# Approximate intersection point of multiple circles

Let we have $$n$$ points $$C_i$$, and also have $$n$$ vectors $$d_i$$ corresponding to each points.
I want to find a point $$X$$ that satisfies $$f(X,C_i)=d_i$$, where $$f$$ is a simple Euclidean distance function.
Problem is, the vectors $$d_i$$ include sort of errors.

I think this problem is equivalent with approximating the intersection point of multiple circles.

We have multiple circles that intersect at a single point.
We already know the centers perfectly, but radii includes some error. (Around 10%. Not so much.)

My question is, how can I approximate the intersection points in this case?
I tried to formulate a kind of optimization problem, but it includes sum of absolute values...

And I also want to ask second problem...
Remember our original problem (finding point).
In fact, I want to find multiple points with multiple distances.
So, the actual goal is finding $$X_i$$s satisfying $$f(X_i,C_j)=d_{ij}$$.
Luckily, we have additional constraints here: $$f(X_i,X_j)=l_{ij}$$
(Of course, these $$l_{ij}$$ also include errors)

Any kind of advise will be really appreciated.

• But $d_i$ are vectors or scalars? If $f$ is a distance I'd expect them to be scalars. Jul 8, 2020 at 16:19

You could obtain an estimate of $$X$$ by searching for the minimum of the weighted sum of the quadratic difference between $$f(X,C_i)$$ and $$d_i$$: $$F(X)=\sum_i\left({f(X,C_i)-d_i\over \sigma_i}\right)^2,$$ where $$\sigma_i$$ is the uncertainty on $$d_i$$.