Re: plot cluster of points without knowing the coordinates.

Paul K. wrote:
Hi!
Is it possible to plot a cluster of points in matlab in 2D given that
we have complete access to all the pairwise distances between points
in a cluster?

For example we have the set of points {p1,p2,p3,p4} with distances:

p1p2=3
p1p3=5
p2p3=1

Say we choose a convention where p1 is the origin of a coord system, and p2 is on the X axis so we have a reference frame. This can later be rotated and/or translated as desired. We now have p1=(0,0) and p2=(3,0) for the example data.

The distance p1-p3=5 defines a circle centered at p1=(0,0) of radius 5. The distance p2-p3=1 defines a circle centered at p2=(3,0) of radius 1. These circles do not intersect, so the example data is inconsistent and represents an impossible situation.

Say the data had p2-p3=2. Then the p1-p3 circle and the p2-p3 circle are tangent at p3=(5,0). That solution would be unique, with p1, p2, and p3 collinear.

Now say the data had 8 > p2-p3 > 2. Then the circles would intersect at two points. Which point is "really" p3?

Do additional points help resolve ambiguity? If we introduce distances p1-p4, p2-p4, and p3-p4, we *could* have three circles which, for consistent data representing a real situation, resolves to a single point for p4 *if* p1, p2, and p3 are not collinear. But we don't have only three circles. There were two possible circles for p3-p4.

OK, then have a convention that the first ambiguity is arbitrarily resolved (effectively choosing one of the two mirror solutions). Then it appears a solution is possible by triple-wise circle intersections, although it would be necessary to handle special cases (inconsistency, tangency, collinearity) for all triples, and probably have to deal with numerical tolerance issues...

....Ron Capelli (Austin)
.

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