Least-Squares Fitting of Circles and Ellipses∗

Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g. computer graphics [1], coordinate metrology [2], petroleum engineering [11], statistics [7]. In the past, algorithms have been given which fit circles and ellipses in some least squares sense without minimizing the geometric distance to the given points [1], [6]. In this paper we present several algorithms which compute the ellipse for which the sum of the squares of the distances to the given points is minimal. These algorithms are compared with classical simple and iterative methods. Circles and ellipses may be represented algebraically i.e. by an equation of the form F (x) = 0. If a point is on the curve then its coordinates x are a zero of the function F . Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances. 1 Preliminaries and Introduction Ellipses, for which the sum of the squares of the distances to the given points is minimal will be referred to as “best fit” or “geometric fit”, and the algorithms will be called “geometric”. Determining the parameters of the algebraic equation F (x) = 0 in the least squares sense will be denoted by “algebraic fit” and the algorithms will be called “algebraic”. We will use the well known Gauss-Newton method to solve the nonlinear least squares problem (cf. [15]). Let u = (u1, . . . , un) T be a vector of unknowns and consider the nonlinear system of m equations f(u) = 0. If m > n, then we want to minimize