Jump to * [Introduction](#intro) * [Additional properties](#additional) * [Multiple shortest geodesics](#multiple) * [Background](#background) * [References](#references) ### Introduction Consider a 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 *s*₁₂ 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. Traditionally two geodesic problems are considered: * the direct problem — given φ₁, λ₁, α₁, *s*₁₂, determine φ₂, λ₂, and α₂; this is solved by {@link module:GeographicLib/Geodesic.Geodesic#Direct Geodesic.Direct}. * the inverse problem — given φ₁, λ₁, φ₂, λ₂, determine *s*₁₂, α₁, and α₂; this is solved by {@link module:GeographicLib/Geodesic.Geodesic#Inverse Geodesic.Inverse}. ### Additional properties The routines also calculate several other quantities of interest * *S*₁₂ 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. * *m*₁₂, the reduced length of the geodesic is defined such that if the initial azimuth is perturbed by *d*α₁ (radians) then the second point is displaced by *m*₁₂ *d*α₁ in the direction perpendicular to the geodesic. *m*₁₂ is given in meters. On a curved surface the reduced length obeys a symmetry relation, *m*₁₂ + *m*₂₁ = 0. On a flat surface, we have *m*₁₂ = *s*₁₂. * *M*₁₂ and *M*₂₁ 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 *M*₁₂ *dt* at point 2. *M*₂₁ is defined similarly (with the geodesics being parallel to one another at point 2). *M*₁₂ and *M*₂₁ are dimensionless quantities. On a flat surface, we have *M*₁₂ = *M*₂₁ = 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: * *s*₁₃ = *s*₁₂ + *s*₂₃ * σ₁₃ = σ₁₂ + σ₂₃ * *S*₁₃ = *S*₁₂ + *S*₂₃ * *m*₁₃ = *m*₁₂*M*₂₃ + *m*₂₃*M*₂₁ * *M*₁₃ = *M*₁₂*M*₂₃ − (1 − *M*₁₂*M*₂₁) *m*₂₃/*m*₁₂ * *M*₃₁ = *M*₃₂*M*₂₁ − (1 − *M*₂₃*M*₃₂) *m*₁₂/*m*₂₃ ### 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], [*M*₁₂,*M*₂₁] ← [*M*₂₁,*M*₁₂], *S*₁₂ ← −*S*₁₂. (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], *S*₁₂ ← −*S*₁₂. (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.) * *s*₁₂ = 0 (coincident points). There are infinitely many geodesics which can be generated by setting [α₁,α₂] ← [α₁,α₂] + [δ,δ], for arbitrary δ. ### 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, {@link https://arxiv.org/abs/0908.1824 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, {@link https://doi.org/10.5281/zenodo.32050 Mathematical and Physical Theories of Higher Geodesy, Vol 1}, (Teubner, Leipzig, 1880), Chaps. 5–7. * T. Vincenty, {@link http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations}, Survey Review **23**(176), 88–93 (1975). * J. Danielsen, {@link https://doi.org/10.1179/003962689791474267 The area under the geodesic}, Survey Review **30**(232), 61–66 (1989). * C. F. F. Karney, {@link https://doi.org/10.1007/s00190-012-0578-z Algorithms for geodesics}, J. Geodesy **87**(1) 43–55 (2013); {@link https://geographiclib.sourceforge.io/geod-addenda.html addenda}. * C. F. F. Karney, {@https://arxiv.org/abs/1102.1215v1 Geodesics on an ellipsoid of revolution}, Feb. 2011; {@link https://geographiclib.sourceforge.io/geod-addenda.html#geod-errata errata}. * {@link https://geographiclib.sourceforge.io/geodesic-papers/biblio.html A geodesic bibliography}. * The wikipedia page, {@link https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid Geodesics on an ellipsoid}.