
         """Quaternion to cubochoric vector."""
-        return Rotation.ho2cu(Rotation.qu2ho(qu))
+        return Rotation._ho2cu(Rotation._qu2ho(qu))
 
 
     #---------- Rotation matrix ----------
     @staticmethod
-    def om2qu(om):
+    def _om2qu(om):
         """
         Rotation matrix to quaternion.
 
-        The original formulation (direct conversion) had (numerical?) issues
+        This formulation is from  www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion.
+        The original formulation had issues.
         """
-        return Rotation.eu2qu(Rotation.om2eu(om))
+        trace = om[...,0,0:1]+om[...,1,1:2]+om[...,2,2:3]
+
+        with np.errstate(invalid='ignore',divide='ignore'):
+            s = [
+                 0.5 / np.sqrt( 1.0 + trace),
+                 2.0 * np.sqrt( 1.0 + om[...,0,0:1] - om[...,1,1:2] - om[...,2,2:3]),
+                 2.0 * np.sqrt( 1.0 + om[...,1,1:2] - om[...,2,2:3] - om[...,0,0:1]),
+                 2.0 * np.sqrt( 1.0 + om[...,2,2:3] - om[...,0,0:1] - om[...,1,1:2] )
+                 ]
+            qu= np.where(trace>0,
+                         np.block([0.25 / s[0],
+                                  (om[...,2,1:2] - om[...,1,2:3] ) * s[0],
+                                  (om[...,0,2:3] - om[...,2,0:1] ) * s[0],
+                                  (om[...,1,0:1] - om[...,0,1:2] ) * s[0]]),
+                         np.where(om[...,0,0:1] > np.maximum(om[...,1,1:2],om[...,2,2:3]),
+                                  np.block([(om[...,2,1:2] - om[...,1,2:3]) / s[1],
+                                            0.25 * s[1],
+                                            (om[...,0,1:2] + om[...,1,0:1]) / s[1],
+                                            (om[...,0,2:3] + om[...,2,0:1]) / s[1]]),
+                                  np.where(om[...,1,1:2] > om[...,2,2:3],
+                                           np.block([(om[...,0,2:3] - om[...,2,0:1]) / s[2],
+                                                     (om[...,0,1:2] + om[...,1,0:1]) / s[2],
+                                                     0.25 * s[2],
+                                                     (om[...,1,2:3] + om[...,2,1:2]) / s[2]]),
+                                           np.block([(om[...,1,0:1] - om[...,0,1:2]) / s[3],
+                                                     (om[...,0,2:3] + om[...,2,0:1]) / s[3],
+                                                     (om[...,1,2:3] + om[...,2,1:2]) / s[3],
+                                                     0.25 * s[3]]),
+                                          )
+                                 )
+                        )*np.array([1,_P,_P,_P])
+            qu[qu[...,0]<0] *=-1
+        return qu
 
     @staticmethod
-    def om2eu(om):
+    def _om2eu(om):
         """Rotation matrix to Bunge-Euler angles."""
-        if len(om.shape) == 2:
-            if not np.isclose(np.abs(om[2,2]),1.0,1.e-4):
-                zeta = 1.0/np.sqrt(1.0-om[2,2]**2)
-                eu = np.array([np.arctan2(om[2,0]*zeta,-om[2,1]*zeta),
-                               np.arccos(om[2,2]),
-                               np.arctan2(om[0,2]*zeta, om[1,2]*zeta)])
-            else:
-                eu = np.array([np.arctan2( om[0,1],om[0,0]), np.pi*0.5*(1-om[2,2]),0.0])            # following the paper, not the reference implementation
-        else:
-            with np.errstate(invalid='ignore',divide='ignore'):
-                zeta = 1.0/np.sqrt(1.0-om[...,2,2:3]**2)
-                eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.0,1e-4),
-                              np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]),
-                                        np.pi*0.5*(1-om[...,2,2:3]),
-                                        np.zeros(om.shape[:-2]+(1,)),
-                                       ]),
-                              np.block([np.arctan2(om[...,2,0:1]*zeta,-om[...,2,1:2]*zeta),
-                                        np.arccos(om[...,2,2:3]),
-                                        np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta)
-                                       ])
-                              )
-        eu[np.abs(eu)<1.e-6] = 0.0
+        with np.errstate(invalid='ignore',divide='ignore'):
+            zeta = 1.0/np.sqrt(1.0-om[...,2,2:3]**2)
+            eu = np.where(np.isclose(np.abs(om[...,2,2:3]),1.0,1e-9),
+                          np.block([np.arctan2(om[...,0,1:2],om[...,0,0:1]),
+                                    np.pi*0.5*(1-om[...,2,2:3]),
+                                    np.zeros(om.shape[:-2]+(1,)),
+                                   ]),
+                          np.block([np.arctan2(om[...,2,0:1]*zeta,-om[...,2,1:2]*zeta),
+                                    np.arccos( om[...,2,2:3]),
+                                    np.arctan2(om[...,0,2:3]*zeta,+om[...,1,2:3]*zeta)
+                                   ])
+                          )
+        eu[np.abs(eu)<1.e-8] = 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):
+    def _om2ax(om):
         """Rotation matrix to axis angle pair."""
-        if len(om.shape) == 2:
-            ax=np.empty(4)
-
-            # first get the rotation angle
-            t = 0.5*(om.trace() -1.0)
-            ax[3] = np.arccos(np.clip(t,-1.0,1.0))
-            if np.abs(ax[3])<1.e-6:
-                ax = np.array([ 0.0, 0.0, 1.0, 0.0])
-            else:
-                w,vr = np.linalg.eig(om)
-                # next, find the eigenvalue (1,0j)
-                i = np.where(np.isclose(w,1.0+0.0j))[0][0]
-                ax[0:3] = np.real(vr[0:3,i])
-                diagDelta = -_P*np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]])
-                diagDelta[np.abs(diagDelta)<1.e-6] = 1.0
-                ax[0:3] = np.where(np.abs(diagDelta)<0, ax[0:3],np.abs(ax[0:3])*np.sign(diagDelta))
-        else:
-            diag_delta = -_P*np.block([om[...,1,2:3]-om[...,2,1:2],
-                                      om[...,2,0:1]-om[...,0,2:3],
-                                      om[...,0,1:2]-om[...,1,0:1]
-                                     ])
-            diag_delta[np.abs(diag_delta)<1.e-6] = 1.0
-            t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.0).reshape(om.shape[:-2]+(1,))
-            w,vr = np.linalg.eig(om)
-            # mask duplicated real eigenvalues
-            w[np.isclose(w[...,0],1.0+0.0j),1:] = 0.
-            w[np.isclose(w[...,1],1.0+0.0j),2:] = 0.
-            vr = np.swapaxes(vr,-1,-2)
-            ax = np.where(np.abs(diag_delta)<0,
-                                 np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,)),
-                          np.abs(np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,))) \
-                          *np.sign(diag_delta))
-            ax = np.block([ax,np.arccos(np.clip(t,-1.0,1.0))])
-            ax[np.abs(ax[...,3])<1.e-6] = [ 0.0, 0.0, 1.0, 0.0]
+        #return Rotation._qu2ax(Rotation._om2qu(om)) # HOTFIX
+        diag_delta = -_P*np.block([om[...,1,2:3]-om[...,2,1:2],
+                                   om[...,2,0:1]-om[...,0,2:3],
+                                   om[...,0,1:2]-om[...,1,0:1]
+                                 ])
+        t = 0.5*(om.trace(axis2=-2,axis1=-1) -1.0).reshape(om.shape[:-2]+(1,))
+        w,vr = np.linalg.eig(om)
+        # mask duplicated real eigenvalues
+        w[np.isclose(w[...,0],1.0+0.0j),1:] = 0.
+        w[np.isclose(w[...,1],1.0+0.0j),2:] = 0.
+        vr = np.swapaxes(vr,-1,-2)
+        ax = np.where(np.abs(diag_delta)<1e-12,
+                             np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,)),
+                      np.abs(np.real(vr[np.isclose(w,1.0+0.0j)]).reshape(om.shape[:-2]+(3,))) \
+                      *np.sign(diag_delta))
+        ax = np.block([ax,np.arccos(np.clip(t,-1.0,1.0))])
+        ax[np.abs(ax[...,3])<1.e-8] = [ 0.0, 0.0, 1.0, 0.0]
         return ax
 
-
     @staticmethod
-    def om2ro(om):
+    def _om2ro(om):
         """Rotation matrix to Rodrigues-Frank vector."""
-        return Rotation.eu2ro(Rotation.om2eu(om))
+        return Rotation._eu2ro(Rotation._om2eu(om))
 
     @staticmethod
-    def om2ho(om):
+    def _om2ho(om):
         """Rotation matrix to homochoric vector."""
-        return Rotation.ax2ho(Rotation.om2ax(om))
+        return Rotation._ax2ho(Rotation._om2ax(om))
 
     @staticmethod
-    def om2cu(om):
+    def _om2cu(om):
         """Rotation matrix to cubochoric vector."""
-        return Rotation.ho2cu(Rotation.om2ho(om))
+        return Rotation._ho2cu(Rotation._om2ho(om))
 
 
     #---------- Bunge-Euler angles ----------
     @staticmethod
-    def eu2qu(eu):
+    def _eu2qu(eu):
         """Bunge-Euler angles to quaternion."""
-        if len(eu.shape) == 1:
-            ee = 0.5*eu
-            cPhi = np.cos(ee[1])
-            sPhi = np.sin(ee[1])
-            qu = np.array([   cPhi*np.cos(ee[0]+ee[2]),
-                           -_P*sPhi*np.cos(ee[0]-ee[2]),
-                           -_P*sPhi*np.sin(ee[0]-ee[2]),
-                           -_P*cPhi*np.sin(ee[0]+ee[2]) ])
-            if qu[0] < 0.0: qu*=-1
-        else:
-            ee = 0.5*eu
-            cPhi = np.cos(ee[...,1:2])
-            sPhi = np.sin(ee[...,1:2])
-            qu = np.block([  cPhi*np.cos(ee[...,0:1]+ee[...,2:3]),
-                          -_P*sPhi*np.cos(ee[...,0:1]-ee[...,2:3]),
-                          -_P*sPhi*np.sin(ee[...,0:1]-ee[...,2:3]),
-                          -_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])])
-            qu[qu[...,0]<0.0]*=-1
+        ee = 0.5*eu
+        cPhi = np.cos(ee[...,1:2])
+        sPhi = np.sin(ee[...,1:2])
+        qu = np.block([    cPhi*np.cos(ee[...,0:1]+ee[...,2:3]),
+                       -_P*sPhi*np.cos(ee[...,0:1]-ee[...,2:3]),
+                       -_P*sPhi*np.sin(ee[...,0:1]-ee[...,2:3]),
+                       -_P*cPhi*np.sin(ee[...,0:1]+ee[...,2:3])])
+        qu[qu[...,0]<0.0]*=-1
         return qu
 
-
     @staticmethod
-    def eu2om(eu):
+    def _eu2om(eu):
         """Bunge-Euler angles to rotation matrix."""
-        if len(eu.shape) == 1:
-            c = np.cos(eu)
-            s = np.sin(eu)
-
-            om = np.array([[+c[0]*c[2]-s[0]*s[2]*c[1], +s[0]*c[2]+c[0]*s[2]*c[1], +s[2]*s[1]],
-                           [-c[0]*s[2]-s[0]*c[2]*c[1], -s[0]*s[2]+c[0]*c[2]*c[1], +c[2]*s[1]],
-                           [+s[0]*s[1],                -c[0]*s[1],                +c[1]     ]])
-        else:
-            c = np.cos(eu)
-            s = np.sin(eu)
-            om = np.block([+c[...,0:1]*c[...,2:3]-s[...,0:1]*s[...,2:3]*c[...,1:2],
-                           +s[...,0:1]*c[...,2:3]+c[...,0:1]*s[...,2:3]*c[...,1:2],
-                           +s[...,2:3]*s[...,1:2],
-                           -c[...,0:1]*s[...,2:3]-s[...,0:1]*c[...,2:3]*c[...,1:2],
-                           -s[...,0:1]*s[...,2:3]+c[...,0:1]*c[...,2:3]*c[...,1:2],
-                           +c[...,2:3]*s[...,1:2],
-                           +s[...,0:1]*s[...,1:2],
-                           -c[...,0:1]*s[...,1:2],
-                           +c[...,1:2]
-                           ]).reshape(eu.shape[:-1]+(3,3))
+        c = np.cos(eu)
+        s = np.sin(eu)
+        om = np.block([+c[...,0:1]*c[...,2:3]-s[...,0:1]*s[...,2:3]*c[...,1:2],
+                       +s[...,0:1]*c[...,2:3]+c[...,0:1]*s[...,2:3]*c[...,1:2],
+                       +s[...,2:3]*s[...,1:2],
+                       -c[...,0:1]*s[...,2:3]-s[...,0:1]*c[...,2:3]*c[...,1:2],
+                       -s[...,0:1]*s[...,2:3]+c[...,0:1]*c[...,2:3]*c[...,1:2],
+                       +c[...,2:3]*s[...,1:2],
+                       +s[...,0:1]*s[...,1:2],
+                       -c[...,0:1]*s[...,1:2],
+                       +c[...,1:2]
+                       ]).reshape(eu.shape[:-1]+(3,3))
         om[np.abs(om)<1.e-12] = 0.0
         return om
 
     @staticmethod
-    def eu2ax(eu):
+    def _eu2ax(eu):
         """Bunge-Euler angles to axis angle pair."""
-        if len(eu.shape) == 1:
-            t = np.tan(eu[1]*0.5)
-            sigma = 0.5*(eu[0]+eu[2])
-            delta = 0.5*(eu[0]-eu[2])
-            tau   = np.linalg.norm([t,np.sin(sigma)])
-            alpha = np.pi if iszero(np.cos(sigma)) else \
-                    2.0*np.arctan(tau/np.cos(sigma))
-
-            if np.abs(alpha)<1.e-6:
-                ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
-            else:
-                ax = -_P/tau * np.array([ t*np.cos(delta), t*np.sin(delta), np.sin(sigma) ])        # passive axis angle pair so a minus sign in front
-                ax = np.append(ax,alpha)
-                if alpha < 0.0: ax *= -1.0                                                          # ensure alpha is positive
-        else:
-            t = np.tan(eu[...,1:2]*0.5)
-            sigma = 0.5*(eu[...,0:1]+eu[...,2:3])
-            delta = 0.5*(eu[...,0:1]-eu[...,2:3])
-            tau   = np.linalg.norm(np.block([t,np.sin(sigma)]),axis=-1,keepdims=True)
-            alpha = np.where(np.abs(np.cos(sigma))<1.e-12,np.pi,2.0*np.arctan(tau/np.cos(sigma)))
-            with np.errstate(invalid='ignore',divide='ignore'):
-                ax = np.where(np.broadcast_to(np.abs(alpha)<1.0e-12,eu.shape[:-1]+(4,)),
-                              [0.0,0.0,1.0,0.0],
-                              np.block([-_P/tau*t*np.cos(delta),
-                                        -_P/tau*t*np.sin(delta),
-                                        -_P/tau*  np.sin(sigma),
-                                         alpha
-                                        ]))
-            ax[(alpha<0.0).squeeze()] *=-1
+        t = np.tan(eu[...,1:2]*0.5)
+        sigma = 0.5*(eu[...,0:1]+eu[...,2:3])
+        delta = 0.5*(eu[...,0:1]-eu[...,2:3])
+        tau   = np.linalg.norm(np.block([t,np.sin(sigma)]),axis=-1,keepdims=True)
+        alpha = np.where(np.abs(np.cos(sigma))<1.e-12,np.pi,2.0*np.arctan(tau/np.cos(sigma)))
+        with np.errstate(invalid='ignore',divide='ignore'):
+            ax = np.where(np.broadcast_to(np.abs(alpha)<1.0e-12,eu.shape[:-1]+(4,)),
+                          [0.0,0.0,1.0,0.0],
+                          np.block([-_P/tau*t*np.cos(delta),
+                                    -_P/tau*t*np.sin(delta),
+                                    -_P/tau*  np.sin(sigma),
+                                     alpha
+                                    ]))
+        ax[(alpha<0.0).squeeze()] *=-1
         return ax
 
     @staticmethod
-    def eu2ro(eu):
+    def _eu2ro(eu):
         """Bunge-Euler angles to Rodrigues-Frank vector."""
-        if len(eu.shape) == 1:
-            ro = Rotation.eu2ax(eu)                                                                 # convert to axis angle pair representation
-            if ro[3] >= np.pi:                                                                      # Differs from original implementation. check convention 5
-                ro[3] = np.inf
-            elif iszero(ro[3]):
-                ro = np.array([ 0.0, 0.0, _P, 0.0 ])
-            else:
-                ro[3] = np.tan(ro[3]*0.5)
-        else:
-            ax = Rotation.eu2ax(eu)
-            ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)])
-            ro[ax[...,3]>=np.pi,3] = np.inf
-            ro[np.abs(ax[...,3])<1.e-16] = [ 0.0, 0.0, _P, 0.0 ]
+        ax = Rotation._eu2ax(eu)
+        ro = np.block([ax[...,:3],np.tan(ax[...,3:4]*.5)])
+        ro[ax[...,3]>=np.pi,3] = np.inf
+        ro[np.abs(ax[...,3])<1.e-16] = [ 0.0, 0.0, _P, 0.0 ]
         return ro
 
     @staticmethod
-    def eu2ho(eu):
+    def _eu2ho(eu):
         """Bunge-Euler angles to homochoric vector."""
-        return Rotation.ax2ho(Rotation.eu2ax(eu))
+        return Rotation._ax2ho(Rotation._eu2ax(eu))
 
     @staticmethod
-    def eu2cu(eu):
+    def _eu2cu(eu):
         """Bunge-Euler angles to cubochoric vector."""
-        return Rotation.ho2cu(Rotation.eu2ho(eu))
+        return Rotation._ho2cu(Rotation._eu2ho(eu))
 
 
     #---------- Axis angle pair ----------
     @staticmethod
-    def ax2qu(ax):
+    def _ax2qu(ax):
         """Axis angle pair to quaternion."""
-        if len(ax.shape) == 1:
-            if np.abs(ax[3])<1.e-6:
-                qu = np.array([ 1.0, 0.0, 0.0, 0.0 ])
-            else:
-                c = np.cos(ax[3]*0.5)
-                s = np.sin(ax[3]*0.5)
-                qu = np.array([ c, ax[0]*s, ax[1]*s, ax[2]*s ])
-        else:
-            c = np.cos(ax[...,3:4]*.5)
-            s = np.sin(ax[...,3:4]*.5)
-            qu = np.where(np.abs(ax[...,3:4])<1.e-6,[1.0, 0.0, 0.0, 0.0],np.block([c, ax[...,:3]*s]))
+        c = np.cos(ax[...,3:4]*.5)
+        s = np.sin(ax[...,3:4]*.5)
+        qu = np.where(np.abs(ax[...,3:4])<1.e-6,[1.0, 0.0, 0.0, 0.0],np.block([c, ax[...,:3]*s]))
         return qu
 
     @staticmethod
-    def ax2om(ax):
+    def _ax2om(ax):
         """Axis angle pair to rotation matrix."""
-        if len(ax.shape) == 1:
-            c = np.cos(ax[3])
-            s = np.sin(ax[3])
-            omc = 1.0-c
-            om=np.diag(ax[0:3]**2*omc + c)
-
-            for idx in [[0,1,2],[1,2,0],[2,0,1]]:
-                q = omc*ax[idx[0]] * ax[idx[1]]
-                om[idx[0],idx[1]] = q + s*ax[idx[2]]
-                om[idx[1],idx[0]] = q - s*ax[idx[2]]
-        else:
-            c = np.cos(ax[...,3:4])
-            s = np.sin(ax[...,3:4])
-            omc = 1. -c
-            om = np.block([c+omc*ax[...,0:1]**2,
-                           omc*ax[...,0:1]*ax[...,1:2] + s*ax[...,2:3],
-                           omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2],
-                           omc*ax[...,0:1]*ax[...,1:2] - s*ax[...,2:3],
-                           c+omc*ax[...,1:2]**2,
-                           omc*ax[...,1:2]*ax[...,2:3] + s*ax[...,0:1],
-                           omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2],
-                           omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1],
-                           c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3))
-        return om if _P < 0.0 else np.swapaxes(om,(-1,-2))
+        c = np.cos(ax[...,3:4])
+        s = np.sin(ax[...,3:4])
+        omc = 1. -c
+        om = np.block([c+omc*ax[...,0:1]**2,
+                       omc*ax[...,0:1]*ax[...,1:2] + s*ax[...,2:3],
+                       omc*ax[...,0:1]*ax[...,2:3] - s*ax[...,1:2],
+                       omc*ax[...,0:1]*ax[...,1:2] - s*ax[...,2:3],
+                       c+omc*ax[...,1:2]**2,
+                       omc*ax[...,1:2]*ax[...,2:3] + s*ax[...,0:1],
+                       omc*ax[...,0:1]*ax[...,2:3] + s*ax[...,1:2],
+                       omc*ax[...,1:2]*ax[...,2:3] - s*ax[...,0:1],
+                       c+omc*ax[...,2:3]**2]).reshape(ax.shape[:-1]+(3,3))
+        return om if _P < 0.0 else np.swapaxes(om,-1,-2)
 
     @staticmethod
-    def ax2eu(ax):
+    def _ax2eu(ax):
         """Rotation matrix to Bunge Euler angles."""
-        return Rotation.om2eu(Rotation.ax2om(ax))
+        return Rotation._om2eu(Rotation._ax2om(ax))
 
     @staticmethod
-    def ax2ro(ax):
+    def _ax2ro(ax):
         """Axis angle pair to Rodrigues-Frank vector."""
-        if len(ax.shape) == 1:
-            if np.abs(ax[3])<1.e-6:
-                ro = [ 0.0, 0.0, _P, 0.0 ]
-            else:
-                ro = [ax[0], ax[1], ax[2]]
-                # 180 degree case
-                ro += [np.inf] if np.isclose(ax[3],np.pi,atol=1.0e-15,rtol=0.0) else \
-                      [np.tan(ax[3]*0.5)]
-            ro = np.array(ro)
-        else:
-            ro = np.block([ax[...,:3],
-                           np.where(np.isclose(ax[...,3:4],np.pi,atol=1.e-15,rtol=.0),
-                                    np.inf,
-                                    np.tan(ax[...,3:4]*0.5))
-                          ])
-            ro[np.abs(ax[...,3])<1.e-6] = [.0,.0,_P,.0]
+        ro = np.block([ax[...,:3],
+                       np.where(np.isclose(ax[...,3:4],np.pi,atol=1.e-15,rtol=.0),
+                                np.inf,
+                                np.tan(ax[...,3:4]*0.5))
+                      ])
+        ro[np.abs(ax[...,3])<1.e-6] = [.0,.0,_P,.0]
         return ro
 
     @staticmethod
-    def ax2ho(ax):
+    def _ax2ho(ax):
         """Axis angle pair to homochoric vector."""
-        if len(ax.shape) == 1:
-            f = (0.75 * ( ax[3] - np.sin(ax[3]) ))**(1.0/3.0)
-            ho = ax[0:3] * f
-        else:
-            f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1.0/3.0)
-            ho = ax[...,:3] * f
+        f = (0.75 * ( ax[...,3:4] - np.sin(ax[...,3:4]) ))**(1.0/3.0)
+        ho = ax[...,:3] * f
         return ho
 
     @staticmethod
-    def ax2cu(ax):
+    def _ax2cu(ax):
         """Axis angle pair to cubochoric vector."""
-        return Rotation.ho2cu(Rotation.ax2ho(ax))
+        return Rotation._ho2cu(Rotation._ax2ho(ax))
 
 
     #---------- Rodrigues-Frank vector ----------
     @staticmethod
-    def ro2qu(ro):
+    def _ro2qu(ro):
         """Rodrigues-Frank vector to quaternion."""
-        return Rotation.ax2qu(Rotation.ro2ax(ro))
+        return Rotation._ax2qu(Rotation._ro2ax(ro))
 
     @staticmethod
-    def ro2om(ro):
+    def _ro2om(ro):
         """Rodgrigues-Frank vector to rotation matrix."""
-        return Rotation.ax2om(Rotation.ro2ax(ro))
+        return Rotation._ax2om(Rotation._ro2ax(ro))
 
     @staticmethod
-    def ro2eu(ro):
+    def _ro2eu(ro):
         """Rodrigues-Frank vector to Bunge-Euler angles."""
-        return Rotation.om2eu(Rotation.ro2om(ro))
+        return Rotation._om2eu(Rotation._ro2om(ro))
 
     @staticmethod
-    def ro2ax(ro):
+    def _ro2ax(ro):
         """Rodrigues-Frank vector to axis angle pair."""
-        if len(ro.shape) == 1:
-            if np.abs(ro[3]) < 1.e-6:
-                ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
-            elif not np.isfinite(ro[3]):
-                ax = np.array([ ro[0], ro[1], ro[2], np.pi ])
-            else:
-                angle = 2.0*np.arctan(ro[3])
-                ta = np.linalg.norm(ro[0:3])
-                ax = np.array([ ro[0]*ta, ro[1]*ta, ro[2]*ta, angle ])
-        else:
-            with np.errstate(invalid='ignore',divide='ignore'):
-                ax = np.where(np.isfinite(ro[...,3:4]),
-                     np.block([ro[...,0:3]*np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True),2.*np.arctan(ro[...,3:4])]),
-                     np.block([ro[...,0:3],np.broadcast_to(np.pi,ro[...,3:4].shape)]))
-            ax[np.abs(ro[...,3]) < 1.e-6]  = np.array([ 0.0, 0.0, 1.0, 0.0 ])
+        with np.errstate(invalid='ignore',divide='ignore'):
+            ax = np.where(np.isfinite(ro[...,3:4]),
+                 np.block([ro[...,0:3]*np.linalg.norm(ro[...,0:3],axis=-1,keepdims=True),2.*np.arctan(ro[...,3:4])]),
+                 np.block([ro[...,0:3],np.broadcast_to(np.pi,ro[...,3:4].shape)]))
+        ax[np.abs(ro[...,3]) < 1.e-8]  = np.array([ 0.0, 0.0, 1.0, 0.0 ])
         return ax
 
     @staticmethod
-    def ro2ho(ro):
+    def _ro2ho(ro):
         """Rodrigues-Frank vector to homochoric vector."""
-        if len(ro.shape) == 1:
-            if np.sum(ro[0:3]**2.0) < 1.e-6:
-                ho = np.zeros(3)
-            else:
-                f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
-                ho = ro[0:3] * (0.75*f)**(1.0/3.0)
-        else:
-            f = np.where(np.isfinite(ro[...,3:4]),2.0*np.arctan(ro[...,3:4]) -np.sin(2.0*np.arctan(ro[...,3:4])),np.pi)
-            ho = np.where(np.broadcast_to(np.sum(ro[...,0:3]**2.0,axis=-1,keepdims=True) < 1.e-6,ro[...,0:3].shape),
-                          np.zeros(3), ro[...,0:3]* (0.75*f)**(1.0/3.0))
+        f = np.where(np.isfinite(ro[...,3:4]),2.0*np.arctan(ro[...,3:4]) -np.sin(2.0*np.arctan(ro[...,3:4])),np.pi)
+        ho = np.where(np.broadcast_to(np.sum(ro[...,0:3]**2.0,axis=-1,keepdims=True) < 1.e-8,ro[...,0:3].shape),
+                      np.zeros(3), ro[...,0:3]* (0.75*f)**(1.0/3.0))
         return ho
 
     @staticmethod
-    def ro2cu(ro):
+    def _ro2cu(ro):
         """Rodrigues-Frank vector to cubochoric vector."""
-        return Rotation.ho2cu(Rotation.ro2ho(ro))
+        return Rotation._ho2cu(Rotation._ro2ho(ro))
 
 
     #---------- Homochoric vector----------
     @staticmethod
-    def ho2qu(ho):
+    def _ho2qu(ho):
         """Homochoric vector to quaternion."""
-        return Rotation.ax2qu(Rotation.ho2ax(ho))
+        return Rotation._ax2qu(Rotation._ho2ax(ho))
 
     @staticmethod
-    def ho2om(ho):
+    def _ho2om(ho):
         """Homochoric vector to rotation matrix."""
-        return Rotation.ax2om(Rotation.ho2ax(ho))
+        return Rotation._ax2om(Rotation._ho2ax(ho))
 
     @staticmethod
-    def ho2eu(ho):
+    def _ho2eu(ho):
         """Homochoric vector to Bunge-Euler angles."""
-        return Rotation.ax2eu(Rotation.ho2ax(ho))
+        return Rotation._ax2eu(Rotation._ho2ax(ho))
 
     @staticmethod
-    def ho2ax(ho):
+    def _ho2ax(ho):
         """Homochoric vector to axis angle pair."""
         tfit = np.array([+1.0000000000018852,      -0.5000000002194847,
                          -0.024999992127593126,    -0.003928701544781374,
@@ -1020,40 +881,25 @@ class Rotation:
                          +0.0001703481934140054,   -0.00012062065004116828,
                          +0.000059719705868660826, -0.00001980756723965647,
                          +0.000003953714684212874, -0.00000036555001439719544])
-        if len(ho.shape) == 1:
-            # normalize h and store the magnitude
-            hmag_squared = np.sum(ho**2.)
-            if iszero(hmag_squared):
-                ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
-            else:
-                hm = hmag_squared
-
-                # convert the magnitude to the rotation angle
-                s = tfit[0] + tfit[1] * hmag_squared
-                for i in range(2,16):
-                    hm *= hmag_squared
-                    s  += tfit[i] * hm
-                ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0)))
-        else:
-            hmag_squared = np.sum(ho**2.,axis=-1,keepdims=True)
-            hm = hmag_squared.copy()
-            s = tfit[0] + tfit[1] * hmag_squared
-            for i in range(2,16):
-                hm *= hmag_squared
-                s  += tfit[i] * hm
-            with np.errstate(invalid='ignore'):
-                ax = np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-6,ho.shape[:-1]+(4,)),
-                              [ 0.0, 0.0, 1.0, 0.0 ],
-                              np.block([ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))]))
+        hmag_squared = np.sum(ho**2.,axis=-1,keepdims=True)
+        hm = hmag_squared.copy()
+        s = tfit[0] + tfit[1] * hmag_squared
+        for i in range(2,16):
+            hm *= hmag_squared
+            s  += tfit[i] * hm
+        with np.errstate(invalid='ignore'):
+            ax = np.where(np.broadcast_to(np.abs(hmag_squared)<1.e-8,ho.shape[:-1]+(4,)),
+                          [ 0.0, 0.0, 1.0, 0.0 ],
+                          np.block([ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0))]))
         return ax
 
     @staticmethod
-    def ho2ro(ho):
+    def _ho2ro(ho):
         """Axis angle pair to Rodrigues-Frank vector."""
-        return Rotation.ax2ro(Rotation.ho2ax(ho))
+        return Rotation._ax2ro(Rotation._ho2ax(ho))
 
     @staticmethod
-    def ho2cu(ho):
+    def _ho2cu(ho):
         """
         Homochoric vector to cubochoric vector.
 
@@ -1063,91 +909,62 @@ class Rotation:
         https://doi.org/10.1088/0965-0393/22/7/075013
 
         """
-        if len(ho.shape) == 1:
-            rs = np.linalg.norm(ho)
+        rs = np.linalg.norm(ho,axis=-1,keepdims=True)
 
-            if np.allclose(ho,0.0,rtol=0.0,atol=1.0e-16):
-                cu = np.zeros(3)
-            else:
-                xyz3 = ho[Rotation._get_pyramid_order(ho,'forward')]
+        xyz3 = np.take_along_axis(ho,Rotation._get_pyramid_order(ho,'forward'),-1)
 
-                # inverse M_3
-                xyz2 = xyz3[0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[2])) )
+        with np.errstate(invalid='ignore',divide='ignore'):
+            # inverse M_3
+            xyz2 = xyz3[...,0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[...,2:3])) )
+            qxy = np.sum(xyz2**2,axis=-1,keepdims=True)
 
-                # inverse M_2
-                qxy = np.sum(xyz2**2)
+            q2 = qxy + np.max(np.abs(xyz2),axis=-1,keepdims=True)**2
+            sq2 = np.sqrt(q2)
+            q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2))
+            tt = np.clip((np.min(np.abs(xyz2),axis=-1,keepdims=True)**2\
+                +np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
+            T_inv = np.where(np.abs(xyz2[...,1:2]) <= np.abs(xyz2[...,0:1]),
+                                np.block([np.ones_like(tt),np.arccos(tt)/np.pi*12.0]),
+                                np.block([np.arccos(tt)/np.pi*12.0,np.ones_like(tt)]))*q
+            T_inv[xyz2<0.0] *= -1.0
+            T_inv[np.broadcast_to(np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-12),T_inv.shape)] = 0.0
+            cu = np.block([T_inv, np.where(xyz3[...,2:3]<0.0,-np.ones_like(xyz3[...,2:3]),np.ones_like(xyz3[...,2:3])) \
+                                  * rs/np.sqrt(6.0/np.pi),
+                          ])/ _sc
 
-                if np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-16):
-                    Tinv = np.zeros(2)
-                else:
-                    q2 = qxy + np.max(np.abs(xyz2))**2
-                    sq2 = np.sqrt(q2)
-                    q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2))*sq2))
-                    tt = np.clip((np.min(np.abs(xyz2))**2+np.max(np.abs(xyz2))*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
-                    Tinv = np.array([1.0,np.arccos(tt)/np.pi*12.0]) if np.abs(xyz2[1]) <= np.abs(xyz2[0]) else \
-                           np.array([np.arccos(tt)/np.pi*12.0,1.0])
-                    Tinv = q * np.where(xyz2<0.0,-Tinv,Tinv)
-
-                # inverse M_1
-                cu = np.array([ Tinv[0], Tinv[1],  (-1.0 if xyz3[2] < 0.0 else 1.0) * rs / np.sqrt(6.0/np.pi) ]) /_sc
-                cu = cu[Rotation._get_pyramid_order(ho,'backward')]
-        else:
-            rs = np.linalg.norm(ho,axis=-1,keepdims=True)
-
-            xyz3 = np.take_along_axis(ho,Rotation._get_pyramid_order(ho,'forward'),-1)
-
-            with np.errstate(invalid='ignore',divide='ignore'):
-                # inverse M_3
-                xyz2 = xyz3[...,0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[...,2:3])) )
-                qxy = np.sum(xyz2**2,axis=-1,keepdims=True)
-
-                q2 = qxy + np.max(np.abs(xyz2),axis=-1,keepdims=True)**2
-                sq2 = np.sqrt(q2)
-                q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2))
-                tt = np.clip((np.min(np.abs(xyz2),axis=-1,keepdims=True)**2\
-                    +np.max(np.abs(xyz2),axis=-1,keepdims=True)*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
-                T_inv = np.where(np.abs(xyz2[...,1:2]) <= np.abs(xyz2[...,0:1]),
-                                    np.block([np.ones_like(tt),np.arccos(tt)/np.pi*12.0]),
-                                    np.block([np.arccos(tt)/np.pi*12.0,np.ones_like(tt)]))*q
-                T_inv[xyz2<0.0] *= -1.0
-                T_inv[np.broadcast_to(np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-12),T_inv.shape)] = 0.0
-                cu = np.block([T_inv, np.where(xyz3[...,2:3]<0.0,-np.ones_like(xyz3[...,2:3]),np.ones_like(xyz3[...,2:3])) \
-                                      * rs/np.sqrt(6.0/np.pi),
-                              ])/ _sc
-
-            cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
-            cu = np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1)
+        cu[np.isclose(np.sum(np.abs(ho),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
+        cu = np.take_along_axis(cu,Rotation._get_pyramid_order(ho,'backward'),-1)
 
         return cu
 
     #---------- Cubochoric ----------
     @staticmethod
-    def cu2qu(cu):
+    def _cu2qu(cu):
         """Cubochoric vector to quaternion."""
-        return Rotation.ho2qu(Rotation.cu2ho(cu))
+        return Rotation._ho2qu(Rotation._cu2ho(cu))
 
     @staticmethod
-    def cu2om(cu):
+    def _cu2om(cu):
         """Cubochoric vector to rotation matrix."""
-        return Rotation.ho2om(Rotation.cu2ho(cu))
+        return Rotation._ho2om(Rotation._cu2ho(cu))
 
     @staticmethod
-    def cu2eu(cu):
+    def _cu2eu(cu):
         """Cubochoric vector to Bunge-Euler angles."""
-        return Rotation.ho2eu(Rotation.cu2ho(cu))
+        return Rotation._ho2eu(Rotation._cu2ho(cu))
 
     @staticmethod
-    def cu2ax(cu):
+    def _cu2ax(cu):
         """Cubochoric vector to axis angle pair."""
-        return Rotation.ho2ax(Rotation.cu2ho(cu))
+        return Rotation._ho2ax(Rotation._cu2ho(cu))
 
     @staticmethod
-    def cu2ro(cu):
+    def _cu2ro(cu):
         """Cubochoric vector to Rodrigues-Frank vector."""
-        return Rotation.ho2ro(Rotation.cu2ho(cu))
+        return Rotation._ho2ro(Rotation._cu2ho(cu))
 
     @staticmethod
-    def cu2ho(cu):
+    def _cu2ho(cu):
         """
         Cubochoric vector to homochoric vector.
 
@@ -1157,64 +974,34 @@ class Rotation:
         https://doi.org/10.1088/0965-0393/22/7/075013
 
         """
-        if len(cu.shape) == 1:
-            # transform to the sphere grid via the curved square, and intercept the zero point
-            if np.allclose(cu,0.0,rtol=0.0,atol=1.0e-16):
-                ho = np.zeros(3)
-            else:
-                # get pyramide and scale by grid parameter ratio
-                XYZ = cu[Rotation._get_pyramid_order(cu,'forward')] * _sc
+        with np.errstate(invalid='ignore',divide='ignore'):
+            # get pyramide and scale by grid parameter ratio
+            XYZ = np.take_along_axis(cu,Rotation._get_pyramid_order(cu,'forward'),-1) * _sc
+            order = np.abs(XYZ[...,1:2]) <= np.abs(XYZ[...,0:1])
+            q = np.pi/12.0 * np.where(order,XYZ[...,1:2],XYZ[...,0:1]) \
+                           / np.where(order,XYZ[...,0:1],XYZ[...,1:2])
+            c = np.cos(q)
+            s = np.sin(q)
+            q = _R1*2.0**0.25/_beta/ np.sqrt(np.sqrt(2.0)-c) \
+              * np.where(order,XYZ[...,0:1],XYZ[...,1:2])
 
-                # intercept all the points along the z-axis
-                if np.allclose(XYZ[0:2],0.0,rtol=0.0,atol=1.0e-16):
-                    ho = np.array([0.0, 0.0, np.sqrt(6.0/np.pi) * XYZ[2]])
-                else:
-                    order = [1,0] if np.abs(XYZ[1]) <= np.abs(XYZ[0]) else [0,1]
-                    q = np.pi/12.0 * XYZ[order[0]]/XYZ[order[1]]
-                    c = np.cos(q)
-                    s = np.sin(q)
-                    q = _R1*2.0**0.25/_beta * XYZ[order[1]] / np.sqrt(np.sqrt(2.0)-c)
-                    T = np.array([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
+            T = np.block([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
 
-                    # transform to sphere grid (inverse Lambert)
-                    # note that there is no need to worry about dividing by zero, since XYZ[2] can not become zero
-                    c = np.sum(T**2)
-                    s = c *         np.pi/24.0 /XYZ[2]**2
-                    c = c * np.sqrt(np.pi/24.0)/XYZ[2]
+            # transform to sphere grid (inverse Lambert)
+            c = np.sum(T**2,axis=-1,keepdims=True)
+            s = c *         np.pi/24.0 /XYZ[...,2:3]**2
+            c = c * np.sqrt(np.pi/24.0)/XYZ[...,2:3]
+            q = np.sqrt( 1.0 - s)
 
-                    q = np.sqrt( 1.0 - s )
-                    ho = np.array([ T[order[1]] * q, T[order[0]] * q, np.sqrt(6.0/np.pi) * XYZ[2] - c ])
+            ho = np.where(np.isclose(np.sum(np.abs(XYZ[...,0:2]),axis=-1,keepdims=True),0.0,rtol=0.0,atol=1.0e-16),
+                          np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6.0/np.pi) *XYZ[...,2:3]]),
+                          np.block([np.where(order,T[...,0:1],T[...,1:2])*q,
+                                    np.where(order,T[...,1:2],T[...,0:1])*q,
+                                    np.sqrt(6.0/np.pi) * XYZ[...,2:3] - c])
+                          )
 
-                ho = ho[Rotation._get_pyramid_order(cu,'backward')]
-        else:
-            with np.errstate(invalid='ignore',divide='ignore'):
-                # get pyramide and scale by grid parameter ratio
-                XYZ = np.take_along_axis(cu,Rotation._get_pyramid_order(cu,'forward'),-1) * _sc
-                order = np.abs(XYZ[...,1:2]) <= np.abs(XYZ[...,0:1])
-                q = np.pi/12.0 * np.where(order,XYZ[...,1:2],XYZ[...,0:1]) \
-                               / np.where(order,XYZ[...,0:1],XYZ[...,1:2])
-                c = np.cos(q)
-                s = np.sin(q)
-                q = _R1*2.0**0.25/_beta/ np.sqrt(np.sqrt(2.0)-c) \
-                  * np.where(order,XYZ[...,0:1],XYZ[...,1:2])
-
-                T = np.block([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
-
-                # transform to sphere grid (inverse Lambert)
-                c = np.sum(T**2,axis=-1,keepdims=True)
-                s = c *         np.pi/24.0 /XYZ[...,2:3]**2
-                c = c * np.sqrt(np.pi/24.0)/XYZ[...,2:3]
-                q = np.sqrt( 1.0 - s)
-
-                ho = np.where(np.isclose(np.sum(np.abs(XYZ[...,0:2]),axis=-1,keepdims=True),0.0,rtol=0.0,atol=1.0e-16),
-                              np.block([np.zeros_like(XYZ[...,0:2]),np.sqrt(6.0/np.pi) *XYZ[...,2:3]]),
-                              np.block([np.where(order,T[...,0:1],T[...,1:2])*q,
-                                        np.where(order,T[...,1:2],T[...,0:1])*q,
-                                        np.sqrt(6.0/np.pi) * XYZ[...,2:3] - c])
-                              )
-
-            ho[np.isclose(np.sum(np.abs(cu),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
-            ho = np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1)
+        ho[np.isclose(np.sum(np.abs(cu),axis=-1),0.0,rtol=0.0,atol=1.0e-16)] = 0.0
+        ho = np.take_along_axis(ho,Rotation._get_pyramid_order(cu,'backward'),-1)
 
         return ho
 
@@ -1238,20 +1025,10 @@ class Rotation:
         https://doi.org/10.1088/0965-0393/22/7/075013
 
         """
-        order = {'forward':np.array([[0,1,2],[1,2,0],[2,0,1]]),
+        order = {'forward': np.array([[0,1,2],[1,2,0],[2,0,1]]),
                  'backward':np.array([[0,1,2],[2,0,1],[1,2,0]])}
-        if len(xyz.shape) == 1:
-            if   np.maximum(abs(xyz[0]),abs(xyz[1])) <= xyz[2] or \
-                 np.maximum(abs(xyz[0]),abs(xyz[1])) <=-xyz[2]:
-                p = 0
-            elif np.maximum(abs(xyz[1]),abs(xyz[2])) <= xyz[0] or \
-                 np.maximum(abs(xyz[1]),abs(xyz[2])) <=-xyz[0]:
-                p = 1
-            elif np.maximum(abs(xyz[2]),abs(xyz[0])) <= xyz[1] or \
-                 np.maximum(abs(xyz[2]),abs(xyz[0])) <=-xyz[1]:
-                p = 2
-        else:
-            p = np.where(np.maximum(np.abs(xyz[...,0]),np.abs(xyz[...,1])) <= np.abs(xyz[...,2]),0,
-                  np.where(np.maximum(np.abs(xyz[...,1]),np.abs(xyz[...,2])) <= np.abs(xyz[...,0]),1,2))
+
+        p = np.where(np.maximum(np.abs(xyz[...,0]),np.abs(xyz[...,1])) <= np.abs(xyz[...,2]),0,
+                     np.where(np.maximum(np.abs(xyz[...,1]),np.abs(xyz[...,2])) <= np.abs(xyz[...,0]),1,2))
 
         return order[direction][p]
diff --git a/python/tests/test_Orientation.py b/python/tests/test_Orientation.py
index 08f060887..a8b8afdac 100644
--- a/python/tests/test_Orientation.py
+++ b/python/tests/test_Orientation.py
@@ -8,13 +8,8 @@ import damask
 from damask import Rotation
 from damask import Orientation
 from damask import Lattice
-   
-n = 1000
 
-@pytest.fixture
-def default():
-    """A set of n random rotations."""
-    return [Rotation.fromRandom() for r in range(n)]
+n = 1000
 
 @pytest.fixture
 def reference_dir(reference_dir_base):
@@ -28,15 +23,15 @@ class TestOrientation:
                                       {'label':'green','RGB':[0,1,0],'direction':[0,1,1]},
                                       {'label':'blue', 'RGB':[0,0,1],'direction':[1,1,1]}])
     @pytest.mark.parametrize('lattice',['fcc','bcc'])
-    def test_IPF_cubic(self,default,color,lattice):
+    def test_IPF_cubic(self,color,lattice):
         cube = damask.Orientation(damask.Rotation(),lattice)
         for direction in set(permutations(np.array(color['direction']))):
-            assert np.allclose(cube.IPFcolor(direction),np.array(color['RGB']))
+            assert np.allclose(cube.IPFcolor(np.array(direction)),np.array(color['RGB']))
 
     @pytest.mark.parametrize('lattice',Lattice.lattices)
     def test_IPF(self,lattice):
         direction = np.random.random(3)*2.0-1
-        for rot in [Rotation.fromRandom() for r in range(n//100)]:
+        for rot in [Rotation.from_random() for r in range(n//100)]:
             R = damask.Orientation(rot,lattice)
             color = R.IPFcolor(direction)
             for equivalent in R.equivalentOrientations():
@@ -45,7 +40,7 @@ class TestOrientation:
     @pytest.mark.parametrize('model',['Bain','KS','GT','GT_prime','NW','Pitsch'])
     @pytest.mark.parametrize('lattice',['fcc','bcc'])
     def test_relationship_forward_backward(self,model,lattice):
-        ori = Orientation(Rotation.fromRandom(),lattice)
+        ori = Orientation(Rotation.from_random(),lattice)
         for i,r in enumerate(ori.relatedOrientations(model)):
             ori2 = r.relatedOrientations(model)[i]
             misorientation = ori.rotation.misorientation(ori2.rotation)
@@ -56,8 +51,8 @@ class TestOrientation:
     def test_relationship_reference(self,update,reference_dir,model,lattice):
         reference = os.path.join(reference_dir,'{}_{}.txt'.format(lattice,model))
         ori = Orientation(Rotation(),lattice)
-        eu = np.array([o.rotation.asEulers(degrees=True) for o in ori.relatedOrientations(model)])
-        if update: 
+        eu = np.array([o.rotation.as_Eulers(degrees=True) for o in ori.relatedOrientations(model)])
+        if update:
             coords = np.array([(1,i+1) for i,x in enumerate(eu)])
             table = damask.Table(eu,{'Eulers':(3,)})
             table.add('pos',coords)
diff --git a/python/tests/test_Rotation.py b/python/tests/test_Rotation.py
index b7442035f..a9768afb8 100644
--- a/python/tests/test_Rotation.py
+++ b/python/tests/test_Rotation.py
@@ -4,6 +4,9 @@ import pytest
 import numpy as np
 
 from damask import Rotation
+from damask import _rotation
+
+
 
 n = 1100
 atol=1.e-4
@@ -12,68 +15,69 @@ scatter=1.e-2
 @pytest.fixture
 def default():
     """A set of n random rotations."""
-    specials = np.array(
-              [np.array([ 1.0, 0.0, 0.0, 0.0]),
-               #-----------------------------------------------
-               np.array([0.0, 1.0, 0.0, 0.0]),
-               np.array([0.0, 0.0, 1.0, 0.0]),
-               np.array([0.0, 0.0, 0.0, 1.0]),
-               np.array([0.0,-1.0, 0.0, 0.0]),
-               np.array([0.0, 0.0,-1.0, 0.0]),
-               np.array([0.0, 0.0, 0.0,-1.0]),
-               #-----------------------------------------------
-               np.array([1.0, 1.0, 0.0, 0.0])/np.sqrt(2.),
-               np.array([1.0, 0.0, 1.0, 0.0])/np.sqrt(2.),
-               np.array([1.0, 0.0, 0.0, 1.0])/np.sqrt(2.),
-               np.array([0.0, 1.0, 1.0, 0.0])/np.sqrt(2.),
-               np.array([0.0, 1.0, 0.0, 1.0])/np.sqrt(2.),
-               np.array([0.0, 0.0, 1.0, 1.0])/np.sqrt(2.),
-               #-----------------------------------------------
-               np.array([1.0,-1.0, 0.0, 0.0])/np.sqrt(2.),
-               np.array([1.0, 0.0,-1.0, 0.0])/np.sqrt(2.),
-               np.array([1.0, 0.0, 0.0,-1.0])/np.sqrt(2.),
-               np.array([0.0, 1.0,-1.0, 0.0])/np.sqrt(2.),
-               np.array([0.0, 1.0, 0.0,-1.0])/np.sqrt(2.),
-               np.array([0.0, 0.0, 1.0,-1.0])/np.sqrt(2.),
-               #-----------------------------------------------
-               np.array([0.0, 1.0,-1.0, 0.0])/np.sqrt(2.),
-               np.array([0.0, 1.0, 0.0,-1.0])/np.sqrt(2.),
-               np.array([0.0, 0.0, 1.0,-1.0])/np.sqrt(2.),
-               #-----------------------------------------------
-               np.array([0.0,-1.0,-1.0, 0.0])/np.sqrt(2.),
-               np.array([0.0,-1.0, 0.0,-1.0])/np.sqrt(2.),
-               np.array([0.0, 0.0,-1.0,-1.0])/np.sqrt(2.),
-               #-----------------------------------------------
-               np.array([1.0, 1.0, 1.0, 0.0])/np.sqrt(3.),
-               np.array([1.0, 1.0, 0.0, 1.0])/np.sqrt(3.),
-               np.array([1.0, 0.0, 1.0, 1.0])/np.sqrt(3.),
-               np.array([1.0,-1.0, 1.0, 0.0])/np.sqrt(3.),
-               np.array([1.0,-1.0, 0.0, 1.0])/np.sqrt(3.),
-               np.array([1.0, 0.0,-1.0, 1.0])/np.sqrt(3.),
-               np.array([1.0, 1.0,-1.0, 0.0])/np.sqrt(3.),
-               np.array([1.0, 1.0, 0.0,-1.0])/np.sqrt(3.),
-               np.array([1.0, 0.0, 1.0,-1.0])/np.sqrt(3.),
-               np.array([1.0,-1.0,-1.0, 0.0])/np.sqrt(3.),
-               np.array([1.0,-1.0, 0.0,-1.0])/np.sqrt(3.),
-               np.array([1.0, 0.0,-1.0,-1.0])/np.sqrt(3.),
-               #-----------------------------------------------
-               np.array([0.0, 1.0, 1.0, 1.0])/np.sqrt(3.),
-               np.array([0.0, 1.0,-1.0, 1.0])/np.sqrt(3.),
-               np.array([0.0, 1.0, 1.0,-1.0])/np.sqrt(3.),
-               np.array([0.0,-1.0, 1.0, 1.0])/np.sqrt(3.),
-               np.array([0.0,-1.0,-1.0, 1.0])/np.sqrt(3.),
-               np.array([0.0,-1.0, 1.0,-1.0])/np.sqrt(3.),
-               np.array([0.0,-1.0,-1.0,-1.0])/np.sqrt(3.),
-               #-----------------------------------------------
-               np.array([1.0, 1.0, 1.0, 1.0])/2.,
-               np.array([1.0,-1.0, 1.0, 1.0])/2.,
-               np.array([1.0, 1.0,-1.0, 1.0])/2.,
-               np.array([1.0, 1.0, 1.0,-1.0])/2.,
-               np.array([1.0,-1.0,-1.0, 1.0])/2.,
-               np.array([1.0,-1.0, 1.0,-1.0])/2.,
-               np.array([1.0, 1.0,-1.0,-1.0])/2.,
-               np.array([1.0,-1.0,-1.0,-1.0])/2.,
-              ])
+    specials = np.array([
+                         [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,-1.0, 0.0, 0.0],
+                         [0.0, 0.0,-1.0, 0.0],
+                         [0.0, 0.0, 0.0,-1.0],
+                        #----------------------
+                         [1.0, 1.0, 0.0, 0.0],
+                         [1.0, 0.0, 1.0, 0.0],
+                         [1.0, 0.0, 0.0, 1.0],
+                         [0.0, 1.0, 1.0, 0.0],
+                         [0.0, 1.0, 0.0, 1.0],
+                         [0.0, 0.0, 1.0, 1.0],
+                        #----------------------
+                         [1.0,-1.0, 0.0, 0.0],
+                         [1.0, 0.0,-1.0, 0.0],
+                         [1.0, 0.0, 0.0,-1.0],
+                         [0.0, 1.0,-1.0, 0.0],
+                         [0.0, 1.0, 0.0,-1.0],
+                         [0.0, 0.0, 1.0,-1.0],
+                        #----------------------
+                         [0.0, 1.0,-1.0, 0.0],
+                         [0.0, 1.0, 0.0,-1.0],
+                         [0.0, 0.0, 1.0,-1.0],
+                        #----------------------
+                         [0.0,-1.0,-1.0, 0.0],
+                         [0.0,-1.0, 0.0,-1.0],
+                         [0.0, 0.0,-1.0,-1.0],
+                        #----------------------
+                         [1.0, 1.0, 1.0, 0.0],
+                         [1.0, 1.0, 0.0, 1.0],
+                         [1.0, 0.0, 1.0, 1.0],
+                         [1.0,-1.0, 1.0, 0.0],
+                         [1.0,-1.0, 0.0, 1.0],
+                         [1.0, 0.0,-1.0, 1.0],
+                         [1.0, 1.0,-1.0, 0.0],
+                         [1.0, 1.0, 0.0,-1.0],
+                         [1.0, 0.0, 1.0,-1.0],
+                         [1.0,-1.0,-1.0, 0.0],
+                         [1.0,-1.0, 0.0,-1.0],
+                         [1.0, 0.0,-1.0,-1.0],
+                        #----------------------
+                         [0.0, 1.0, 1.0, 1.0],
+                         [0.0, 1.0,-1.0, 1.0],
+                         [0.0, 1.0, 1.0,-1.0],
+                         [0.0,-1.0, 1.0, 1.0],
+                         [0.0,-1.0,-1.0, 1.0],
+                         [0.0,-1.0, 1.0,-1.0],
+                         [0.0,-1.0,-1.0,-1.0],
+                        #----------------------
+                         [1.0, 1.0, 1.0, 1.0],
+                         [1.0,-1.0, 1.0, 1.0],
+                         [1.0, 1.0,-1.0, 1.0],
+                         [1.0, 1.0, 1.0,-1.0],
+                         [1.0,-1.0,-1.0, 1.0],
+                         [1.0,-1.0, 1.0,-1.0],
+                         [1.0, 1.0,-1.0,-1.0],
+                         [1.0,-1.0,-1.0,-1.0],
+                        ])
+    specials /= np.linalg.norm(specials,axis=1).reshape(-1,1)
     specials_scatter = specials + np.broadcast_to(np.random.rand(4)*scatter,specials.shape)
     specials_scatter /= np.linalg.norm(specials_scatter,axis=1).reshape(-1,1)
     specials_scatter[specials_scatter[:,0]<0]*=-1
@@ -88,22 +92,518 @@ def reference_dir(reference_dir_base):
     return os.path.join(reference_dir_base,'Rotation')
 
 
+####################################################################################################
+# 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.
+####################################################################################################
+_P = _rotation._P
+
+# parameters for conversion from/to cubochoric
+_sc   = _rotation._sc
+_beta = _rotation._beta
+_R1   = _rotation._R1
+
+def iszero(a):
+    return np.isclose(a,0.0,atol=1.0e-12,rtol=0.0)
+
+#---------- Quaternion ----------
+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[0,1] = 2.0*(qu[2]*qu[1]+qu[0]*qu[3])
+    om[1,0] = 2.0*(qu[1]*qu[2]-qu[0]*qu[3])
+    om[1,2] = 2.0*(qu[3]*qu[2]+qu[0]*qu[1])
+    om[2,1] = 2.0*(qu[2]*qu[3]-qu[0]*qu[1])
+    om[2,0] = 2.0*(qu[1]*qu[3]+qu[0]*qu[2])
+    om[0,2] = 2.0*(qu[3]*qu[1]-qu[0]*qu[2])
+    return om if _P < 0.0 else np.swapaxes(om,-1,-2)
+
+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   np.abs(q12) < 1.e-8:
+        eu = np.array([np.arctan2(-_P*2.0*qu[0]*qu[3],qu[0]**2-qu[3]**2), 0.0,   0.0])
+    elif np.abs(q03) < 1.e-8:
+        eu = np.array([np.arctan2(   2.0*qu[1]*qu[2],qu[1]**2-qu[2]**2), np.pi, 0.0])
+    else:
+        eu = np.array([np.arctan2((-_P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]-qu[2]*qu[3])*chi ),
+                       np.arctan2( 2.0*chi, q03-q12 ),
+                       np.arctan2(( _P*qu[0]*qu[2]+qu[1]*qu[3])*chi, (-_P*qu[0]*qu[1]+qu[2]*qu[3])*chi )])
+    # reduce Euler angles to definition range
+    eu[np.abs(eu)<1.e-6] = 0.0
+    eu = np.where(eu<0, (eu+2.0*np.pi)%np.array([2.0*np.pi,np.pi,2.0*np.pi]),eu)
+    return eu
+
+def qu2ax(qu):
+    """
+    Quaternion to axis angle pair.
+
+    Modified version of the original formulation, should be numerically more stable
+    """
+    if np.isclose(qu[0],1.,rtol=0.0):                                                              # set axis to [001] if the angle is 0/360
+        ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
+    elif qu[0] > 1.e-8:
+        s = np.sign(qu[0])/np.sqrt(qu[1]**2+qu[2]**2+qu[3]**2)
+        omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
+        ax = ax = np.array([ qu[1]*s, qu[2]*s, qu[3]*s, omega ])
+    else:
+        ax = ax = np.array([ qu[1], qu[2], qu[3], np.pi])
+    return ax
+
+def qu2ro(qu):
+    """Quaternion to Rodrigues-Frank vector."""
+    if iszero(qu[0]):
+        ro = np.array([qu[1], qu[2], qu[3], np.inf])
+    else:
+        s = np.linalg.norm(qu[1:4])
+        ro = np.array([0.0,0.0,_P,0.0] if iszero(s) else \
+                      [ qu[1]/s,  qu[2]/s,  qu[3]/s, np.tan(np.arccos(np.clip(qu[0],-1.0,1.0)))])
+    return ro
+
+def qu2ho(qu):
+    """Quaternion to homochoric vector."""
+    omega = 2.0 * np.arccos(np.clip(qu[0],-1.0,1.0))
+    if np.abs(omega) < 1.0e-12:
+        ho = np.zeros(3)
+    else:
+        ho = np.array([qu[1], qu[2], qu[3]])
+        f  = 0.75 * ( omega - np.sin(omega) )
+        ho = ho/np.linalg.norm(ho) * f**(1./3.)
+    return ho
+
+
+#---------- Rotation matrix ----------
+def om2qu(a):
+    trace = a[0,0] + a[1,1] + a[2,2]
+    if trace > 0:
+        s = 0.5 / np.sqrt(trace+ 1.0)
+        qu = np.array([0.25 / s,( a[2,1] - a[1,2] ) * s,( a[0,2] - a[2,0] ) * s,( a[1,0] - a[0,1] ) * s])
+    else:
+        if ( a[0,0] > a[1,1] and a[0,0] > a[2,2] ):
+            s = 2.0 * np.sqrt( 1.0 + a[0,0] - a[1,1] - a[2,2])
+            qu = np.array([ (a[2,1] - a[1,2]) / s,0.25 * s,(a[0,1] + a[1,0]) / s,(a[0,2] + a[2,0]) / s])
+        elif (a[1,1] > a[2,2]):
+            s = 2.0 * np.sqrt( 1.0 + a[1,1] - a[0,0] - a[2,2])
+            qu = np.array([ (a[0,2] - a[2,0]) / s,(a[0,1] + a[1,0]) / s,0.25 * s,(a[1,2] + a[2,1]) / s])
+        else:
+            s = 2.0 * np.sqrt( 1.0 + a[2,2] - a[0,0] - a[1,1] )
+            qu = np.array([ (a[1,0] - a[0,1]) / s,(a[0,2] + a[2,0]) / s,(a[1,2] + a[2,1]) / s,0.25 * s])
+    if qu[0]<0: qu*=-1
+    return qu*np.array([1.,_P,_P,_P])
+
+def om2eu(om):
+    """Rotation matrix to Bunge-Euler angles."""
+    if not np.isclose(np.abs(om[2,2]),1.0,1.e-9):
+        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
+    eu[np.abs(eu)<1.e-8] = 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
+
+def om2ax(om):
+    """Rotation matrix to axis angle pair."""
+    #return qu2ax(om2qu(om)) # HOTFIX
+    ax=np.empty(4)
+
+    # first get the rotation angle
+    t = 0.5*(om.trace() -1.0)
+    ax[3] = np.arccos(np.clip(t,-1.0,1.0))
+    if np.abs(ax[3])<1.e-8:
+        ax = np.array([ 0.0, 0.0, 1.0, 0.0])
+    else:
+        w,vr = np.linalg.eig(om)
+        # next, find the eigenvalue (1,0j)
+        i = np.where(np.isclose(w,1.0+0.0j))[0][0]
+        ax[0:3] = np.real(vr[0:3,i])
+        diagDelta = -_P*np.array([om[1,2]-om[2,1],om[2,0]-om[0,2],om[0,1]-om[1,0]])
+        ax[0:3] = np.where(np.abs(diagDelta)<1e-12, ax[0:3],np.abs(ax[0:3])*np.sign(diagDelta))
+    return ax
+
+#---------- Bunge-Euler angles ----------
+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
+
+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.abs(om)<1.e-12] = 0.0
+    return om
+
+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 np.abs(alpha)<1.e-6:
+        ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
+    else:
+        ax = -_P/tau * np.array([ t*np.cos(delta), t*np.sin(delta), np.sin(sigma) ])                # passive axis angle pair so a minus sign in front
+        ax = np.append(ax,alpha)
+        if alpha < 0.0: ax *= -1.0                                                                  # ensure alpha is positive
+    return ax
+
+def eu2ro(eu):
+    """Bunge-Euler angles to Rodrigues-Frank vector."""
+    ro = eu2ax(eu)                                                                                  # convert to axis angle pair representation
+    if ro[3] >= np.pi:                                                                              # Differs from original implementation. check convention 5
+        ro[3] = np.inf
+    elif iszero(ro[3]):
+        ro = np.array([ 0.0, 0.0, _P, 0.0 ])
+    else:
+        ro[3] = np.tan(ro[3]*0.5)
+    return ro
+
+#---------- Axis angle pair ----------
+def ax2qu(ax):
+    """Axis angle pair to quaternion."""
+    if np.abs(ax[3])<1.e-6:
+        qu = np.array([ 1.0, 0.0, 0.0, 0.0 ])
+    else:
+        c = np.cos(ax[3]*0.5)
+        s = np.sin(ax[3]*0.5)
+        qu = np.array([ c, ax[0]*s, ax[1]*s, ax[2]*s ])
+    return qu
+
+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
+
+def ax2ro(ax):
+    """Axis angle pair to Rodrigues-Frank vector."""
+    if np.abs(ax[3])<1.e-6:
+        ro = [ 0.0, 0.0, _P, 0.0 ]
+    else:
+        ro = [ax[0], ax[1], ax[2]]
+        # 180 degree case
+        ro += [np.inf] if np.isclose(ax[3],np.pi,atol=1.0e-15,rtol=0.0) else \
+              [np.tan(ax[3]*0.5)]
+    ro = np.array(ro)
+    return ro
+
+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
+
+
+#---------- Rodrigues-Frank vector ----------
+def ro2ax(ro):
+    """Rodrigues-Frank vector to axis angle pair."""
+    if np.abs(ro[3]) < 1.e-8:
+        ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
+    elif not np.isfinite(ro[3]):
+        ax = np.array([ ro[0], ro[1], ro[2], np.pi ])
+    else:
+        angle = 2.0*np.arctan(ro[3])
+        ta = np.linalg.norm(ro[0:3])
+        ax = np.array([ ro[0]*ta, ro[1]*ta, ro[2]*ta, angle ])
+    return ax
+
+def ro2ho(ro):
+    """Rodrigues-Frank vector to homochoric vector."""
+    if np.sum(ro[0:3]**2.0) < 1.e-8:
+        ho = np.zeros(3)
+    else:
+        f = 2.0*np.arctan(ro[3]) -np.sin(2.0*np.arctan(ro[3])) if np.isfinite(ro[3]) else np.pi
+        ho = ro[0:3] * (0.75*f)**(1.0/3.0)
+    return ho
+
+#---------- Homochoric vector----------
+def ho2ax(ho):
+    """Homochoric vector to axis angle pair."""
+    tfit = np.array([+1.0000000000018852,      -0.5000000002194847,
+                     -0.024999992127593126,    -0.003928701544781374,
+                     -0.0008152701535450438,   -0.0002009500426119712,
+                     -0.00002397986776071756,  -0.00008202868926605841,
+                     +0.00012448715042090092,  -0.0001749114214822577,
+                     +0.0001703481934140054,   -0.00012062065004116828,
+                     +0.000059719705868660826, -0.00001980756723965647,
+                     +0.000003953714684212874, -0.00000036555001439719544])
+    # normalize h and store the magnitude
+    hmag_squared = np.sum(ho**2.)
+    if iszero(hmag_squared):
+        ax = np.array([ 0.0, 0.0, 1.0, 0.0 ])
+    else:
+        hm = hmag_squared
+
+        # convert the magnitude to the rotation angle
+        s = tfit[0] + tfit[1] * hmag_squared
+        for i in range(2,16):
+            hm *= hmag_squared
+            s  += tfit[i] * hm
+        ax = np.append(ho/np.sqrt(hmag_squared),2.0*np.arccos(np.clip(s,-1.0,1.0)))
+    return ax
+
+def ho2cu(ho):
+    """
+    Homochoric vector to cubochoric vector.
+
+    References
+    ----------
+    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
+    https://doi.org/10.1088/0965-0393/22/7/075013
+
+    """
+    rs = np.linalg.norm(ho)
+
+    if np.allclose(ho,0.0,rtol=0.0,atol=1.0e-16):
+        cu = np.zeros(3)
+    else:
+        xyz3 = ho[_get_pyramid_order(ho,'forward')]
+
+        # inverse M_3
+        xyz2 = xyz3[0:2] * np.sqrt( 2.0*rs/(rs+np.abs(xyz3[2])) )
+
+        # inverse M_2
+        qxy = np.sum(xyz2**2)
+
+        if np.isclose(qxy,0.0,rtol=0.0,atol=1.0e-16):
+            Tinv = np.zeros(2)
+        else:
+            q2 = qxy + np.max(np.abs(xyz2))**2
+            sq2 = np.sqrt(q2)
+            q = (_beta/np.sqrt(2.0)/_R1) * np.sqrt(q2*qxy/(q2-np.max(np.abs(xyz2))*sq2))
+            tt = np.clip((np.min(np.abs(xyz2))**2+np.max(np.abs(xyz2))*sq2)/np.sqrt(2.0)/qxy,-1.0,1.0)
+            Tinv = np.array([1.0,np.arccos(tt)/np.pi*12.0]) if np.abs(xyz2[1]) <= np.abs(xyz2[0]) else \
+                   np.array([np.arccos(tt)/np.pi*12.0,1.0])
+            Tinv = q * np.where(xyz2<0.0,-Tinv,Tinv)
+
+        # inverse M_1
+        cu = np.array([ Tinv[0], Tinv[1],  (-1.0 if xyz3[2] < 0.0 else 1.0) * rs / np.sqrt(6.0/np.pi) ]) /_sc
+        cu = cu[_get_pyramid_order(ho,'backward')]
+    return cu
+
+#---------- Cubochoric ----------
+def cu2ho(cu):
+    """
+    Cubochoric vector to homochoric vector.
+
+    References
+    ----------
+    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
+    https://doi.org/10.1088/0965-0393/22/7/075013
+
+    """
+    # transform to the sphere grid via the curved square, and intercept the zero point
+    if np.allclose(cu,0.0,rtol=0.0,atol=1.0e-16):
+        ho = np.zeros(3)
+    else:
+        # get pyramide and scale by grid parameter ratio
+        XYZ = cu[_get_pyramid_order(cu,'forward')] * _sc
+
+        # intercept all the points along the z-axis
+        if np.allclose(XYZ[0:2],0.0,rtol=0.0,atol=1.0e-16):
+            ho = np.array([0.0, 0.0, np.sqrt(6.0/np.pi) * XYZ[2]])
+        else:
+            order = [1,0] if np.abs(XYZ[1]) <= np.abs(XYZ[0]) else [0,1]
+            q = np.pi/12.0 * XYZ[order[0]]/XYZ[order[1]]
+            c = np.cos(q)
+            s = np.sin(q)
+            q = _R1*2.0**0.25/_beta * XYZ[order[1]] / np.sqrt(np.sqrt(2.0)-c)
+            T = np.array([ (np.sqrt(2.0)*c - 1.0), np.sqrt(2.0) * s]) * q
+
+            # transform to sphere grid (inverse Lambert)
+            # note that there is no need to worry about dividing by zero, since XYZ[2] can not become zero
+            c = np.sum(T**2)
+            s = c *         np.pi/24.0 /XYZ[2]**2
+            c = c * np.sqrt(np.pi/24.0)/XYZ[2]
+
+            q = np.sqrt( 1.0 - s )
+            ho = np.array([ T[order[1]] * q, T[order[0]] * q, np.sqrt(6.0/np.pi) * XYZ[2] - c ])
+
+        ho = ho[_get_pyramid_order(cu,'backward')]
+    return ho
+
+def _get_pyramid_order(xyz,direction=None):
+    """
+    Get order of the coordinates.
+
+    Depending on the pyramid in which the point is located, the order need to be adjusted.
+
+    Parameters
+    ----------
+    xyz : numpy.ndarray
+       coordinates of a point on a uniform refinable grid on a ball or
+       in a uniform refinable cubical grid.
+
+    References
+    ----------
+    D. Roşca et al., Modelling and Simulation in Materials Science and Engineering 22:075013, 2014
+    https://doi.org/10.1088/0965-0393/22/7/075013
+
+    """
+    order = {'forward':np.array([[0,1,2],[1,2,0],[2,0,1]]),
+             'backward':np.array([[0,1,2],[2,0,1],[1,2,0]])}
+    if   np.maximum(abs(xyz[0]),abs(xyz[1])) <= xyz[2] or \
+         np.maximum(abs(xyz[0]),abs(xyz[1])) <=-xyz[2]:
+        p = 0
+    elif np.maximum(abs(xyz[1]),abs(xyz[2])) <= xyz[0] or \
+         np.maximum(abs(xyz[1]),abs(xyz[2])) <=-xyz[0]:
+        p = 1
+    elif np.maximum(abs(xyz[2]),abs(xyz[0])) <= xyz[1] or \
+         np.maximum(abs(xyz[2]),abs(xyz[0])) <=-xyz[1]:
+        p = 2
+    return order[direction][p]
+
+####################################################################################################
+####################################################################################################
+
+def mul(me, other):
+    """
+    Multiplication.
+
+    Parameters
+    ----------
+    other : numpy.ndarray or Rotation
+        Vector, second or fourth order tensor, or rotation object that is rotated.
+
+    Todo
+    ----
+    Document details active/passive)
+    consider rotation of (3,3,3,3)-matrix
+
+    """
+    if me.quaternion.shape != (4,):
+        raise NotImplementedError('Support for multiple rotations missing')
+    if isinstance(other, Rotation):
+        me_q  = me.quaternion[0]
+        me_p  = me.quaternion[1:]
+        other_q = other.quaternion[0]
+        other_p = other.quaternion[1:]
+        R = me.__class__(np.append(me_q*other_q - np.dot(me_p,other_p),
+                                   me_q*other_p + other_q*me_p + _P * np.cross(me_p,other_p)))
+        return R.standardize()
+    elif isinstance(other, np.ndarray):
+        if other.shape == (3,):
+            A = me.quaternion[0]**2.0 - np.dot(me.quaternion[1:],me.quaternion[1:])
+            B = 2.0 * np.dot(me.quaternion[1:],other)
+            C = 2.0 * _P*me.quaternion[0]
+
+            return A*other + B*me.quaternion[1:] + C * np.cross(me.quaternion[1:],other)
+
+        elif other.shape == (3,3,):
+            R = me.as_matrix()
+            return np.dot(R,np.dot(other,R.T))
+        elif other.shape == (3,3,3,3,):
+            R = me.as_matrix()
+            return np.einsum('ia,jb,kc,ld,abcd->ijkl',R,R,R,R,other)
+            RR = np.outer(R, R)
+            RRRR = np.outer(RR, RR).reshape(4 * (3,3))
+            axes = ((0, 2, 4, 6), (0, 1, 2, 3))
+            return np.tensordot(RRRR, other, axes)
+        else:
+            raise ValueError('Can only rotate vectors, 2nd order ternsors, and 4th order tensors')
+    else:
+        raise TypeError('Cannot rotate {}'.format(type(other)))
+
+
 class TestRotation:
 
-    def test_Eulers(self,default):
+    @pytest.mark.parametrize('forward,backward',[(Rotation._qu2om,Rotation._om2qu),
+                                                 (Rotation._qu2eu,Rotation._eu2qu),
+                                                 (Rotation._qu2ax,Rotation._ax2qu),
+                                                 (Rotation._qu2ro,Rotation._ro2qu),
+                                                 (Rotation._qu2ho,Rotation._ho2qu),
+                                                 (Rotation._qu2cu,Rotation._cu2qu)])
+    def test_quaternion_internal(self,default,forward,backward):
+        """Ensure invariance of conversion from quaternion and back."""
         for rot in default:
             m = rot.as_quaternion()
-            o = Rotation.from_Eulers(rot.as_Eulers()).as_quaternion()
+            o = backward(forward(m))
             ok = np.allclose(m,o,atol=atol)
             if np.isclose(rot.as_quaternion()[0],0.0,atol=atol):
                 ok = ok or np.allclose(m*-1.,o,atol=atol)
             print(m,o,rot.as_quaternion())
             assert ok and np.isclose(np.linalg.norm(o),1.0)
 
-    def test_AxisAngle(self,default):
+    @pytest.mark.parametrize('forward,backward',[(Rotation._om2qu,Rotation._qu2om),
+                                                 (Rotation._om2eu,Rotation._eu2om),
+                                                 (Rotation._om2ax,Rotation._ax2om),
+                                                 (Rotation._om2ro,Rotation._ro2om),
+                                                 (Rotation._om2ho,Rotation._ho2om),
+                                                 (Rotation._om2cu,Rotation._cu2om)])
+    def test_matrix_internal(self,default,forward,backward):
+        """Ensure invariance of conversion from rotation matrix and back."""
+        for rot in default:
+            m = rot.as_matrix()
+            o = backward(forward(m))
+            ok = np.allclose(m,o,atol=atol)
+            print(m,o,rot.as_quaternion())
+            assert ok and np.isclose(np.linalg.det(o),1.0)
+
+    @pytest.mark.parametrize('forward,backward',[(Rotation._eu2qu,Rotation._qu2eu),
+                                                 (Rotation._eu2om,Rotation._om2eu),
+                                                 (Rotation._eu2ax,Rotation._ax2eu),
+                                                 (Rotation._eu2ro,Rotation._ro2eu),
+                                                 (Rotation._eu2ho,Rotation._ho2eu),
+                                                 (Rotation._eu2cu,Rotation._cu2eu)])
+    def test_Eulers_internal(self,default,forward,backward):
+        """Ensure invariance of conversion from Euler angles and back."""
         for rot in default:
             m = rot.as_Eulers()
-            o = Rotation.from_axis_angle(rot.as_axis_angle()).as_Eulers()
+            o = backward(forward(m))
             u = np.array([np.pi*2,np.pi,np.pi*2])
             ok = np.allclose(m,o,atol=atol)
             ok = ok or np.allclose(np.where(np.isclose(m,u),m-u,m),np.where(np.isclose(o,u),o-u,o),atol=atol)
@@ -113,7 +613,187 @@ class TestRotation:
             print(m,o,rot.as_quaternion())
             assert ok and (np.zeros(3)-1.e-9 <= o).all() and (o <= np.array([np.pi*2.,np.pi,np.pi*2.])+1.e-9).all()
 
-    def test_Matrix(self,default):
+    @pytest.mark.parametrize('forward,backward',[(Rotation._ax2qu,Rotation._qu2ax),
+                                                 (Rotation._ax2om,Rotation._om2ax),
+                                                 (Rotation._ax2eu,Rotation._eu2ax),
+                                                 (Rotation._ax2ro,Rotation._ro2ax),
+                                                 (Rotation._ax2ho,Rotation._ho2ax),
+                                                 (Rotation._ax2cu,Rotation._cu2ax)])
+    def test_axis_angle_internal(self,default,forward,backward):
+        """Ensure invariance of conversion from axis angle angles pair and back."""
+        for rot in default:
+            m = rot.as_axis_angle()
+            o = backward(forward(m))
+            ok = np.allclose(m,o,atol=atol)
+            if np.isclose(m[3],np.pi,atol=atol):
+                ok = ok or np.allclose(m*np.array([-1.,-1.,-1.,1.]),o,atol=atol)
+            print(m,o,rot.as_quaternion())
+            assert ok and np.isclose(np.linalg.norm(o[:3]),1.0) and o[3]<=np.pi+1.e-9
+
+    @pytest.mark.parametrize('forward,backward',[(Rotation._ro2qu,Rotation._qu2ro),
+                                                 #(Rotation._ro2om,Rotation._om2ro),
+                                                 #(Rotation._ro2eu,Rotation._eu2ro),
+                                                 (Rotation._ro2ax,Rotation._ax2ro),
+                                                 (Rotation._ro2ho,Rotation._ho2ro),
+                                                 (Rotation._ro2cu,Rotation._cu2ro)])
+    def test_Rodrigues_internal(self,default,forward,backward):
+        """Ensure invariance of conversion from Rodrigues-Frank vector and back."""
+        cutoff = np.tan(np.pi*.5*(1.-1e-4))
+        for rot in default:
+            m = rot.as_Rodrigues()
+            o = backward(forward(m))
+            ok = np.allclose(np.clip(m,None,cutoff),np.clip(o,None,cutoff),atol=atol)
+            ok = ok or np.isclose(m[3],0.0,atol=atol)
+            print(m,o,rot.as_quaternion())
+            assert ok and np.isclose(np.linalg.norm(o[:3]),1.0)
+
+    @pytest.mark.parametrize('forward,backward',[(Rotation._ho2qu,Rotation._qu2ho),
+                                                 (Rotation._ho2om,Rotation._om2ho),
+                                                 #(Rotation._ho2eu,Rotation._eu2ho),
+                                                 (Rotation._ho2ax,Rotation._ax2ho),
+                                                 (Rotation._ho2ro,Rotation._ro2ho),
+                                                 (Rotation._ho2cu,Rotation._cu2ho)])
+    def test_homochoric_internal(self,default,forward,backward):
+        """Ensure invariance of conversion from homochoric vector and back."""
+        for rot in default:
+            m = rot.as_homochoric()
+            o = backward(forward(m))
+            ok = np.allclose(m,o,atol=atol)
+            print(m,o,rot.as_quaternion())
+            assert ok and np.linalg.norm(o) < _R1 + 1.e-9
+
+    @pytest.mark.parametrize('forward,backward',[(Rotation._cu2qu,Rotation._qu2cu),
+                                                 (Rotation._cu2om,Rotation._om2cu),
+                                                 (Rotation._cu2eu,Rotation._eu2cu),
+                                                 (Rotation._cu2ax,Rotation._ax2cu),
+                                                 (Rotation._cu2ro,Rotation._ro2cu),
+                                                 (Rotation._cu2ho,Rotation._ho2cu)])
+    def test_cubochoric_internal(self,default,forward,backward):
+        """Ensure invariance of conversion from cubochoric vector and back."""
+        for rot in default:
+            m = rot.as_cubochoric()
+            o = backward(forward(m))
+            ok = np.allclose(m,o,atol=atol)
+            if np.count_nonzero(np.isclose(np.abs(o),np.pi**(2./3.)*.5)):
+                ok = ok or np.allclose(m*-1.,o,atol=atol)
+            print(m,o,rot.as_quaternion())
+            assert ok and np.max(np.abs(o)) < np.pi**(2./3.) * 0.5 + 1.e-9
+
+    @pytest.mark.parametrize('vectorized, single',[(Rotation._qu2om,qu2om),
+                                                   (Rotation._qu2eu,qu2eu),
+                                                   (Rotation._qu2ax,qu2ax),
+                                                   (Rotation._qu2ro,qu2ro),
+                                                   (Rotation._qu2ho,qu2ho)])
+    def test_quaternion_vectorization(self,default,vectorized,single):
+        """Check vectorized implementation for quaternion against single point calculation."""
+        qu = np.array([rot.as_quaternion() for rot in default])
+        vectorized(qu.reshape(qu.shape[0]//2,-1,4))
+        co = vectorized(qu)
+        for q,c in zip(qu,co):
+            print(q,c)
+            assert np.allclose(single(q),c) and np.allclose(single(q),vectorized(q))
+
+
+    @pytest.mark.parametrize('vectorized, single',[(Rotation._om2qu,om2qu),
+                                                   (Rotation._om2eu,om2eu),
+                                                   (Rotation._om2ax,om2ax)])
+    def test_matrix_vectorization(self,default,vectorized,single):
+        """Check vectorized implementation for rotation matrix against single point calculation."""
+        om = np.array([rot.as_matrix() for rot in default])
+        vectorized(om.reshape(om.shape[0]//2,-1,3,3))
+        co = vectorized(om)
+        for o,c in zip(om,co):
+            print(o,c)
+            assert np.allclose(single(o),c) and np.allclose(single(o),vectorized(o))
+
+    @pytest.mark.parametrize('vectorized, single',[(Rotation._eu2qu,eu2qu),
+                                                   (Rotation._eu2om,eu2om),
+                                                   (Rotation._eu2ax,eu2ax),
+                                                   (Rotation._eu2ro,eu2ro)])
+    def test_Eulers_vectorization(self,default,vectorized,single):
+        """Check vectorized implementation for Euler angles against single point calculation."""
+        eu = np.array([rot.as_Eulers() for rot in default])
+        vectorized(eu.reshape(eu.shape[0]//2,-1,3))
+        co = vectorized(eu)
+        for e,c in zip(eu,co):
+            print(e,c)
+            assert np.allclose(single(e),c) and np.allclose(single(e),vectorized(e))
+
+    @pytest.mark.parametrize('vectorized, single',[(Rotation._ax2qu,ax2qu),
+                                                   (Rotation._ax2om,ax2om),
+                                                   (Rotation._ax2ro,ax2ro),
+                                                   (Rotation._ax2ho,ax2ho)])
+    def test_axis_angle_vectorization(self,default,vectorized,single):
+        """Check vectorized implementation for axis angle pair against single point calculation."""
+        ax = np.array([rot.as_axis_angle() for rot in default])
+        vectorized(ax.reshape(ax.shape[0]//2,-1,4))
+        co = vectorized(ax)
+        for a,c in zip(ax,co):
+            print(a,c)
+            assert np.allclose(single(a),c) and np.allclose(single(a),vectorized(a))
+
+
+    @pytest.mark.parametrize('vectorized, single',[(Rotation._ro2ax,ro2ax),
+                                                   (Rotation._ro2ho,ro2ho)])
+    def test_Rodrigues_vectorization(self,default,vectorized,single):
+        """Check vectorized implementation for Rodrigues-Frank vector against single point calculation."""
+        ro = np.array([rot.as_Rodrigues() for rot in default])
+        vectorized(ro.reshape(ro.shape[0]//2,-1,4))
+        co = vectorized(ro)
+        for r,c in zip(ro,co):
+            print(r,c)
+            assert np.allclose(single(r),c) and np.allclose(single(r),vectorized(r))
+
+    @pytest.mark.parametrize('vectorized, single',[(Rotation._ho2ax,ho2ax),
+                                                   (Rotation._ho2cu,ho2cu)])
+    def test_homochoric_vectorization(self,default,vectorized,single):
+        """Check vectorized implementation for homochoric vector against single point calculation."""
+        ho = np.array([rot.as_homochoric() for rot in default])
+        vectorized(ho.reshape(ho.shape[0]//2,-1,3))
+        co = vectorized(ho)
+        for h,c in zip(ho,co):
+            print(h,c)
+            assert np.allclose(single(h),c) and np.allclose(single(h),vectorized(h))
+
+    @pytest.mark.parametrize('vectorized, single',[(Rotation._cu2ho,cu2ho)])
+    def test_cubochoric_vectorization(self,default,vectorized,single):
+        """Check vectorized implementation for cubochoric vector against single point calculation."""
+        cu = np.array([rot.as_cubochoric() for rot in default])
+        vectorized(cu.reshape(cu.shape[0]//2,-1,3))
+        co = vectorized(cu)
+        for u,c in zip(cu,co):
+            print(u,c)
+            assert np.allclose(single(u),c) and np.allclose(single(u),vectorized(u))
+
+    @pytest.mark.parametrize('degrees',[True,False])
+    def test_Eulers(self,default,degrees):
+        for rot in default:
+            m = rot.as_quaternion()
+            o = Rotation.from_Eulers(rot.as_Eulers(degrees),degrees).as_quaternion()
+            ok = np.allclose(m,o,atol=atol)
+            if np.isclose(rot.as_quaternion()[0],0.0,atol=atol):
+                ok = ok or np.allclose(m*-1.,o,atol=atol)
+            print(m,o,rot.as_quaternion())
+            assert ok and np.isclose(np.linalg.norm(o),1.0)
+
+    @pytest.mark.parametrize('P',[1,-1])
+    @pytest.mark.parametrize('normalise',[True,False])
+    @pytest.mark.parametrize('degrees',[True,False])
+    def test_axis_angle(self,default,degrees,normalise,P):
+        c = np.array([P*-1,P*-1,P*-1,1.])
+        for rot in default:
+            m = rot.as_Eulers()
+            o = Rotation.from_axis_angle(rot.as_axis_angle(degrees)*c,degrees,normalise,P).as_Eulers()
+            u = np.array([np.pi*2,np.pi,np.pi*2])
+            ok = np.allclose(m,o,atol=atol)
+            ok = ok or np.allclose(np.where(np.isclose(m,u),m-u,m),np.where(np.isclose(o,u),o-u,o),atol=atol)
+            if np.isclose(m[1],0.0,atol=atol) or np.isclose(m[1],np.pi,atol=atol):
+                sum_phi = np.unwrap([m[0]+m[2],o[0]+o[2]])
+                ok = ok or np.isclose(sum_phi[0],sum_phi[1],atol=atol)
+            print(m,o,rot.as_quaternion())
+            assert ok and (np.zeros(3)-1.e-9 <= o).all() and (o <= np.array([np.pi*2.,np.pi,np.pi*2.])+1.e-9).all()
+
+    def test_matrix(self,default):
         for rot in default:
             m = rot.as_axis_angle()
             o = Rotation.from_axis_angle(rot.as_axis_angle()).as_axis_angle()
@@ -121,48 +801,66 @@ class TestRotation:
             if np.isclose(m[3],np.pi,atol=atol):
                 ok = ok or np.allclose(m*np.array([-1.,-1.,-1.,1.]),o,atol=atol)
             print(m,o,rot.as_quaternion())
-            assert ok and np.isclose(np.linalg.norm(o[:3]),1.0) and o[3]<=np.pi++1.e-9
+            assert ok and np.isclose(np.linalg.norm(o[:3]),1.0) and o[3]<=np.pi+1.e-9
 
-    def test_Rodrigues(self,default):
+    @pytest.mark.parametrize('P',[1,-1])
+    @pytest.mark.parametrize('normalise',[True,False])
+    def test_Rodrigues(self,default,normalise,P):
+        c = np.array([P*-1,P*-1,P*-1,1.])
         for rot in default:
             m = rot.as_matrix()
-            o = Rotation.from_Rodrigues(rot.as_Rodrigues()).as_matrix()
+            o = Rotation.from_Rodrigues(rot.as_Rodrigues()*c,normalise,P).as_matrix()
             ok = np.allclose(m,o,atol=atol)
             print(m,o)
             assert ok and np.isclose(np.linalg.det(o),1.0)
 
-    def test_Homochoric(self,default):
+    @pytest.mark.parametrize('P',[1,-1])
+    def test_homochoric(self,default,P):
         cutoff = np.tan(np.pi*.5*(1.-1e-4))
         for rot in default:
             m = rot.as_Rodrigues()
-            o = Rotation.from_homochoric(rot.as_homochoric()).as_Rodrigues()
+            o = Rotation.from_homochoric(rot.as_homochoric()*P*-1,P).as_Rodrigues()
             ok = np.allclose(np.clip(m,None,cutoff),np.clip(o,None,cutoff),atol=atol)
             ok = ok or np.isclose(m[3],0.0,atol=atol)
             print(m,o,rot.as_quaternion())
             assert ok and np.isclose(np.linalg.norm(o[:3]),1.0)
 
-    def test_Cubochoric(self,default):
+    @pytest.mark.parametrize('P',[1,-1])
+    def test_cubochoric(self,default,P):
         for rot in default:
             m = rot.as_homochoric()
-            o = Rotation.from_cubochoric(rot.as_cubochoric()).as_homochoric()
+            o = Rotation.from_cubochoric(rot.as_cubochoric()*P*-1,P).as_homochoric()
             ok = np.allclose(m,o,atol=atol)
             print(m,o,rot.as_quaternion())
             assert ok and np.linalg.norm(o) < (3.*np.pi/4.)**(1./3.) + 1.e-9
 
-    def test_Quaternion(self,default):
+    @pytest.mark.parametrize('P',[1,-1])
+    @pytest.mark.parametrize('accept_homomorph',[True,False])
+    def test_quaternion(self,default,P,accept_homomorph):
+        c = np.array([1,P*-1,P*-1,P*-1]) * (-1 if accept_homomorph else 1)
         for rot in default:
             m = rot.as_cubochoric()
-            o = Rotation.from_quaternion(rot.as_quaternion()).as_cubochoric()
+            o = Rotation.from_quaternion(rot.as_quaternion()*c,accept_homomorph,P).as_cubochoric()
             ok = np.allclose(m,o,atol=atol)
+            if np.count_nonzero(np.isclose(np.abs(o),np.pi**(2./3.)*.5)):
+                ok = ok or np.allclose(m*-1.,o,atol=atol)
             print(m,o,rot.as_quaternion())
             assert ok and o.max() < np.pi**(2./3.)*0.5+1.e-9
 
+    @pytest.mark.parametrize('reciprocal',[True,False])
+    def test_basis(self,default,reciprocal):
+        for rot in default:
+            om = rot.as_matrix() + 0.1*np.eye(3)
+            rot = Rotation.from_basis(om,False,reciprocal=reciprocal)
+            assert np.isclose(np.linalg.det(rot.as_matrix()),1.0)
+
     @pytest.mark.parametrize('function',[Rotation.from_quaternion,
                                          Rotation.from_Eulers,
                                          Rotation.from_axis_angle,
                                          Rotation.from_matrix,
                                          Rotation.from_Rodrigues,
-                                         Rotation.from_homochoric])
+                                         Rotation.from_homochoric,
+                                         Rotation.from_cubochoric])
     def test_invalid_shape(self,function):
         invalid_shape = np.random.random(np.random.randint(8,32,(3)))
         with pytest.raises(ValueError):
@@ -176,117 +874,12 @@ class TestRotation:
                                                  (Rotation.from_matrix,     np.random.rand(3,3)),
                                                  (Rotation.from_Rodrigues,  np.array([1,0,0,-1])),
                                                  (Rotation.from_Rodrigues,  np.array([1,1,0,1])),
-                                                 (Rotation.from_homochoric, np.array([2,2,2]))  ])
-    def test_invalid(self,function,invalid):
+                                                 (Rotation.from_homochoric, np.array([2,2,2])),
+                                                 (Rotation.from_cubochoric, np.array([1.1,0,0]))  ])
+    def test_invalid_value(self,function,invalid):
         with pytest.raises(ValueError):
             function(invalid)
 
-    @pytest.mark.parametrize('conversion',[Rotation.qu2om,
-                                           Rotation.qu2eu,
-                                           Rotation.qu2ax,
-                                           Rotation.qu2ro,
-                                           Rotation.qu2ho,
-                                           Rotation.qu2cu
-                                          ])
-    def test_quaternion_vectorization(self,default,conversion):
-        qu = np.array([rot.as_quaternion() for rot in default])
-        conversion(qu.reshape(qu.shape[0]//2,-1,4))
-        co = conversion(qu)
-        for q,c in zip(qu,co):
-            print(q,c)
-            assert np.allclose(conversion(q),c)
-
-    @pytest.mark.parametrize('conversion',[Rotation.om2qu,
-                                           Rotation.om2eu,
-                                           Rotation.om2ax,
-                                           Rotation.om2ro,
-                                           Rotation.om2ho,
-                                           Rotation.om2cu
-                                          ])
-    def test_matrix_vectorization(self,default,conversion):
-        om = np.array([rot.as_matrix() for rot in default])
-        conversion(om.reshape(om.shape[0]//2,-1,3,3))
-        co = conversion(om)
-        for o,c in zip(om,co):
-            print(o,c)
-            assert np.allclose(conversion(o),c)
-
-    @pytest.mark.parametrize('conversion',[Rotation.eu2qu,
-                                           Rotation.eu2om,
-                                           Rotation.eu2ax,
-                                           Rotation.eu2ro,
-                                           Rotation.eu2ho,
-                                           Rotation.eu2cu
-                                          ])
-    def test_Euler_vectorization(self,default,conversion):
-        eu = np.array([rot.as_Eulers() for rot in default])
-        conversion(eu.reshape(eu.shape[0]//2,-1,3))
-        co = conversion(eu)
-        for e,c in zip(eu,co):
-            print(e,c)
-            assert np.allclose(conversion(e),c)
-
-    @pytest.mark.parametrize('conversion',[Rotation.ax2qu,
-                                           Rotation.ax2om,
-                                           Rotation.ax2eu,
-                                           Rotation.ax2ro,
-                                           Rotation.ax2ho,
-                                           Rotation.ax2cu
-                                          ])
-    def test_axisAngle_vectorization(self,default,conversion):
-        ax = np.array([rot.as_axis_angle() for rot in default])
-        conversion(ax.reshape(ax.shape[0]//2,-1,4))
-        co = conversion(ax)
-        for a,c in zip(ax,co):
-            print(a,c)
-            assert np.allclose(conversion(a),c)
-
-
-    @pytest.mark.parametrize('conversion',[Rotation.ro2qu,
-                                           Rotation.ro2om,
-                                           Rotation.ro2eu,
-                                           Rotation.ro2ax,
-                                           Rotation.ro2ho,
-                                           Rotation.ro2cu
-                                          ])
-    def test_Rodrigues_vectorization(self,default,conversion):
-        ro = np.array([rot.as_Rodrigues() for rot in default])
-        conversion(ro.reshape(ro.shape[0]//2,-1,4))
-        co = conversion(ro)
-        for r,c in zip(ro,co):
-            print(r,c)
-            assert np.allclose(conversion(r),c)
-
-    @pytest.mark.parametrize('conversion',[Rotation.ho2qu,
-                                           Rotation.ho2om,
-                                           Rotation.ho2eu,
-                                           Rotation.ho2ax,
-                                           Rotation.ho2ro,
-                                           Rotation.ho2cu
-                                          ])
-    def test_homochoric_vectorization(self,default,conversion):
-        ho = np.array([rot.as_homochoric() for rot in default])
-        conversion(ho.reshape(ho.shape[0]//2,-1,3))
-        co = conversion(ho)
-        for h,c in zip(ho,co):
-            print(h,c)
-            assert np.allclose(conversion(h),c)
-
-    @pytest.mark.parametrize('conversion',[Rotation.cu2qu,
-                                           Rotation.cu2om,
-                                           Rotation.cu2eu,
-                                           Rotation.cu2ax,
-                                           Rotation.cu2ro,
-                                           Rotation.cu2ho
-                                          ])
-    def test_cubochoric_vectorization(self,default,conversion):
-        cu = np.array([rot.as_cubochoric() for rot in default])
-        conversion(cu.reshape(cu.shape[0]//2,-1,3))
-        co = conversion(cu)
-        for u,c in zip(cu,co):
-            print(u,c)
-            assert np.allclose(conversion(u),c)
-
     @pytest.mark.parametrize('direction',['forward',
                                           'backward'])
     def test_pyramid_vectorization(self,direction):
@@ -300,3 +893,55 @@ class TestRotation:
         f = Rotation._get_pyramid_order(a,'forward')
         b = Rotation._get_pyramid_order(a,'backward')
         assert np.all(np.take_along_axis(np.take_along_axis(a,f,-1),b,-1) == a)
+
+
+    @pytest.mark.parametrize('data',[np.random.rand(5,3),
+                                     np.random.rand(5,3,3),
+                                     np.random.rand(5,3,3,3,3)])
+    def test_rotate_vectorization(self,default,data):
+        for rot in default:
+            v = rot.broadcast_to((5,)) @ data
+            for i in range(data.shape[0]):
+                print(i-data[i])
+                assert np.allclose(mul(rot,data[i]),v[i])
+
+
+    @pytest.mark.parametrize('data',[np.random.rand(3),
+                                     np.random.rand(3,3),
+                                     np.random.rand(3,3,3,3)])
+    def test_rotate_identity(self,data):
+        R = Rotation()
+        assert np.allclose(data,R*data)
+
+    @pytest.mark.parametrize('data',[np.random.rand(3),
+                                     np.random.rand(3,3),
+                                     np.random.rand(3,3,3,3)])
+    def test_rotate_360deg(self,data):
+        phi_1 = np.random.random() * np.pi
+        phi_2 = 2*np.pi - phi_1
+        R_1 = Rotation.from_Eulers(np.array([phi_1,0.,0.]))
+        R_2 = Rotation.from_Eulers(np.array([0.,0.,phi_2]))
+        assert np.allclose(data,R_2*(R_1*data))
+
+    @pytest.mark.parametrize('data',[np.random.rand(3),
+                                     np.random.rand(3,3),
+                                     np.random.rand(3,3,3,3)])
+    def test_rotate_inverse(self,data):
+        R = Rotation.from_random()
+        assert np.allclose(data,R.inversed()*(R*data))
+
+    @pytest.mark.parametrize('data',[np.random.rand(4),
+                                     np.random.rand(3,2),
+                                     np.random.rand(3,2,3,3)])
+    def test_rotate_invalid_shape(self,data):
+        R = Rotation.from_random()
+        with pytest.raises(ValueError):
+            R*data
+
+    @pytest.mark.parametrize('data',['does_not_work',
+                                     (1,2),
+                                     5])
+    def test_rotate_invalid_type(self,data):
+        R = Rotation.from_random()
+        with pytest.raises(TypeError):
+            R*data