fit a sphere using non linear least squares method in matlab

To fit a sphere using the nonlinear least squares method in Matlab, you can use the lsqnonlin function. Here's an example code for fitting a sphere using lsqnonlin:

main.m
% Generate some sample data
N = 100; % Number of data points
x = 10*(rand(N, 1)-0.5);
y = 10*(rand(N, 1)-0.5);
z = 10*(rand(N, 1)-0.5);

% Define the function to be optimized
fun = @(c)(sqrt((x-c(1)).^2 + (y-c(2)).^2 + (z-c(3)).^2) - c(4));

% Define initial guess
c0 = [0; 0; 0; 1];

% Call lsqnonlin to optimize
[c,resnorm] = lsqnonlin(fun,c0);

% Extract the sphere parameters
xc = c(1);
yc = c(2);
zc = c(3);
R = c(4);

% Plot the results
figure;
scatter3(x,y,z);
hold on;
[xs,ys,zs] = sphere;
surf(xc+R*xs,yc+R*ys,zc+R*zs,'FaceAlpha',0.5,'EdgeAlpha',0.2);
axis equal;
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In this example code, we generate some random sample data representing points in 3D space. We define the function fun that calculates the distance between each data point and the fitted sphere, and subtracts the sphere radius R. We then define an initial guess for the sphere center and radius, and call lsqnonlin to optimize the parameters c. Finally, we extract the optimized sphere parameters and plot the data points and fitted sphere.