TITLE: Adding two polar vectors
QUESTION [55 upvotes]: Is there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?

REPLY [3 votes]: While correct mathematically, these are quite hard to code properly due to sign ambiguities and branch cuts in phase. This can be seen in that the argument of the arccos() does not depend on sign of the angle difference ($\phi_2-\phi_1$), it gives the same answer regardless. Thus when adding back $\phi_1$ to $\psi_2$ to get the angle of the result $\phi$, the resulting vector has the correct magnitude, but the angle is wrong if $\phi_2 \lt \phi_1$.
You can go through and code in the signs, but then you get another set of signs if the angle between the input vectors is more than $\pi$, and even this is complicated by the branch cut angle (difference between angles should be less than ±$\pi$). Essentially you need to know whether $\vec{r}_1 \times \vec{r}_2$ is left or right handed (negative or positive), and handle correctly.
You can make this work, as I worked through this afternoon in a fit of pique. BUT, the answer is less accurate than the cartesian result due to phase errors accumulating when the vectors nearly add to zero. So less accurate and lots of fiddly logic that is going to be slow. If you do go down this road I hope you are unit testing the bejesus out of it. 
That said, you can make a chisq fitting algorithm work well, because that just wants the square of the vector distance, which avoids all the sign issues. Just use law of cosines and go to town.