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I'm doing the following simple code, that aims to find a point position knowing distances in 2D

but the outputs is far from the actual values:

the gap for x is around 0.4! which is huge, why is it so? normally the code is correct and one solution should match almost exactly the actual position

x =

-22.5971 -70.9058

x3 =

-22.1513

y =

34.5832 35.4659

y3 =

34.5751

----code---

X =[ -47.0124 47.5995;

-47.6018 15.3411;

-22.1513 34.5751];

x1 = X(1,1);

y1 = X(1,2);

x2 = X(2,1);

y2 = X(2,2);

x3 = X(3,1);

y3 = X(3,2);

r1 = norm(X(3,:) - X(1,:));

r2 = norm(X(3,:) - X(2,:));

B = (sum(X(2,:).^2)-sum(X(1,:).^2)-r2^2+r1^2)/2;

% trying to retrieve X(3,:) knowing the distances only

% (x3-x1)^2 + (y3-y1)^2 =r1^2 (1)

% (x3-x2)^2 + (y3-y2)^2 =r2^2 (2)

% (2) - (1) to have a relation between x3 and y3

% x3(x2-x1) + y3(y2-y1) = B

% we replace in (1)

if y2-y1~=0

'we should be there'

C = y1-B/(y2-y1);

tau = (x2-x1)/(y2-y1);

a = 1+tau^2;

b = -2*(x1+ tau*C);

c = x1^2+C^2-r1^2;

delta = b^2-4*a*c

x = (sqrt(delta)*[1 -1] - b)/(2*a) % the 2 solutions for x

x3 % check actual value

y = (B-x*(x2-x1)) / (y2-y1)

y3 % actual value

end

Geoff Hayes
on 24 May 2014

cyril - I think that there is a sign mismatch in the calculation of C. Rather than

C = y1-B/(y2-y1);

it should be

C = -y1+B/(y2-y1);

due to the re-arrangement of

x3(x2-x1) + y3(y2-y1) = B

=> y3 = [B - x3(x2-x1)]/(y2-y1)

=> y3 = B/(y2-y1) - x3(x2-x1)/(y2-y1)

and the above substitution into

(x3-x1)^2 + (y3-y1)^2 =r1^2

=> (x3-x1)^2 + (B/(y2-y1) - x3(x2-x1)/(y2-y1) - y1)^2 = r1^2

=> (x3-x1)^2 + (C - x3*tau)^2

where

C = B/(y2-y1) - y1

tau = (x2-x1)/(y2-y1)

Note that a couple of other things you could do is to replace the norm calls with just

r1 = (x3-x1)^2 + (y3-y1)^2; %norm(X(3,:) - X(1,:));

r2 = (x3-x2)^2 + (y3-y2)^2; %norm(X(3,:) - X(2,:));

to avoid the square root of norm and subsequent squaring (where needed) of r1 and r2. (Note that if you use the above two replacements, then you will have to remove the squares for r1 and r2.)

When I re-run your code with the above changes, I get the expected answer:

x =

-22.1513 -72.3327

x3 =

-22.1513

y =

34.5751 35.4920

y3 =

34.5751

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