Least-squares fitting of circles and ellipses

Abstract

Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g., computer graphics, coordinate meteorology, petroleum engineering, statistics. In the past, algorithms have been given which fit circles and ellipses insome least-squares sense without minimizing the geometric distance to the given points.

In this paper we present several algorithms which compute the ellipse for which thesum 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 formF(x)=0. If a point is on the curve, then its coordinates x are a zero of the functionF. Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances.

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Dedicated to Åke Björck on the occasion of his 60th birthday

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Gander, W., Golub, G.H. & Strebel, R. Least-squares fitting of circles and ellipses. BIT 34, 558–578 (1994). https://doi.org/10.1007/BF01934268

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AMS subject classifications

  • 65H10
  • 65F20

Key words

  • Least squares
  • curve fitting
  • singular value decomposition