Geodesics on an ellipsoid

The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks.

The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry (Euler 1755).

If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry.

However, Newton (1687) showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid: in this case, the equator and the meridians are the only simple closed geodesics.

By the end of the 18th century, an ellipsoid of revolution (the term spheroid is also used) was a well-accepted approximation to the figure of the Earth.

The adjustment of triangulation networks entailed reducing all the measurements to a reference ellipsoid and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry (Bomford 1952, Chap.

The full solution for the direct problem (complete with computational tables and a worked out example) is given by Bessel (1825).

The shortest path or geodesic entails finding that function φ(λ) which minimizes s12.

There are also several ways of approximating geodesics on a terrestrial ellipsoid (with small flattening) (Rapp 1991, §6); some of these are described in the article on geographical distance.

7 shows the simple closed geodesics which consist of the meridians (green) and the equator (red).

13 shows geodesics (in blue) emanating A with α1 a multiple of 15° up to the point at which they cease to be shortest paths.

On an oblate ellipsoid (shown here), it is a segment of the circle of latitude centered on the point antipodal to A, φ = −φ1.

This is useful in trigonometric adjustments (Ehlert 1993), determining the physical properties of signals which follow geodesics, etc.

Gauss (1828) showed that t(s) obeys the Gauss-Jacobi equation where K(s) is the Gaussian curvature at s. As a second order, linear, homogeneous differential equation, its solution may be expressed as the sum of two independent solutions where The quantity m(s1, s2) = m12 is the so-called reduced length, and M(s1, s2) = M12 is the geodesic scale.

5, Napier's rules for quadrantal triangles can be employed, The mapping of the geodesic involves evaluating the integrals for the distance, s, and the longitude, λ, Eqs.

In geodetic applications, where f is small, the integrals are typically evaluated as a series (Legendre 1806) (Oriani 1806) (Bessel 1825) (Helmert 1880) (Rainsford 1955) (Rapp 1993).

Vincenty (1975) provides solutions for the direct and inverse problems; these are based on a series expansion carried out to third order in the flattening and provide an accuracy of about 0.1 mm for the WGS84 ellipsoid; however the inverse method fails to converge for nearly antipodal points.

Karney (2013) continues the expansions to sixth order which suffices to provide full double precision accuracy for |f| ≤ 1⁄50 and improves the solution of the inverse problem so that it converges in all cases.

Karney (2013, addendum) extends the method to use elliptic integrals which can be applied to ellipsoids with arbitrary flattening.

By the early 19th century (with the work of Legendre, Oriani, Bessel, et al.), there was a complete understanding of the properties of geodesics on an ellipsoid of revolution.

On the other hand, geodesics on a triaxial ellipsoid (with three unequal axes) have no obvious constant of the motion and thus represented a challenging unsolved problem in the first half of the 19th century.

In a remarkable paper, Jacobi (1839) discovered a constant of the motion allowing this problem to be reduced to quadrature also (Klingenberg 1982, §3.5).

These lines meet at four umbilical points (two of which are visible in this figure) where the principal radii of curvature are equal.

Here and in the other figures in this section the parameters of the ellipsoid are a:b:c = 1.01:1:0.8, and it is viewed in an orthographic projection from a point above φ = 40°, λ = 30°.

A derivation of Jacobi's result is given by Darboux (1894, §§583–584); he gives the solution found by Liouville (1846) for general quadratic surfaces.

All tangents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at β = β1 (Chasles 1846) (Hilbert & Cohn-Vossen 1952, pp.

All tangents to a transpolar geodesic touch the confocal double-sheeted hyperboloid which intersects the ellipsoid at ω = ω1.

However, on each circuit the angle at which it intersects Y = 0 becomes closer to 0° or 180° so that asymptotically the geodesic lies on the ellipse Y = 0 (Hart 1849) (Arnold 1989, p. 265), as shown in Fig.

The direct and inverse geodesic problems no longer play the central role in geodesy that they once did.

Indeed, the geodesic problem is equivalent to the motion of a particle constrained to move on the surface, but otherwise subject to no forces (Laplace 1799a) (Hilbert & Cohn-Vossen 1952, p. 222).

A geodesic on an oblate ellipsoid
Fig. 1. A geodesic AB on an ellipsoid of revolution. N is the north pole and EFH lie on the equator.
Fig. 6. Differential element of a geodesic on a sphere.
Fig. 7. Meridians and the equator are the only closed geodesics. (For the very flattened ellipsoids, there are other closed geodesics; see Figs. 11 and 12).
Fig. 10. Geodesic on a prolate ellipsoid ( f = − 1 ⁄ 50 ) with α 0 = 45° . Compare with Fig. 8.
Fig. 13. Geodesics (blue) from a single point for f = 1 ⁄ 10 , φ 1 = −30° ; geodesic circles are shown in green and the cut locus in red.
Fig. 14. Definition of reduced length and geodesic scale.
Fig. 17. Triaxial ellipsoidal coordinates.
Fig. 22. An umbilical geodesic, β 1 = 90° , ω 1 = 0° , α 1 = 135° .