A boundary can be described by the orientation of the boundary to the two grains and the 3-D rotation required to bring the grains into coincidence. Thus a boundary has 5 macroscopic degrees of freedom. However, it is common to describe a boundary only as the orientation relationship of the neighbouring grains. Generally, the convenience of ignoring the boundary plane orientation, which is very difficult to determine, outweighs the reduced information.
The relative orientation of the two grains is described using the rotation matrix:
Using this system the rotation angle θ is:
while the direction [uvw] of the rotation axis is:
The nature of the crystallography involved limits the misorientation of the boundary. A completely random polycrystal, with no texture, thus has a characteristic distribution of boundary misorientations (see figure). However, such cases are rare and most materials will deviate from this ideal to a greater or lesser degree.