Least squares fitting of circles and ellipses to measured data (Q1283256)

Given are some coordinate vectors that are obtained by a coordinate measuring machine. This can be described as a stick with a touch probe that measures the coordinates of e.g., the circumference of some object. For a number of points the coordinate vectors \(x_i\) and the corresponding probe direction \(v_i\) are given. The problem is to fit in a least squares sense the best possible circle or ellipse through the data points. The objective function to be minimized is the sum of squares of the distances between the measured points \(x_i\) and the intersection in the direction \(v_i\) with the approximating conic. The parameters to be found is the coordinate vector of the center of the conic and the radius (or the length of the axes in the case of an ellipse). For an ellipse, the fit can be improved by also allowing a rotation of the axes. The elements needed in the Gauss-Newton method to solve the problem are provided. The problem is not convex, so that global convergence can not be guaranteed.