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

Luis Manuel Cruz-Orive

doi:10.5566/ias.3794

Authors

Luis Manuel Cruz-Orive University of Cantabria (E-Santander)

DOI:

https://doi.org/10.5566/ias.3794

Keywords:

Elliptic integrals, ellipsoidal section, flower formula, heat map, integral geometry, invariator, pivotal plane, pivotal point, stereology, support set

Abstract

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 L³_2[P] through P, whose normal direction u is uniform random on the unit hemisphere S²₊. In the ellipse of section K ∩ L³_2[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°.

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