So basically I’m trying to create an optimization equation for finding the closest point to the intersection between 3 spheres that don’t actually intersect. My idea was to first find the circle created by the intersection of two spheres and then minimizing the distance from a point on the 3rd sphere to that circle.
If this method is wrong, I would greatly appreciate any ideas. For context, this is for GPS trilateration, where you derive position using distances from 3-4 satellites. Due to error these distances sometimes don’t fully match up so I’m trying to find the closest point
Thank you so much for your time
Best Answer
How about $$\min_{x,y_i} \left\{ \sum_{i=1}^3 ||x-y_i|| : y_i \in S_i \right\},$$ where $S_i$ are the spheres. At the optimum, $x$ is the point you are interested in, and $||x-y_i||$ is the distance between $x$ and sphere $i$. Alternatively, you can minimize the maximum distance ($t$) instead of the sum of the distances: $$\min_{t,x,y_i} \left\{ t : \sum_{i=1}^3 ||x-y_i|| \leq t, \; y_i \in S_i \right\},$$ Both optimization problems are second order cone problems, for which many methods are available.