I'm doing some research on time-difference-of-arrival geolocation and what this amounts to in the absence of noise is finding the intersection points of a collection of hyperbolas.
I've noticed that visualizing the intersection points of three or more hyperbolas is easy, however, if you want an analytic solution for the intersection points I can't figure out a good way to get it. Online resources only discuss solving for the intersection points of two hyperbolas. I'm wondering if there is even a closed form solution at this point?
Assume there are $N$ equations defining $N$ hyperbolas given by $$ \frac{x^2}{a_i^2}-\frac{y^2}{b_i^2} = 1\,,\quad 3\leq i\leq N \,,$$ where $a_i$ and $b_i$ are known and given for all $i$. At what values of $x$ and $y$ are all these equations satisfied? If you simply plot each hyperbola the intersection point is usually easy to spot by eye, but is there a closed form analytic solution?
For two hyperbolas this reduces to system of linear equations in $x^2$, $y^2$. These give you up to 4 solutions (you can pick arbitrary signs).
If you have more hyperbolas then if you want points to lie on all of them just test those 4 solutions against your other equations, if you want them to lie on at least two then just do this for any pair.