Two Integrals of Geodetic Lines in Oblate Ellipsoidal Coordinates

TL;DR

This paper derives series expansions for integrals related to geodetic lines on an oblate ellipsoid, aiming to facilitate analytical computation of longitude, latitude, and path length without relying solely on numerical methods.

Contribution

It introduces a power series expansion for core geodetic integrals in oblate ellipsoidal coordinates, reducing them to sums involving inverse trigonometric functions, square roots, and elliptic integrals.

Findings

Series expansion simplifies geodetic integral calculations.

Reduces reliance on numerical integration methods.

Provides analytical expressions for geodetic line properties.

Abstract

The manuscript establishes a series expansion of the core integral that relates changes in longitude and latitude along the geodetic line in oblate elliptical coordinates, and of a companion integral which is the path length along this line as a function of latitude. The expansion is a power series in the scaled (constant) altitude of the trajectory over the surface of the ellipsoid. Each term of this series is reduced to sums over inverse trigonometric functions, square roots and Elliptic Integrals. The aim is to avoid purely numerical means of integration.