Finding coordinates of a fourth point from 3 known points of fixed distance in 3D?

I have three moving points that are tracked through time A(x,y,z), B(x,y,z) and C(x,y,z) through 150 frames (150 x 3). There is a fourth point that can be located in one frame (x,y,z) but not the others. If this fourth point is at a fixed distance from the other three is there a simple way to find the coordinates for this point in all 149 other frames. Intersection of three spheres has been recommended but I cannot find any functions that match this problem.

[Rik]

What have you tried so far? What problems are you having? Note that the precision might not be enough to get an exact match, so you might have to compare with a tolerance (see doc eps).

[HC1992]

I've tried various functions below but haven't been able to solve error messages

in frame 1, A = [1673.7 85.1 279] B = [1579.4 51.2 279] C = [1674.4 86.7 249] and X = [1718.3 36.9 246.3]

https://uk.mathworks.com/matlabcentral/fileexchange/54680-trilateration-code

http://www.mathworks.com/matlabcentral/fileexchange/26186-absolute-orientation-horn-s-method

https://uk.mathworks.com/matlabcentral/fileexchange/26148-intersection-of-three-spheres

[HC1992]

I also have the distances AX, BX, and CX

Answer 1

[Rik]

I don't get any errors for the code below. Note that there may be two solutions and that this code returns the result in the form of [x y z 1]. It also reports its precision, which is fairly close to machine precision, if you take the input magnitude into account.

A = [1673.7 85.1 279];
B = [1579.4 51.2 279];
C = [1674.4 86.7 249];
X = [1718.3 36.9 246.3];
%Pythagoras: dist^2=deltaX^2+deltaY^2+deltaZ^2
AX=sqrt(sum(diff([A(:),X(:)],1,2).^2));
BX=sqrt(sum(diff([B(:),X(:)],1,2).^2));
CX=sqrt(sum(diff([C(:),X(:)],1,2).^2));
%loop for each pair of (A,B,C):
result=interx(A,B,C,AX,BX,CX)
%result is [x y z 1] of the intersection (of which there may be two)

[Rik]

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