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Let O∈ SO(3) be a random rotation from a UARS distribution with central rotation S. Then O=S P, where P is obtained by rotating the 3×3 identity matrix, I 3×3, about an axis (mathbf {U} in mathbb {R}^{3}) by a random angle r∈(−π,π].
The central rotation was fixed at the identity matrix, since choice of center does not have a bearing on the rotational symmetry.
In many applications involving 3-D rotation data, the rotational distributions used are symmetric (or isotropic) so that they have rotationally invariant densities about a central rotation.
The central rotation was fixed at the identity matrix and the preferred axis of rotation V was generated from the uniform distribution on the sphere for each simulation.
Recent works have focused on estimation of the central rotation (Will and Bingham 2015; Stanfil et al. 2015), but no attention has been given to using bootstrapping for estimating spread.
Stanfill et al. (2015) and Will and Bingham (2016) focus on estimating the central rotation through bootstrapping, while Bero and Bingham (2015) explore permutation tests for comparing the central rotations in two data sets.
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