  We study the problem of fitting circles (or circular arcs) to data points
observed with errors in both variables. A detailed error analysis for all
popular circle fitting methods -- geometric fit, Kasa fit, Pratt fit, and
Taubin fit -- is presented. Our error analysis goes deeper than the traditional
expansion to the leading order. We obtain higher order terms, which show
exactly why and by how much circle fits differ from each other. Our analysis
allows us to construct a new algebraic (non-iterative) circle fitting algorithm
that outperforms all the existing methods, including the (previously regarded
as unbeatable) geometric fit.
