The well-known technique of triangulation in $\mathbb{R}^2$ consists of identifying the position of a test point $x$ using the distances from $x$ to three known points $A,B,$ and $C$. This relies on the fact that, given $A,B,$ and $C$, the map $T: \mathbb{R}^2 \to \mathbb{R}^3$ sending $T: x \to (d(A,x),d(B,x),d(C,x))$ is injective.
Suppose now that we know the distances $d(A,x), d(B,x), d(C,x)$ and we also know the points $A,B,$ and $C$, but we do not know which distance corresponds to which point. Formally, we have post-composed the map $T$ with the action of the symmetric group $S_{3}$ on $\mathbb{R}^3$ that rearranges the coordinates. When is the map $\hat{T} : \mathbb{R}^2 \to \mathbb{R}^3/S_{3}$ injective?
Clearly, if $A,B,$ and $C$ are arranged in such a way that there is a Euclidean symmetry exchanging one point for another, there is no hope for $\hat{T}$ to be injective. But what about when there is no such symmetry? Can we say it is injective for a generic choice of $A,B,$ and $C$? Lastly, to lead into a possible generalization, what happens if we add more points, say $D,E, \cdots$, or more dimensions? More points means more information, but also more symmetries. How does the tension between these two affect injectivity? How can we leverage the triangle equality and the Euclidean metric to be able to deduce these injectivity results?