I have 2 points in 3-dimensional space. I need to know the distance between them. The problem is, I do not know their coordinates.

The only thing I have to go on are 6 other points in this space. The distance to these points from both the points in question is know. Their coordinates are not know, only the distances between each point and the 2 original points are known. Distances between any of the 6 individually are not known.

Is it possible to extrapolate the distance between the first 2 points, using only the distances to other points they have in common?

An example, let's call the first 2 points a1 and a2, the 6 other point A to F:

From a1 to:

  • A: 91cm
  • B: 91cm
  • C: 89cm
  • D: 67cm
  • E: 67cm
  • F: 95cm

From a2 to:

  • A: 86cm
  • B: 87cm
  • C: 87cm
  • D: 91cm
  • E: 89cm
  • F: 95cm
  • $\begingroup$ One cannot say anything beyond what may be implied by the Triangle Inequality. This will put upper and lower bounds on the distance between your two points, but these bounds generally do not agree. $\endgroup$ – André Nicolas Jan 7 '12 at 1:27

We cannot even do it in $2$-dimensional space, and we cannot do it with $1006$ distance pairs, let alone $6$.

Let $A=(-1,0)$, $B=(1,0)$, $A'=(-2,0)$, $B'=(2,0)$. For every positive integer $n$, we can find a point $P_n$ such that $d(P_n,A)=d(P_n, B)=10+n$. For every positive integer $n$, we can also find a point $Q_n$ such that $d(Q_n,A')=d(Q_n, B')=10+n$.

Thus knowing the infinite set $\{(10+n, 10+n)\}$ of distance pairs cannot tell us whether our two points are distance $2$ or $4$ apart. Bounds on the distance between our points can be obtained from the Triangle Inequality, but that's all that can be done in general.


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