Direct and Indirect Estimation of the Variance–Covariance Matrix of the Parameters of a Fitted Ellipse and a Triaxial Ellipsoid

This work deals with the estimation of the variance–covariance matrix of the parameters of a fitted ellipse and an ellipsoid by a direct and an indirect procedure. In the direct approach, the Cartesian equation of an ellipsoid was expressed in terms of the coordinates of the ellipsoid center, the three ellipsoid semiaxes, and the three rotation angles. The general least-squares method was applied to estimate these parameters and their variance–covariance matrix. In the indirect approach, the Cartesian equation of an ellipsoid was expressed as a polynomial. The coefficients of this polynomial equation and their variance–covariance matrix were estimated using the general least-squares method. Then these coefficients were transformed into the parameters of the ellipsoid through an analytical diagonalization of a suitable matrix. The variance–covariance matrix of these parameters was estimated applying the law of propagation of variances. Both approaches are applied to the special case of an ellipse. The numerical examples in both cases indicated that the two procedures produce almost identical results.