# Simple calculations giving high imprecisions

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cyril on 24 May 2014
Edited: cyril on 24 May 2014
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
cyril on 24 May 2014
thanks very very much Geoff for the error