Geodesics on an ellipsoid

Introduction

Consider an ellipsoid of revolution with equatorial radius a, polar semi-axis b, and flattening f = (a − b)/a . Points on the surface of the ellipsoid are characterized by their latitude φ and longitude λ. (Note that latitude here means the geographical latitude, the angle between the normal to the ellipsoid and the equatorial plane).

The shortest path between two points on the ellipsoid at (φ₁, λ₁) and (φ₂, λ₂) is called the geodesic. Its length is s₁₂ and the geodesic from point 1 to point 2 has forward azimuths α₁ and α₂ at the two end points. In this figure, we have λ₁₂ = λ₂ − λ₁.

A geodesic can be extended indefinitely by requiring that any sufficiently small segment is a shortest path; geodesics are also the straightest curves on the surface.

Solution of geodesic problems

Traditionally two geodesic problems are considered:

• the direct problem — given φ₁, λ₁, α₁, s₁₂, determine φ₂, λ₂, and α₂; this is solved by Geodesic.Direct.

• the inverse problem — given φ₁, λ₁, φ₂, λ₂, determine s₁₂, α₁, and α₂; this is solved by Geodesic.Inverse.

Additional properties

The routines also calculate several other quantities of interest

• S₁₂ is the area between the geodesic from point 1 to point 2 and the equator; i.e., it is the area, measured counter-clockwise, of the quadrilateral with corners (φ₁,λ₁), (0,λ₁), (0,λ₂), and (φ₂,λ₂). It is given in meters².

• m₁₂, the reduced length of the geodesic is defined such that if the initial azimuth is perturbed by dα₁ (radians) then the second point is displaced by m₁₂ dα₁ in the direction perpendicular to the geodesic. m₁₂ is given in meters. On a curved surface the reduced length obeys a symmetry relation, m₁₂ + m₂₁ = 0. On a flat surface, we have m₁₂ = s₁₂.

• M₁₂ and M₂₁ are geodesic scales. If two geodesics are parallel at point 1 and separated by a small distance dt, then they are separated by a distance M₁₂ dt at point 2. M₂₁ is defined similarly (with the geodesics being parallel to one another at point 2). M₁₂ and M₂₁ are dimensionless quantities. On a flat surface, we have M₁₂ = M₂₁ = 1.

• σ₁₂ is the arc length on the auxiliary sphere. This is a construct for converting the problem to one in spherical trigonometry. The spherical arc length from one equator crossing to the next is always 180°.

If points 1, 2, and 3 lie on a single geodesic, then the following addition rules hold:

• s₁₃ = s₁₂ + s₂₃

• σ₁₃ = σ₁₂ + σ₂₃

• S₁₃ = S₁₂ + S₂₃

• m₁₃ = m₁₂M₂₃ + m₂₃M₂₁

• M₁₃ = M₁₂M₂₃ − (1 − M₁₂M₂₁) m₂₃/m₁₂

• M₃₁ = M₃₂M₂₁ − (1 − M₂₃M₃₂) m₁₂/m₂₃

Multiple shortest geodesics

The shortest distance found by solving the inverse problem is (obviously) uniquely defined. However, in a few special cases there are multiple azimuths which yield the same shortest distance. Here is a catalog of those cases:

• φ₁ = −φ₂ (with neither point at a pole). If α₁ = α₂, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [α₁,α₂] ← [α₂,α₁], [M₁₂,M₂₁] ← [M₂₁,M₁₂], S₁₂ ← −S₁₂. (This occurs when the longitude difference is near ±180° for oblate ellipsoids.)

• λ₂ = λ₁ ± 180° (with neither point at a pole). If α₁ = 0° or ±180°, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [α₁,α₂] ← [−α₁,−α₂], S₁₂ ← −S₁₂. (This occurs when φ₂ is near −φ₁ for prolate ellipsoids.)

• Points 1 and 2 at opposite poles. There are infinitely many geodesics which can be generated by setting [α₁,α₂] ← [α₁,α₂] + [δ,−δ], for arbitrary δ. (For spheres, this prescription applies when points 1 and 2 are antipodal.)

• s₁₂ = 0 (coincident points). There are infinitely many geodesics which can be generated by setting [α₁,α₂] ← [α₁,α₂] + [δ,δ], for arbitrary δ.

Area of a polygon

The area of a geodesic polygon can be determined by summing S₁₂ for successive edges of the polygon (S₁₂ is negated so that clockwise traversal of a polygon gives a positive area). However, if the polygon encircles a pole, the sum must be adjusted by ±A/2, where A is the area of the full ellipsoid, with the sign chosen to place the result in (-A/2, A/2].

Background

The algorithms implemented by this package are given in Karney (2013) and are based on Bessel (1825) and Helmert (1880); the algorithm for areas is based on Danielsen (1989). These improve on the work of Vincenty (1975) in the following respects:

• The results are accurate to round-off for terrestrial ellipsoids (the error in the distance is less than 15 nanometers, compared to 0.1 mm for Vincenty).

• The solution of the inverse problem is always found. (Vincenty’s method fails to converge for nearly antipodal points.)

• The routines calculate differential and integral properties of a geodesic. This allows, for example, the area of a geodesic polygon to be computed.

References

• F. W. Bessel, The calculation of longitude and latitude from geodesic measurements (1825), Astron. Nachr. 331(8), 852–861 (2010), translated by C. F. F. Karney and R. E. Deakin.

• F. R. Helmert, Mathematical and Physical Theories of Higher Geodesy, Vol 1, (Teubner, Leipzig, 1880), Chaps. 5–7.

• T. Vincenty, Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations, Survey Review 23(176), 88–93 (1975).

• J. Danielsen, The area under the geodesic, Survey Review 30(232), 61–66 (1989).

• C. F. F. Karney, Algorithms for geodesics, J. Geodesy 87(1) 43–55 (2013); addenda.

• C. F. F. Karney, Geodesics on an ellipsoid of revolution, Feb. 2011; errata.

• A geodesic bibliography.

• The wikipedia page, Geodesics on an ellipsoid.