I agree with you on three dimensional vector products. It's too special, too cute and doesn't generalize to all dimensions, and as you said, you have to keep track of the two types of vectors.
On complex multiplications though, I disagree. It's a great way to do Euclidean manipulations on the 2d plane. Rotations, translations and reflections (via conjugates) are simple. You rarely need calls to trigonometric functions.
If you have runtime support, it's sorta criminal not to use complex multiplication when applicable.
BTW there is another, equivalent, way of deriving the solution which to me seems more intuitive (and not limited to sum of powers to 1):
The angular travel from z1 to z2 is
z2 / z1.
√(z2/z1).
√(z2/z1) * z1
If the need was to continue to travel angularly (rotate) beyond z2, say double the subtended angle, that's easy too. No need for the constraint the sum of powers be 1.
@chinjut yes you are exactly right about sum to 1 bit.
For a moment I had got distracted by the exponential between Lie group and algebra.
What magnitude are you expecting your angle bisector to have afterwards?
There was only the need for the point on the bisector (on the unit circle). There was no need for the magnitude of the angle.
The only thing that needed care was which sign of the sqrt bisects the internal angle as opposed to the external angle.
In general I prefer not to deal with angles when dealing with 2D rotation. Get inputs in angles if need be and from then onwards use the (cos,sin) tuple or, equivalently, use complex numbers. One can get rid of calls to trascendentals as long as you are happy to call sqrt.
In other words angle is a tuple.
if they're unit vectors then yes that makes a lot of sense.
The same calculation works in R^n, incidentally, using the geometric product. This is pretty much the ideal usecase for it, for constructing operators between vectors.
Maybe that's my missing link for
https://news.ycombinator.com/item?id=48619191
You probably know this, but this is one way to generalize beyond 2D
https://en.wikipedia.org/wiki/Angle_bisector_theorem