Least-squares fitting of ellipses and hyperbolas (Q1382873)

The problem of orthogonal regression for curve fitting to given \(R^2\) data points is considered. Let the data be \((x_i,y_i)\), \(i=1,\dots,n\), and the curve to be fitted described by the parametric model \(x=x(t,\alpha)\), \(y=y(t,\alpha)\), where \(\alpha\) is a vector parameter to be estimated. Then the problem is to minimize \[ S(\alpha,t_1,\dots,t_n)=\sum_{k=1}^n((y_k-y(t_k,\alpha))^2+ (x_k-x(t_k,\alpha))^2). \] The author proposes an iterative algorithm for minimization of \(S\) which consists of two steps. At step 1 \(S\) is minimized by \(\alpha\) with \(t_k\) fixed, at step 2 \(S\) is minimized by \(t_k\) with \(\alpha\) fixed. For ellipses and hyperbolas these steps are reduced to solving of simple algebraic equations. Numerical examples are presented.