What's your context for averaging multiple rotations?
Averages are something that you can do with sums, where order of composition doesn't matter.
Rotations don't commute, so the concept of average as we understand it doesn't normally apply to them.
If you're holding a phone: rotate the screen away from you 180° (so you'd be seeing the back), then clockwise 90° relative to the ground (around vertical axis).
Your camera will be facing your left.
Now hold the phone normally, and do the same rotations in the opposite order. Your camera will now be facing your right.
Question: where should the camera be facing in the "average" of these two rotations?
There's no single answer — it depends on what you need from the average.
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
> Rotations don't commute, so the concept of average as we understand it doesn't normally apply to them.
It does apply, considering that the Euclidean mean minimizes the sum of squared lengths to samples there's a fairly obvious generalization to Riemannian manifolds using geodesic distance.
There are some reasonable assumptions to be made on the manifold and/or sample distribution to ensure convergence but in practice it works pretty well.
If you start with the phone upright and rotate the screen away from you by turning the phone around the vertical axis, then both rotations are around the same axis and of course they do commute.
My guess is that romwell is holding the phone flat, so that the rotation away from you is about a horizontal axis; then you should experience the noncommutativity.
(The resulting orientations are 180 degrees apart, which indeed makes it difficult to say that any one orientation should be the unique average. But this is due to the geodesic structure of the space of rotations, not the noncommutative product that happened to construct these points, see above.)
Draw a line on the screen from top to bottom. You interpreted "rotate away" as turning it around this axis, which is the same axis you used for the 90-degree clockwise turn. You end up with the screen right-side-up, just facing away from you. It's the same thing I intuitively did.
Now draw a line on the screen from left to right. "Rotate away" by turning the phone around this new axis - so the top of the phone moves away from you, and the bottom of the phone moves closer to you. You end up with the screen upside-down, and also facing away from you.
Can confirm. Followed instructions, got the camera facing the same in both cases (left, with phone upside down). I interpreted the 180 part as flipping the phone around the horizontal axis.
The 180 first part was right. Make sure you are rotating 90 degrees in the same direction both times from your frame of reference (clockwise looking from the top).
I'm making a video game, and I want to map a joystick position to a rotation (of a sword). I have a set of keyframes for certain joystick positions, and I want to interpolate between the three nearest ones to create a continuous function from joystick position to sword rotation.
I.e., is it a point on a hemisphere (equivalently, X/Y controller)?
And what kind of rotations are you talking about for the sword? Are they constrained in any way? I'm assuming it's attached to something - is that point moving too?
Finally why do you need to interpolate? Why three? What are you interpolating between?
I.e. why not simply convert the input of the joystick to a rotation - you're literally rotating a stick when you're moving a joystick in the first place.
You don't need interpolation to convert a rotation (of the joystick relative to its base) to the rotation of the sword unless you're doing something that I don't understand.
If you describe what you're doing in detail, I think we might save you a deep dive into deeper math than necessary for this (..or dive into even deeper math :).
Averages are something that you can do with sums, where order of composition doesn't matter.
Rotations don't commute, so the concept of average as we understand it doesn't normally apply to them.
If you're holding a phone: rotate the screen away from you 180° (so you'd be seeing the back), then clockwise 90° relative to the ground (around vertical axis).
Your camera will be facing your left.
Now hold the phone normally, and do the same rotations in the opposite order. Your camera will now be facing your right.
Question: where should the camera be facing in the "average" of these two rotations?
There's no single answer — it depends on what you need from the average.