Precision of the Invariator Estimator of the Surface Area of an Ellipsoid

Consider a triaxial ellipsoid K of surface area S. Fix an arbitrary point P in the interior of K, that is P ∈ K°, and generate a sectioning plane L32[P] through P, whose normal direction u is uniform random on the unit hemisphere S2+. In the ellipse of section K ∩ L32[P], let M and m denote the lengths of the major and minor principal semiaxes, respectively, and let r denote the distance of P from the ellipse centre. Then Ŝ = 2π(M² + m² + r²) is an unbiased estimator of S. The purpose of this paper is to express Ŝ in terms of the eight parameters involved, namely the lengths of the three principal semiaxes of K, the three Cartesian coordinates of P, and the two spherical polar coordinates (φ, θ) of u. Then Var(Ŝ) is accessible via a double definite integral in (φ, θ), which can be evaluated quickly with available software for any K and any choice of point P ∈ K°.