Circle least squares fit for 3D data

Hi everyone,

I have 6000 x coordinates, y coordinates and z cooridinates that form a circle that does not perfectly occupy one plane. I am looking to fit the data with a circle but I can only find functions online that do it for x and y coordinates and not including z. I would appreciate any help in creating some code for this as I am not sure where to start (I am a beginner in MATLAB!)

Thank you in advance

Ciara

[Bruno Luong]

Start to find the plane that approximate your 3D points then project point on this plane then use your prefered 2D circle fitting method.

[KSSV]

If you have z also , it should be a sphere right?

[Ciara Gibbs]

@Bruno Luong, thank you for replying firstly! Please could you help me in terms of finding a plane, I am not sure how to go about this (sorry I am only beginners level :) ). Any help would be much appreciated!

[Ciara Gibbs]

@ KSSV thank you for the response! I was thinking this also but this is how my data appears

I need to find the best fit for the blue data and the purple data - they appear more like a slanted circle / plane, so I worry about how accurate using a sphere will be.

My main interest is determining the circle centre as I need it to determine some offset in my data. Any advice/help would be greatly appreciated!

[Torsten]

@Bruno Luong, thank you for replying firstly! Please could you help me in terms of finding a plane, I am not sure how to go about this (sorry I am only beginners level :) ). Any help would be much appreciated!

https://de.mathworks.com/help/vision/ref/pcfitplane.html

https://de.mathworks.com/matlabcentral/fileexchange/43305-plane-fit

https://stackoverflow.com/questions/48200261/fit-plane-to-n-dimensional-points-in-matlab

[Bruno Luong]

Can you attach your data. From the picture it is not clear any circle shape would emerge from this perspertive.

[Ciara Gibbs]

@Bruno Luong , please find the data attached in the order of x, y ,x . Thanks again

[Ciara Gibbs]

@Torsten, thank you for the links I will give them a read!

Answer 1

[Bruno Luong]

X=csvread('data.csv');

XC = mean(X,1);
Y=X-XC;
[~,~,V]=svd(Y,0);
Q = V(:,[1 2]); % basis of the plane
Y=Y*Q;

xc=Y(:,1);
yc=Y(:,2);
M=[xc.^2+yc.^2,-2*xc,-2*yc];
% Fit ellipse through (xc,yc)
P = M\ones(size(xc));

a=P(1);
P = P/a;
r=sqrt(P(2)^2+P(3)^2+1/a); % radius
xyzc = XC' + Q*P(2:3); % center

theta = linspace(0,2*pi);
c = xyzc + r*Q*[cos(theta); sin(theta)]; % fit circle

close all
plot3(X(:,1),X(:,2),X(:,3),'.');
hold on
plot3(c(1,:),c(2,:),c(3,:),'r','LineWidth', 2);
axis equal

[Ciara Gibbs]

Thank you this is just what I was looking for !! If you don't mind, could you explain at little the steps you have done just so I can understand more? Thank you again !

[Bruno Luong]

The first block using the well-known technique to find the plane that fits your 3D data using the SVD. The smallest eigen vector V(:,3) correcponds to the direction orthogonal to the plane and the 2 other directions V(:,[1 2]) are the (orthonormal) basis of the plane;I call it Q.

Y is the projection of X on the plane after moving the origin to the barycenter of the points (XC).

Next I look at the equation of the circle on the plane by using least square fit to the eqt

P(1)*(x.^2 + y.^2) -2*P(2)*x - -2*P(2)*y = 1

After transformation you 'll find the 2D equation of the circle as

(x-xc).^2  + (y-yc).^2 = r^2

with xc = P(2)/P(1) and yc = P(3)/P(1) and r = sqrt((P(2)^2+P(3)^2+1)/P(1));

The last step just bring back the 2D coordinates in the plance to 3D coodinates

[Ciara Gibbs]

Amazing! Thank you very much it is greatly appreciated and you have saved me a lot of time!

[Martin Fuchs]

This script is just brilliant!