MTEX offers the following functions to define rotations
At the end all functions return a variable of type
rotation which represents a list of rotations that are internaly stored as quaternions. An overview of different rotation representations by three dimensional vectors and their properties can be found in the section Representations.
One of the most common ways to describe a rotation is as three subsequent rotations about fixed axes, e.g., first around the z axis, second around the x axis and third again around the z. The corresponding rotational angles are commonly called Euler angles. Beside the most common
ZXZ covention other choices of the axes are sometimes used. Sorted by popularity in the texture analysis community these are
- Bunge (phi1,Phi,phi2) - ZXZ
- Matthies (alpha,beta,gamma) - ZYZ
- Roe (Psi,Theta,Phi)
- Kocks (Psi,Theta,phi)
- Canova (omega,Theta,phi)
The default Euler angle convention in MTEX are the Bunge Euler angles, with axes Z, X, and Z. The following command defines a rotation by its three Bunge Euler angles
Note that the angles needs to be multiplied with degree since all commands in MTEX expect the input in radiant. Furthermore, the order of the first and the third Euler angle are interchanged in comparison to standard notation for reasons explained here.
In order to define a rotation by a Euler angle convention different to the default Euler angle convention you to specify the convention as an additional parameter, e.g.
This does not change the way MTEX displays the rotation on the screen. The default Euler angle convention for displaying a rotation can be changed by the command setMTEXpref, for a permanent change the mtex_settings should be edited. Compare
Axis angle parametrisation and Rodrigues Frank vector
A very simple possibility to specify a rotation is to specify the rotation axis and the rotation angle.
Conversely, we can extract the rotational axis and the rotation angle of a rotation by
Closely related to the axis angle parameterisation of a rotation is the Rodriguess Frank vector.
This is the rotational axis scaled by \(\tan \omega/2\), where \(\omega\) is the rotational angle.
We can also define a rotation by a Rodrigues Frank vector by
Another common way to represent rotations is by 3x3 matrices. The column of such a rotation matrix coincide with the new positions of the x, y and z vector after the rotation. For a given rotation we may compute the matrix by
Conversely, we may define a rotation by its matrix with the command
Four vectors defining a rotation
Another useful method to define a rotation is by describing how in acts on two given directions. More precisely, given four vectors u1, v1, u2, v2 there is a unique rotation
rot such that
rot * u1 = v1 and
rot * u2 = v2. E.g., to find the rotation the maps the x-axis onto the y-axis and keeps the z-axis we do
The above definition require that the angle between u1 and u2 is the same as between v1 and v2. The function gives an error if this condition is not meet. If only two vectors are specified, then the rotation with the smallest angle is returned that rotates the first vector onto the second one.
More generaly, one can fit a rotation
rot to a list of left and right vectors
r such that
rot * l is the best approximation of
r. This is done by the function
MTEX offers several ways for generating random rotations. In the most
Logarithm, Exponential Mapping, Spin Tensor
A last possibility to define a rotation is by quaternion coordinates a, b, c, d.