Problem:
Let there be $N $ points $P_i(x_i,y_i,z_i)$
and a $\mathbf M$ the $NxN$ matrix of distances $d_{ij}$ between each point.
Lastly, let's consider that the 3 of the N points $P_1$, $P_2$ and $P_3$ are know.
How can I solve the trilateration (multilateration?) problem to solve for the other unknow $P_i$ points?
Coments:
The problem I am interested is not about solving individual trilateration problems!
For example, with points $P_1$, $P_2$ and $P_3$ and the distances from them each to all other $P_i$ I can find each $P_i$ individually minimizing the mean square error of the distance from $P_i$ to each $P_1$, $P_2$ and $P_3$. However, this approach only uses a subset of the available information I have (only the distances between the know points and the unknow points).
I am looking for a solution that uses all available information in the hopes that it returns a better more stable solution over different noise levels.
Thanks for the help!