Set of points to Cartesian system

Hello,

what whould be the fastest way to calculate the Cartesian coordinates from a distance matrix? First vertex would be at x=0 y=0 and second vertex at x=0 y=dist(V1V2).

THX

Answer 1

[Matt Tearle]

You can use cmdscale to reconstruct coordinates from distances. You will need a further restriction to determine the locations uniquely (eg y2 > 0). But here's a way to get what you've specified:

% Make some Euclidean coord data
x = [0 0;0 2*rand - 1;2*rand(5,2) - 1];
% Construct distance matrix
d = squareform(pdist(x))

% Reconstruct Euclidean coords from distances
X = cmdscale(d);

% Shift center to first point
X = bsxfun(@minus,X,X(1,:));
% Rotate so x_2 = 0
alpha = atan2(X(2,1),-X(2,2));
R = [cos(alpha),-sin(alpha);sin(alpha),cos(alpha)];
X = X*R

If you compare X and the original x, you'll see that there can be sign ambiguities. You could nail that down, though, if it matters.

[Walter Roberson]

cmdspace? Interesting; I had never seen that before.

[Matt Tearle]

Did Walter just say "I had never seen that before"? I think I get some kind of award now...

[Walter Roberson]

Well, it is part of the Stats toolbox, and I do prefer to use software that works more than 19 times out of 20.

Answer 2

[Walter Roberson]

x = zeros(1+length(Dists),1);
y = [0; cumsum(Dists(:))];

Ah, the joys of not providing sufficient examples...

Answer 3

[Denny Milakara]

Thank you guys!

I wasn't aware of 'cmdscale'. Anyways, I need it in 2D so I use from now on mdscale.

I always say that the worst part of Matlab is its help: it prevents people of being aware how powerful Matlab really is.

Cheers