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What can we say about the configuration space of $n$ points in $\mathbb{R}^{m}$ with fixed distances between the points ?

e.g. for 2 points in $\mathbb{R}^{3}$ with fixed distance between them,the configuration space is $S^{2}$ and for 3 points in $\mathbb{R}^{3}$ with known distances between them, the configuration space is $SO(3)$.

A similar problem is to ask the configuration space of a point in $\mathbb{R}^{m}$ such that its distances from $n$ fixed points is known.

e.g. Consider a point in $\mathbb{R}^{3}$ such that its distances from 2 fixed points is $r_{1}$ and $r_{2}$. Then the point can lie on a sphere of radius $r_{1}$ around point 1 and also on a sphere of radius $r_{2}$ around point 2. Thus the configuration space is the intersection of these 2 sphere, which generically is a circle. (Correct if I'm wrong.)

What is known about these problems in general for any $n,m$ ?

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Some keywords to try: polygon spaces, configuration spaces of linkages. – Mark Grant Nov 22 '11 at 22:45
Other key words are: Kapovich, Millson, Knutson, Hausmann – Igor Rivin Nov 22 '11 at 22:49
The setting I'm interested is slightly different from the polygon spaces, in this case every point is connected to every other by an edge, whereas in polygon spaces we've edges between consecutive points i.e. a 4-gon is a quadrilateral, but what I am interested in is a tetrahedron. After some googling it seems that what I'm looking for is configuration spaces of complete graphs. Also what I'm asking is - is it a well established theory e.g. Do we know the general configuration space for such problems. Like google tells me that for weighted graphs configuration space is some sphere. – J Verma Nov 23 '11 at 21:13

Your question is close to the setup for rigidity theory, though the results there are typically phrased in terms of a perhaps more modest goal:

Let $G=(V,E)$ be a graph. Is the space of embeddings $p:V\rightarrow\mathbb{R}^n$ of the vertex set which preserve the distances between vertices joined by edges in $E$, modulo isometries, discrete (local rigidity), or perhaps even a single point (corresponding to the concept of global rigidity)?

Some basic results are described in this paper of Asimow and Roth, and there are many other references on a variety of aspects of this theory on this REU webpage of Dylan Thurston's.

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The real setting for rigidity theory is understanding when the space of embeddings is regular, so it certainly sets itself much more modest goals than the OP... – Igor Rivin Nov 23 '11 at 10:06

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