INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 38 (2005) 3475–3482 doi:10.1088/0305-4470/38/16/001
Distribution of distance in the spheroid
Ricardo Garcı´a-Pelayo
ETS de Ingenierı´a Aerona´utica, Plaza del Cardenal Cisneros, 3, Universidad Polite´cnica
de Madrid, Madrid 28040, Spain
E-mail: r.garcia-pelayo@upm.es
Received 14 July 2004, in final form 4 March 2005
Published 6 April 2005
Online at stacks.iop.org/JPhysA/38/3475
Abstract
The distribution of distance in the sphere is reviewed. The distribution of
distance in the ellipsoid is given as an integral which can be done in terms
of elementary functions for the spheroid. As an application, Maclaurin’s ratio
of the polar to equatorial radius of the Earth due to its rotation is rederived
using the distribution found here.
PACS numbers: 02.50.−r, 04.40.−b, 46.15.−x, 91.10.−v
1. Introduction
In physics, the potential energy of a body is, very often, the sum of the potential energies of
its pairs of parts or particles. From a computational point of view, this means that we have
to do a six-dimensional integral. But if the distribution of distance of the body is known
then only a one-dimensional integral is necessary. This increases the numerical precision
by many orders of magnitude and allows the computation of potential energies in instances
where a straightforward approach would be too costly. Applications to nuclear physics and
electrostatics based on this idea have been discussed recently [1–4]. Some of these authors
have also applied the distribution of distance to the testing of random number generators
[5, 4].
Computation of potential energy is the main application of the distribution of distance to
physics, but not the only one, the analysis of mobile radio systems [6] being another.
The topic has, of course, intrinsic mathematical interest and has been attracting
mathematicians for some time. There is a branch of mathematics which is, loosely speaking,
an outgrowth of Buffon’s needle problem (1777) and Bertrand’s paradox (1907) and which
is called integral geometry or geometric probability [7–9]. This branch, which attracted the
attention of Poincare´ and Mark Kac (see the foreword to the book by Santalo´ [8]), has dealt
with the problem of distribution of distance.
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