Least Square Linear Regression Help

Question

I am given 10 points in the R^3 and my job is to see if the points fit a circle, and if so what is the radius and center. I was not quite sure how to start. Originally, I thought the couldn't because they all don't share a dimension but then I realized that the points can lie on a slanted plane. I am thinking that this will have something to do with using an orthonormal basis and then projecting the points onto a plane and seeing if the points fit an equation for a circle, using maybe least square linear regress? Not really sure, would appreciate some direction.

I am given the 10 points. in (x,y,z) form

Answer 1

The first thing to be done is to find the equation of the plane where the given points are located around.

Several methods of regression are discussed pp.13-26 in the paper (*) : https://fr.scribd.com/doc/31477970/Regressions-et-trajectoires-3D . Numerical examples are presented.

An alternative approach is the "Principal Component Analysis" : https://en.wikipedia.org/wiki/Principal_component_analysis . A numerical example is given p.26 in the paper referenced above (*).

Once one have got the equation of the plane, two ways are possible :

First method :

Change of axes system. Define a 2-D base of axes on the plane. Compute the coordinates of the orthogonal projections of the given points on the plane. Proceed to a circular regression. For example see pp.11-13 in the paper https://fr.scribd.com/doc/14819165/Regressions-coniques-quadriques-circulaire-spherique

Second method :

Again in the paper referenced above (*), the problem is treated pp.28-34 not only for circles but for general conic curves in 3-D. This gives the kind of conic with it's characteristics : Hyperbola, parabola or ellipse. In the case of ellipse one can see if it is reduced to circle : The radius and coordinates of center are obtained.

I am sorry to limit my answer to some links to papers published elsewhere. The full typing should be much too large to be edited here.

Answer 2

If your points do fit a circle, you can pick any three of them and compute the equation of the circle they define (take the intersection of their plane and the bissector planes of two edges to get the center).

Then check that the remaining points are coplanar and compare the distances to the center.

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If you expect your data points to be noisy or if there are outliers, it may be a good idea to redo the computation with several point triples and keep the configuration with the smallest sum of distances.