Finding the coordinates of points from distance matrix

Question

I have a set of points (with unknow coordinates) and the distance matrix. I need to find the coordinates of these points in order to plot them and show the solution of my algorithm.

I can set one of these points in the coordinate (0,0) to simpify, and find the others. Can anyone tell me if it's possible to find the coordinates of the other points, and if yes, how?

Thanks in advance!

EDIT Forgot to say that I need the coordinates on x-y only

Answer 1

The answers based on angles are cumbersome to implement and can't be easily generalized to data in higher dimensions. A better approach is that mentioned in my and WimC's answers here: given the distance matrix D(i, j), define

M(i, j) = 0.5*(D(1, j)^2 + D(i, 1)^2 - D(i, j)^2)

which should be a positive semi-definite matrix with rank equal to the minimal Euclidean dimension k in which the points can be embedded. The coordinates of the points can then be obtained from the k eigenvectors v(i) of M corresponding to non-zero eigenvalues q(i): place the vectors sqrt(q(i))*v(i) as columns in an n x k matrix X; then each row of X is a point. In other words, sqrt(q(i))*v(i) gives the ith component of all of the points.

The eigenvalues and eigenvectors of a matrix can be obtained easily in most programming languages (e.g., using GSL in C/C++, using the built-in function eig in Matlab, using Numpy in Python, etc.)

Note that this particular method always places the first point at the origin, but any rotation, reflection, or translation of the points will also satisfy the original distance matrix.

Answer 2

Step 1, arbitrarily assign one point P1 as (0,0).

Step 2, arbitrarily assign one point P2 along the positive x axis. (0, Dp1p2)

Step 3, find a point P3 such that

Dp1p2 ~= Dp1p3+Dp2p3
Dp1p3 ~= Dp1p2+Dp2p3
Dp2p3 ~= Dp1p3+Dp1p2

and set that point in the "positive" y domain (if it meets any of these criteria, the point should be placed on the P1P2 axis).
Use the cosine law to determine the distance:

cos (A) = (Dp1p2^2 + Dp1p3^2 - Dp2p3^2)/(2*Dp1p2* Dp1p3)
P3 = (Dp1p3 * cos (A), Dp1p3 * sin(A))

You have now successfully built an orthonormal space and placed three points in that space.

Step 4: To determine all the other points, repeat step 3, to give you a tentative y coordinate. (Xn, Yn).
Compare the distance {(Xn, Yn), (X3, Y3)} to Dp3pn in your matrix. If it is identical, you have successfully identified the coordinate for point n. Otherwise, the point n is at (Xn, -Yn).

Note there is an alternative to step 4, but it is too much math for a Saturday afternoon