Comment by wyager | Hacker News Reader

Comment by wyager | Hacker News Reader

> unit quaternions also don’t form a vector space

Unit quaternions are the lie group. If you want something you can add willy-nilly, you want the lie algebra of all quaternions, which represent rotational velocities, just as axis-angle represents rotational velocities.

Comparing unit quaternions to axis-angle is a bit of a category error - it would be more appropriate to compare unit quaternions to rotation matrices, and all quaternions to axis-angle.

> One advantage of using quaternions is how easy the exponential map is to compute—if you don’t need a rotation matrix, it’s a good option.

You rarely need a rotation matrix at all when using quaternions. As mentioned in the article, you can compute rotations using `pqp^-1`.

I think the easiest way to understand quaternions is just to read about geometric algebra. It took hundreds of years to invent quaternions, but once you understand geometric algebra (which is shockingly simple), you can invent quaternions in just a few minutes. I found this article to be a good intro several years ago: https://crypto.stanford.edu/~blynn/haskell/ga.html

I think the exponential map is still the most robust way to think about rotations, since it gives you tools to deal with the Lie group of SO(3) in the most straightforward way (switching between coordinates, dealing with differentiation and tangent spaces, etc.)

Even when going through all the formulations with geometric algebra, you still land on using rotors (isomorphic to quaternions) and motors (isomorphic to dual quaternions) to represent SO(3) / SE(3) spaces - but I think for that purpose 3x3 rotation matrices / 4x4 transformation matrices with exponential maps are still much more useful. (Quaternions do have an advantage that it stores up less space and also faster to multiply with each other - but when you want to transform points with it then matrices are still faster. In overall in terms of efficiency it really depends on the situation.)

When you say "all quaternions" did you mean "pure imaginary quaternions"?

Oops, yes I did.

For curious readers: `2` is a quaternion, but `exp(2)` doesn't generate a rotation, it generates 7.389... just like normal exponentiation on reals. So if you have a real term in your lie algebra, it will generate things that scale the object as well as rotating it.

So instead you want only `ai + bj + ck`, as `exp(ai + bj + ck)` will always have magnitude 1

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