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Comment by Certhas | Hacker News Reader

I don't get the love of quaternions in this context. How are quaternions more intuitive than matrices?

A matrix acts on a vector. Rotations act on vectors. What could be more natural than to look at rotations as matrices?

The matrix exponential then makes all this intuitive to work with, if you connect it to solutions of ODEs: dx/dt = Ax has solution exp(t A). If A is skew symmetric, then the change in x is orthogonal to x at all times. So it's a rotation that doesn't change the length. (Alternative: apply a small orthogonal change over and over: (1 + delta A)^N x)

Lie Groups/Algebras vastly generalize this, and you can call skew symmetric matrices geometric algebra, but that's not important. Important is that you generate rotations by continuously making orthogonal changes, and that the exponential map describes this process. I find this picture far more geometric and intuitive than anything else.

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Might be worth pointing out that quaternions themselves are often represented as matrices as well: That’s the Pauli matrices, widely used to model quantum spin.

One advantage of quaternions is that it’s (IMO) easier to calculate things with them using only pen and paper than it is to do equivalent matrix multiplications by hand.

Personally I find them “intuitive” in the same sense as complex numbers are intuitive: They seemed weird on first exposure, but they now feel simpler to use and reason about than the alternatives I know.

The article explained a major advantage of quaternions over matrices. Quaternions interpolate well and matrices don’t. This property is very important for animation and other things in computer graphics, like computing frames along a 3d spline curve.

If you are referring to slerp, the matrix version is trivial using the log and exp maps.

    slerp(A, B, t) = exp(t log(B A^T)) A

Something more complex like splines, I am not familiar with the algorithms used for quaternions. I would bet they use Lie algebraic operations, but do they actually use anything specific to quaternions?

Not disagreeing that quaternions have some computational advantages over matrices, but most of the points people repeat as "advantages of quaternions" are really "advantages of proper Lie group/algebra operations over Euler angles" that are independent of matrix vs. quaternion representation.

Very well put.

The article also explains that the natural way to look at this property is in terms of matrix exponentials/logarithm.

The space of rotation matrices is not a linear space. It's obvious that interpolation is meaningless there. The space of skew symmetric matrices is linear, so you can interpolate the generators of rotations naturally.

Without this perspective, how do you even begin to understand what this interpolation does or what it actually means?

One advantage of quaternians from a computational perspective is that they are 4 numbers vs 9 for a 3x3 matrix (and applying the rotations has a similar reduction in flops)

The main advantage of quaternions is composing rotations.

The analogy would be to 2D rotations with complex numbers. When you multiply two complex numbers, you're composing the rotations (in the 2D case: it ends up just adding the arg, or the phase angle). Likewise, multiplying two quaternions lets you compose the 3D rotations. This is a lot more efficient than multiplying two 3x3 matrices.

For intuition, quaternions are closely related to the axis-angle representation which is the same as the Lie algebra so(3).

As for acting on vectors, you can just think of different rotation parameterizations as implementations of the same abstract Rotation trait. A Rotation acts on vectors, composes, etc, in exactly the same way regardless if the underlying implementation is a matrix, a quaternion, euler vector, euler angles, gibbs vector, etc.

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