I always wondered why bother with quaternions and abstruse math when at the end of the day they behave like some matrices of floats.
The advantages I found are:
Unit quaternions represent rotations with less redundancy than matrices
It's easier and more intuitive to derive, manipulate, simplify, interpolate, and solve quaternions than matrices.
They're less abstract/more concrete than matrices.
Rotating with quaternions takes fewer multiplications than matrices.
Dual quaternion inverse kinematics are easier to derive and faster than matrices
Unit and Dual quaternions have an efficient implicit form, which further speeds up IK. See: https://www.researchgate.net/profile/Neil-Dantam/publication...
Normally it happens because we have to solve a rotation into potentially redundant roll, pitch and yaw factors.
In Q, you can just write the axis of rotation as a unit vector quaternion v = v1 i + v2 j + v3 k. That pure vector quaternion represents a 180 degree rotation around v. Other rotations around v are interpolations between v and the identity rotation scalar 1:
w = a1 + bv, where a^2 + b^2 = 1.
This is a circular analogue of linear interpolation. Indeed, if t in [0, 1] is our rotation angle, we observe:
a = cos(pi t) b = sin(pi t)
And we can reduce
w = a + sqrt(1-a^2)*v
Where a \in [-1, 1].
This representation is so nice. There is no need to solve roll, pitch, and yaw. Just pick unit rotation axis v, then twist the scalar knob a to set the rotation amount.
This is quite human!
Imaginary numbers aren't "imaginary" they are how to do consistent math in a 2D framework. Quaternions? Well that's about 4D. Poincaré once said that math isn't about numbers, but relationships between numbers. For some reason we don't talk about mathematical structure (explicitly) until the late stages. I wouldn't necessarily call these concepts "abstruse" but they are a bit more abstract. A big part of the problem though, we often ignore the ground we are building upon and so when you finally look at it, it is new and confusing. But then again, the success of Bourbaki's New Math is arguable[0]
[0] https://en.wikipedia.org/wiki/New_Math#In_other_countries
He described math more poetically and profoundly: "the art of giving the same name to different things"
Quaternions are about 3D. You need 4 numbers for it, but the space that they operate on is 3D.
(Unit) quaternions happens to work as rotations in 3D space. But quaternions' algebraic structure is indeed 4D, just like imaginary numbers' is 2D.
In math, this is called a representation of a Lie group, and there are representations in all dimensions.
https://en.wikipedia.org/wiki/Transformation_matrix#Examples...
I think most of the difference in mastery of a given topic in an academic setting comes down to the skill and interactivity with a competent and expert lecturer who also has expertise in the additional domains of public speaking and teaching. Ken Joy and Sean Davis were the virtuosos of teaching at UC Davis: massive and accessible brains. Intellectual curiosity and an semi-extroverted personality help too.
This. If you only want to understand *how to apply* quaternions to represent rotations, it's not too hard to learn. However, if you want to understand *why* they work, that's a totally different story.
Many of my colleagues still use Euler angles to represent rotations. I always use quaternions, and am a bit baffled why people are so averse to them.
Quaternions are the generators of SU(2) which is a double covering of SO(3). The latter describes rotations in 3 dimensions. Thus you can express any rotation in 3d with quaternions.
The idea is that quaternions are the wrong way to look at the problem and that a better approach would be to use geometric algebra, bivectors and rotors. The formulas are essentially the same as with quaternions, but at least for me, this approach make more intuitive sense. It also work in dimensions other than 3, which matter to the author as he is the author of "4D toys" and hopefully, Miegakure.
Here is another talk on the subject, by a different author: https://www.youtube.com/watch?v=htYh-Tq7ZBI It is not specifically about quaternions, it is about multiplying vectors and what you get from that, and it includes quaternions.
This is essentially because quaternions are a remarkably good representation of rotations.
Rotation quaternions (i.e. unit quaternions) are simply the unit sphere in 4 dimensional space.