A NOTE ON THE LEAST-SQUARES FITTING OF ELLIPSES

The characteristics of two normalisations for the general conic equation are investigated for use in least squares fitting: either setting F = 1 or A + C = 1. The normalisations vary in three main areas: curvature bias, singularities, transformational invariance. It is shown that setting F = 1 is the more appropriate for ellipse fitting since it is less heavily curvature biased. Setting A + C = 1 produces more eccentric conics, resulting either in over-elongated ellipses or hyperbolae. Although the F = 1 normalisation is less well suited than the A + C = 1 normalisation with respect to singularities and transformational invariance both these problems are solved by normalising the data, shifting it so that it is centred on the origin before fitting, and then re-expressing the fit in the original frame of reference.