How to find a pointset with unique distances

Question

Is there a way to arrange N number of 2D points within a box so that the distances between the points are unique?

I have an application where I can measure the distances between points with some accuracy. At the time of measurement I don't know between which points I am measuring. I thus need to match the measured value against an exptected distance and see which is closest. Taking more and more measurements into account I should be able to rule out which point has a certain id.

But is there a mathematical method to actualy place the points optimally?

Answer 1

You can easily put the points on a line so that each distance is unique:

If $x_1,x_2,..,x_n$ are your $n$ points, put them on a ray starting at $O$, in this order such that $d(x_{i},O)=2^{i-1}$. It follows that $$d(x_{i},x_j)=|2^{i-1}-2^{j-1}| \,.$$

And it is easy to prove that $2^a-2^b=2^c-2^d$ implies $(a=c)$ and $(b=d)$.

Answer 2

Best known solutions for choosing up to 26 points in a lattice can be found at No Repeated Distances. It is a very hard combinatorial problem.