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Comment by nraynaud | Hacker News Reader

I had a question about quaternions: does anyone use them for anything else than multiplying a rotation by a scalar?

In particular, it feels a bit like a waste of coding space to always use unit ones.

The remarkable thing about quaternions is that you can describe rotation around any arbitrary axis! Try doing that with real rotation matrices and you will bump into a major problem:

https://en.wikipedia.org/wiki/Gimbal_lock

Because matrix multiplication is not commutative, you can't easily compose a matrix representing rotation around an arbitrary axis from "component rotation matrices".

With quaternions/clifford algebras, you can say "here's the vector I want to rotate about and here's this is how much I want to rotate", and it just magically works.

I find that many articles explaining how to describe the orientation of a rigid body rotated along an axis by an angle in a confusing manner.

Normally in 3d space you will have to construct quaternions for rotations along yaw, pitch and roll, and then take their product to get the quaternion of the orientation of the rigid body.

That is, when using quaternions to describe orientations,we are actually describing the rotations done to bring the body from its default orientation to its present orientation.

> Normally in 3d space you will have to construct quaternions for rotations along yaw, pitch and roll, and then take their product to get the quaternion of the orientation of the rigid body

Is this not just using Euler angles via quaternions? If I understand correctly, tracking rotation via yaw, pitch, and roll will still run into issues of gimbal lock because it's the same parameterization just using quaternions.

> The remarkable thing about quaternions is that you can describe rotation around any arbitrary axis! Try doing that with real rotation matrices and you will bump into a major problem

https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matri...

I understand you can represent any rotation with matrices, but it's very awkward, and when you compose rotations, you quickly lose information about the resulting axis of rotation.

The information is still there. The axis of rotation is the eigenvector with eigenvalue 1. (That vector doesn't move when you apply the rotation)

Okay let me rephrase: the axis of rotation becomes relatively obfuscated!

I guess it doesn't matter if you're doing everything on the computer, but it's pretty cool that with quarternions you can immediately see the composite rotation's axis and angle without any extra work!

how is that an example of using quaternion for something else than rotations?

Not quite...

Gimbal lock is no problem of rotation matrices but of euler angles. Just look up the wikipedia page you posted: "The problem of gimbal lock appears when one uses Euler angles in applied mathematics"

You can represent any (any!) rotation with rotation matrices.

Quaternions are neither commutative. Fortunately you don't need commutativity to build up arbitrary rotations.

Thanks, I realize now that the non-commutativity I was noticing is more a property of SO(3), which can be seen both in quaternion and matrix representations of rotations.

The non-commutativity of rotations can also be seen directly: rotating something 90 degrees about the x then the y axis is not the same as the y then the x.

I was just doing this earlier with my arms

Some control systems for air and space craft are designed with quaternions - here the problem being solved isn't a programming problem but a math problem involving the dynamics of the craft and/or orbital dynamics.

As far as I know they are also only doing SLERP for attitude control.

I am not sure what you mean.

Attitude control is not an example of using quaternions for something else than rotations.

There does not exist a R^3 parameterization of the 3d rotations without some problem. https://en.m.wikipedia.org/wiki/Charts_on_SO(3)#Parametrizat...

But quaternions are pretty good. They're a double cover but angles are an "infinity cover" of SO(2) and everyone understands angles. For animation purposes, being a single-cover is arguably a problem. An example in 2D is to suppose you have a rotation sequence

    0° -> 90° -> 180° -> 270° -> X

and you want to return to the original state. Well, you can do that either by continuing to rotate in the same direction, corresponding to X=360° or by going back along the path you came corresponding to X=0°. These alternatives are unrepresentable in a single cover. (For angles you also have X=-360°, 720°, etc. corresponding to making any number of revolutions in either direction before coming to rest at the desired target, which if you think is weird makes quaternions an even better chart on SO(3) than angles are on SO(2)).

I agree quaternions are a good parameterization of SO(3). But they can still cause unexpected problems, e.g. any machine learning task involving quaternions will be made harder because the learned function must deal with the double covering.

Of course, the same problem exists for angles in SO(2), and from what I've seen people usually deal with that by using the [sin, cos] pair instead.

You can also use the for making pretty pictures of Quaternion Julia fractals. It uses non-unit quaternions, but typically intersects a plane with the 4D solid:

http://paulbourke.net/fractals/quatjulia/

Apparently there's also an application in multiantenna radio transmission: http://www.ece.ualberta.ca/~hcdc/Library/StCommClass/Hugh00....

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