Fit a circle to 3d dots

Hi, I have dots spread like a tube in a 3d space. I want to fit a circle to this data (the data is attached). The plane that the circle should be in is:

   S1=[-15.8,8.6,1.3];
   S2=[-10.5,10.3,-3.3];
   S3=[-15.2,8.9,1.6];
   normal = cross(S1-S2, S1-S3);
   A = normal(1); B = normal(2); C = normal(3);
   D = dot(normal,S1);

This is what I was trying to do:

    plot3(n1,n2,n3,'*')   %The dots
    hold on
    a2=[mean(n1),mean(n2),mean(n3)]; %Center of the dots
    plot3(a2(1),a2(2),a2(3),'*','color','g')
    a3=dist(a2,[n1;n2;n3]); %calculate the radius
    t = linspace(0,2*pi,40);
    x=(mean(a3).*sin(t)+mean(n1));
    y=(mean(a3).*cos(t)+mean(n2));
    z = (A * x + B * y - D)/(-C);
    plot3(x,y,z,'*','color','r')

But I can't set the circle right in place (it is grater than what it should be). could you help?

[ELI]

The dots data are attached.

Answer 1

[Mischa Kim]

There is probably something not working with the way you rotate the circle back into the plane. This works:

S1=[-15.8,8.6,1.3];
S2=[-10.5,10.3,-3.3];
S3=[-15.2,8.9,1.6];

normal  = cross(S1-S2, S1-S3);                % compute reference frame axes
nz      = normal'/norm(normal);
normal2 = cross(nz, [0; 1; 0]);
nx      = normal2/norm(normal2);
ny      = cross(nz, nx)/norm(cross(nz, nx));  % end compute reference frame axes
R       = [nx ny nz];                         % compute rotation matrix 

plot3(n1,n2,n3,'*')   %The dots
hold on; grid; box;

a2=[mean(n1),mean(n2),mean(n3)]; %Center of the dots
plot3(a2(1),a2(2),a2(3),'*','color','g')
a3=dist(a2,[n1;n2;n3]); %calculate the radius
t = linspace(0,2*pi,40);

x0=(mean(a3).*sin(t));                        % get circle coordinates in x-y plane...
y0=(mean(a3).*cos(t));
z0      = zeros(1,length(x0));
xyz     = bsxfun(@plus, R*[x0; y0; z0], a2'); % rotate circle plane and adjust center

plot3(xyz(1,:),xyz(2,:),xyz(3,:),'-','color','r')

Answer 2

[Iain]

Without looking at the data, it looks like the size of your circle would be larger than it should be because of the spots being off the plane where your circle exists.

Either you should reform your formula to fit the best circle in 3D space (that should be a page or two of working out the formulae), or, you should calculate the closest point on your plane to each point, and then use those points to fit your circle.

[ELI]

The circle should be in the edge of the tube like dots.