I have been thinking about spatial transforms.

Given $n$ points, there are $\frac{n!}{(n-2)!2!}$ combinations of selecting two points, so for 64 points in space, there are 2016 single point-to-point relationships (e.g. distances between points involved).

Can a set of points be uniquely defined by the corresponding full set of relationships. That is to say, if I had points A,B and C, could I derive a unique point arrangements to satisfy a list of distances, AB, AC, BC. Could I do this generally for any list of point to point distances given $n$ points? How is it done?

I understand that given only relative position, placement of the points in an absolute coordinate system is impossible. I understand also, that with few points, e.g. 2,3.., the solution is not unique - but what about larger numbers?

Edit - Addition of a physical constraint.

Let us define a 16 by 16 by 16 grid $G$, based on the Cartesian coordinate system. Let us calculate/define a set of distances $D_R$, that represent the physical distance between points in $G$. If we have a nice continuous function without a singularity - to represent gradient in the space for example - then we could correct our distance $D_R$, to some effective distance $D_E$, by multiply each distance by the appropriate mean gradient for example. The task is to define the points in some "Effective Distance Space". My point is that in physical reality there are constraints on the sets of distances - I don't expect to find a solution to an arbitrary set of distances. Does this change things?

This post imported from StackExchange Mathematics at 2014-06-16 11:28 (UCT), posted by SE-user Andrewb