Quaternions (The Class quaternion)

class representing orientations

Calculating with three dimensional vectors

Class Description

The class quaternion allows working with rotations in MTEX, as they occur e.g. as crystal orientation or symmetries. Quaternions may be multiplied with three-dimensional vecotors which means rotating the vector or may be multiplied with another quaternion which means to concatenate both rotations.

Defining quaternions

The standard way is to define a quaternion q is to give its coordinates (a,b,c,d). However, making use of one of the following conversion methods is much more human readable.

q = quaternion(a,b,c,d)          % by coordinates
q = axis2quat(axis,angle);       % by rotational axis and rotational angle
q = euler2quat(alpha,beta,gamma) % by Euler angles
q = Miller2quat([h k l],[u v w],symmetry); % by Miller indece
q = quaternion.id;                % identical quaternion
q = vec42quat(u1,v1,u2,v2);      % by four vectors

Additional methods to define a rotation are hr2quat and vec42quat. Using the brackets q = [q1,q2] two quaternions can be concatened. Now each single quaternion is accesable via q(1) and q(2).

Calculating with three dimensional vectors

Besides the standard linear algebra operations there are also the following functions available in MTEX.

angle(q); % rotational angle
axis(q);  % rotational axis
inverse(q);  % inverse rotation

Conversion

There are methods to transform quaternion in almost any other parameterization of rotations as they are:

Euler(q)     % in Euler angle
Rodrigues(q) % in Rodrigues parameter

Plotting quaternions

The plot function allows you to visualize an quaternion by plotting how the standard basis x,y,z transforms under the rotation.

plot(quaternion.rand(100))

Complete Function list

q = quaternion(varargin)