I want to find the solution space of coordinates for point $p$ that satisfies the following system: $$ \begin{cases} [distance(p,a) - distance(p,b)] = k_1\\ [distance(p,c) - distance(p,d)] = k_2 \end{cases} $$
where the points $a,b,c,d,q$ have arbitrary known positions, $k_1 = k - [distance(q,a) - distance(q,b)]$ and $k_2 = k - [distance(q,c) - distance(q,d)]$
I'm trying to solve it on 3D space, but for simplicity I started to analyse it on 2D to further extend my solution. I'm using Cartesian coordinates and $distance(a,b)$ represents the Euclidean distance between points $a$ and $b$.
1) Does system with this shape have a specific name?
2) How can I find an exact solution for this problem?
This is a non-linear system, and from what I searched usually non-linear systems can be solved by either linearization or numeric methods. I'm interested in an exact solution, so I would like to avoid numeric methods. I read that is not always possible to linearize systems of equation, but I couldn't find which are the conditions to know if it is possible or not.
3) How do I know if it is possible to linearize this system?
4) Would make the problem easier to solve if we add two additional arbitrary points $e,f$ to generate a third equation like $[distance(p,e) - distance(p,f)] = k_3, \mbox{ where } k_3 = k - [distance(q,e) - distance(q,f)]$?
Hyperbolas can be defined as the locus of points where the absolute value of the difference of the distances to the two foci is a constant equal to 2a, the distance between its two vertices. In this case I could see the problem as the intersection of hyperbolas $H_1$ and $H_2$ with foci $a,b$ and $c,d$, respectively.
$$ H_1 := [distance(p,a) - distance(p,b)] = k_1\\ H_2 :=[distance(p,c) - distance(p,d)] = k_2 $$
5) Is there any method to find intersections of conics section?
6) Would change the coordinate system make it easier to solve (ex. polar coordinates)?