In MTEX a crystal orientation is defined as the rotation that transforms crystal coordinates, i.e., a description of a vector or a tensor with respect to the crystal reference frame, into specimen coordinates, i.e., a desciption of the same object with respect to a specimen fixed reference frame.
In MTEX any orientation consists of two incredients. A rotation
and a description of the crystal lattice, which are represented in MTEX by variables of type crystalSymmetry
Combining both incredients allows us to define an orientation
Crystal coordinates to specimen coordinates
Let us consider to following direction with respect to the crystal reference system
Then in a grain with orientation
ori this direction
h has with respect to the specimen reference system the coordinates
Similarly, orientations transform tensors given with respect to the crystal reference frame, e.g., the following single crystal stiffness tensor
into a stiffness tensor with respect to the specimen reference frame
Objects that can be translated by orientations from crystal into specimen cooordinates and vice verca include
Specimen coordinates into crystal coordinates
Conversely, we can go back from specimen coordinates to crystal coordinates by multiplying with the inverse orientation
Note, that in literature orientations are often defined to transform specimen coordinates into crystal coordinates, i.e., to coincide with the inverse orientations in MTEX. The consequences of this differences are exhaustivly discussed in the topic orientation convemtions.
Rotations of the specimen ,i.e., changing the specimen reference frame, do also change the orientation. Assume the specimen is rotated about the X-axis about 60 degree. We may define this rotation by
Then an orientation
ori is updated to the rotated reference frame by
It should also be noted, that orientations are sensitiv with respect to the alignment of the Euclidean reference frame \(\vec X\), \(\vec Y\), \(\vec Z\) with respect to the crystal axes \(\vec a\), \(\vec b\) and \(\vec c\). This issue is discussed in more detail in the topic The crystal reference system.