> >Distances are canonically defined in the space of rotations
> I am sorry, but this is simply not true.
It is true, there is a canonical choice given by the bi-invariant Riemannian metric on compact Lie groups, such as rotations (in this case the shortest angle between rotations)
Whether or not you want this metric in practice is another problem, of course.
> The article we are discussing does not provide any means of "averaging" any more than two rotations,
The Karcher/Fréchet mean described in the original article does average more than two rotations
> I am sorry, but this is simply not true.
It is true, there is a canonical choice given by the bi-invariant Riemannian metric on compact Lie groups, such as rotations (in this case the shortest angle between rotations)
Whether or not you want this metric in practice is another problem, of course.
> The article we are discussing does not provide any means of "averaging" any more than two rotations,
The Karcher/Fréchet mean described in the original article does average more than two rotations