The easiest way to understand quaternions is as quotients of 3-dimensional vectors.
That is, if Q = u/v, then Q is the “quaternion” (complex sum of a scalar and a bivector) which rotates and scales v into u. Or written out, Qv = (u/v)v = u. The bivector part of Q has the same orientation as the plane spanning u and v.
If we want to use that quaternion as a representation of a general rotation of 3-dimensional space, we need to use a “sandwich product” because just multiplying Q directly by an arbitrary vector will rotate and scale the portion of the vector parallel to the plane of u–v, but then also produce a trivector component from the perpendicular part of the vector, which is not what we wanted.
If we define R = √Q, then we can multiply any vector a like RaR`, where R` is the conjugate of R, and get the scaling & rotation operation we were looking for. Multiplication of any vector perpendicular to the plane of u–v by this sandwich will just scale it by the magnitude of Q, but not change its direction, that is, for a vector b perpendicular to the plane of u–v, RbR` = |Q|b, whereas for a vector c parallel to the plane of u–v, RcR` = Qc, or in other words c gets rotated by the angle between v and u, as intended.
The square root here is where the half angle measures come from.
> The easiest way to understand quaternions is as quotients of 3-dimensional vectors.
Only for those who already know geometric algebra, I think. People coming from vector geometry and complex numbers will find other explanations much simpler.
Note also that your explanation in terms of a vector quotient is effectively a definition of vector quotients, and hence doesn't really explain.
Sure. You can’t only drop literally only my comment on someone who has never seen this subject before.
But in my experience, trying to explain the basics of 2- and 3-dimensional geometric algebra to someone followed by explaining quaternion rotation results in a lot less confusion (and might even save time overall, while resulting in dramatically richer understanding) versus trying to explain quaternions qua quaternions. It’s also very helpful for people who have worked with quaternions before in code but previously treated them as a mysterious black box.
If you explain how (multi)vector multiplication (Clifford product) works, then the behavior of quaternions can be clearly explained, and makes straight-forward geometrical sense.
If you just dive into talking about quaternions, they seem entirely arbitrary and mystical.
In neither case will the typical person be able to figure out what is going on instantly, without thinking about it for themselves.
Disclaimer: the above is based on my pretty limited, anecdotal experience. I would be interested to see a study try teaching e.g. undergraduates using two different formalisms, and see which group comes out with better understanding after a semester.
That is, if Q = u/v, then Q is the “quaternion” (complex sum of a scalar and a bivector) which rotates and scales v into u. Or written out, Qv = (u/v)v = u. The bivector part of Q has the same orientation as the plane spanning u and v.
If we want to use that quaternion as a representation of a general rotation of 3-dimensional space, we need to use a “sandwich product” because just multiplying Q directly by an arbitrary vector will rotate and scale the portion of the vector parallel to the plane of u–v, but then also produce a trivector component from the perpendicular part of the vector, which is not what we wanted.
If we define R = √Q, then we can multiply any vector a like RaR`, where R` is the conjugate of R, and get the scaling & rotation operation we were looking for. Multiplication of any vector perpendicular to the plane of u–v by this sandwich will just scale it by the magnitude of Q, but not change its direction, that is, for a vector b perpendicular to the plane of u–v, RbR` = |Q|b, whereas for a vector c parallel to the plane of u–v, RcR` = Qc, or in other words c gets rotated by the angle between v and u, as intended.
The square root here is where the half angle measures come from.