intersection of three spheres in matlab

To find the intersection of three spheres in Matlab, we can use the following steps:

1. Define the three spheres using their center coordinates and radii.

   main.m
   % Sphere 1
   c1 = [x1, y1, z1];  % center coordinates
   r1 = radius1;        % radius
   
   % Sphere 2
   c2 = [x2, y2, z2];  % center coordinates
   r2 = radius2;        % radius
   
   % Sphere 3
   c3 = [x3, y3, z3];  % center coordinates
   r3 = radius3;        % radius
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2. Use the trilateration function from the File Exchange to find the intersection points of the three spheres.

   main.m
   % Define the input and output matrices
   input_matrix = [c1; c2; c3];
   output_matrix = [r1; r2; r3];
   
   % Find the intersection points
   [x,y,z] = trilateration(input_matrix, output_matrix);
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3. If there are two intersection points, we can choose the one that lies within all three spheres by checking the distances to each sphere center.

   main.m
   % Calculate the distances from the intersection point to each sphere center
   d1 = norm([x,y,z]-c1);
   d2 = norm([x,y,z]-c2);
   d3 = norm([x,y,z]-c3);
   
   % Check if the intersection point lies within all three spheres
   if abs(d1-r1) < 1e-12 && abs(d2-r2) < 1e-12 && abs(d3-r3) < 1e-12
       % The intersection point is valid
   else
       % The intersection point is invalid
   end
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Here's an example implementation:

main.m
% Define the spheres
c1 = [0,0,0];
r1 = 2;
c2 = [3,0,0];
r2 = 2;
c3 = [1.5,2.598,0];
r3 = 2;

% Find the intersection points
input_matrix = [c1; c2; c3];
output_matrix = [r1; r2; r3];
[x,y,z] = trilateration(input_matrix, output_matrix);

% Check which intersection point is valid
d1 = norm([x,y,z]-c1);
d2 = norm([x,y,z]-c2);
d3 = norm([x,y,z]-c3);
if abs(d1-r1) < 1e-12 && abs(d2-r2) < 1e-12 && abs(d3-r3) < 1e-12
    % The intersection point is valid
    disp(['Intersection point: (', num2str(x), ',', num2str(y), ',', num2str(z), ')'])
else
    % The intersection point is invalid
    disp('No valid intersection point')
end
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