The most widely-used concept of "average" is surely a point that minimizes the sum of squared distances to each of a list of input points. Distances are canonically defined in the space of rotations, and so are averages in this sense (not always uniquely).
Commutativity has nothing to do with this; do not confuse the typical formula for averaging with the reason for doing so! Of course, there are other senses of "average" (which generally do continue to apply to the space of rotations as well).
The application for this given by GreedCtrl's reference is to spline interpolation. Another is in robotics, when combining multiple noisy observations of the orientation of an object.
>Distances are canonically defined in the space of rotations
I am sorry, but this is simply not true.
There are many distance/metric definitions that are applicable to the space of rotations, and the best choice of metric is defined by the application, which is why I asked that question.
None of them is any more "canonical" than the other. See [1][2][3] for an introduction and comparison.
One will find there at least four "canonical" distance definitions, and applications ranging from optometry to computer graphics to IK (which is what you referred to).
>The most widely-used concept of "average" is surely a point that minimizes the sum of squared distances...Of course, there are other senses of "average" (which generally do continue to apply to the space of rotations as well).
I know this, not all of the readers may.
What I don't know is what context the parent is coming from.
Maybe all they need is interpolating between two camera positions - which is a much simpler problem than the paper they found (and what we're discussing) is addressing.
>The application for this given by GreedCtrl's reference is to spline interpolation.
It is not clear that the reference that they have found is actually the best for their application - they only said it was something they found, and that the article we're discussing looks "simpler" for their level of mathematics.
The article we are discussing does not provide any means of "averaging" any more than two rotations, though, which motivated my question.
The bi-invariant metric as pointed out by chombier is what I have in mind. I agree that a non-canonical metric may be the right one for some applications, but those are the exceptional ones. The bi-invariant metric (which has a simple, geometric meaning given by Euler's rotation theorem) is the right starting point for thinking about distances in this space.
(Your reference [2] supports this point of view: read "simplest" as "canonical". Your reference [1] claims five distinct bi-invariant metrics, but this is wrong. The argument given is that any metric related to a bi-invariant metric by a "positive continuous strictly increasing function" is itself bi-invariant, which is not true.)
> >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
Commutativity has nothing to do with this; do not confuse the typical formula for averaging with the reason for doing so! Of course, there are other senses of "average" (which generally do continue to apply to the space of rotations as well).
The application for this given by GreedCtrl's reference is to spline interpolation. Another is in robotics, when combining multiple noisy observations of the orientation of an object.