Geodesics on an ellipsoid ========================= Jump to * :ref:`intro` * :ref:`additional` * :ref:`multiple` * :ref:`background` * :ref:`references` .. _intro: 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 (φ\ :sub:`1`, λ\ :sub:`1`) and (φ\ :sub:`2`, λ\ :sub:`2`) is called the geodesic. Its length is *s*\ :sub:`12` and the geodesic from point 1 to point 2 has forward azimuths α\ :sub:`1` and α\ :sub:`2` at the two end points. In this figure, we have λ\ :sub:`12` = λ\ :sub:`2` − λ\ :sub:`1`. .. raw:: html
Figure from wikipedia
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 φ\ :sub:`1`, λ\ :sub:`1`, α\ :sub:`1`, *s*\ :sub:`12`, determine φ\ :sub:`2`, λ\ :sub:`2`, and α\ :sub:`2`; this is solved by :meth:`Geodesic.Direct `. * the inverse problem — given φ\ :sub:`1`, λ\ :sub:`1`, φ\ :sub:`2`, λ\ :sub:`2`, determine *s*\ :sub:`12`, α\ :sub:`1`, and α\ :sub:`2`; this is solved by :meth:`Geodesic.Inverse `. .. _additional: Additional properties --------------------- The routines also calculate several other quantities of interest * *S*\ :sub:`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 (φ\ :sub:`1`,λ\ :sub:`1`), (0,λ\ :sub:`1`), (0,λ\ :sub:`2`), and (φ\ :sub:`2`,λ\ :sub:`2`). It is given in meters\ :sup:`2`. * *m*\ :sub:`12`, the reduced length of the geodesic is defined such that if the initial azimuth is perturbed by *d*\ α\ :sub:`1` (radians) then the second point is displaced by *m*\ :sub:`12` *d*\ α\ :sub:`1` in the direction perpendicular to the geodesic. *m*\ :sub:`12` is given in meters. On a curved surface the reduced length obeys a symmetry relation, *m*\ :sub:`12` + *m*\ :sub:`21` = 0. On a flat surface, we have *m*\ :sub:`12` = *s*\ :sub:`12`. * *M*\ :sub:`12` and *M*\ :sub:`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*\ :sub:`12` *dt* at point 2. *M*\ :sub:`21` is defined similarly (with the geodesics being parallel to one another at point 2). *M*\ :sub:`12` and *M*\ :sub:`21` are dimensionless quantities. On a flat surface, we have *M*\ :sub:`12` = *M*\ :sub:`21` = 1. * σ\ :sub:`12` 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*\ :sub:`13` = *s*\ :sub:`12` + *s*\ :sub:`23` * σ\ :sub:`13` = σ\ :sub:`12` + σ\ :sub:`23` * *S*\ :sub:`13` = *S*\ :sub:`12` + *S*\ :sub:`23` * *m*\ :sub:`13` = *m*\ :sub:`12`\ *M*\ :sub:`23` + *m*\ :sub:`23`\ *M*\ :sub:`21` * *M*\ :sub:`13` = *M*\ :sub:`12`\ *M*\ :sub:`23` − (1 − *M*\ :sub:`12`\ *M*\ :sub:`21`) *m*\ :sub:`23`/*m*\ :sub:`12` * *M*\ :sub:`31` = *M*\ :sub:`32`\ *M*\ :sub:`21` − (1 − *M*\ :sub:`23`\ *M*\ :sub:`32`) *m*\ :sub:`12`/*m*\ :sub:`23` .. _multiple: 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: * φ\ :sub:`1` = −φ\ :sub:`2` (with neither point at a pole). If α\ :sub:`1` = α\ :sub:`2`, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [α\ :sub:`1`,α\ :sub:`2`] ← [α\ :sub:`2`,α\ :sub:`1`], [*M*\ :sub:`12`,\ *M*\ :sub:`21`] ← [*M*\ :sub:`21`,\ *M*\ :sub:`12`], *S*\ :sub:`12` ← −\ *S*\ :sub:`12`. (This occurs when the longitude difference is near ±180° for oblate ellipsoids.) * λ\ :sub:`2` = λ\ :sub:`1` ± 180° (with neither point at a pole). If α\ :sub:`1` = 0° or ±180°, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [α\ :sub:`1`,α\ :sub:`2`] ← [−α\ :sub:`1`,−α\ :sub:`2`], *S*\ :sub:`12` ← −\ *S*\ :sub:`12`. (This occurs when φ\ :sub:`2` is near −φ\ :sub:`1` for prolate ellipsoids.) * Points 1 and 2 at opposite poles. There are infinitely many geodesics which can be generated by setting [α\ :sub:`1`,α\ :sub:`2`] ← [α\ :sub:`1`,α\ :sub:`2`] + [δ,−δ], for arbitrary δ. (For spheres, this prescription applies when points 1 and 2 are antipodal.) * *s*\ :sub:`12` = 0 (coincident points). There are infinitely many geodesics which can be generated by setting [α\ :sub:`1`,α\ :sub:`2`] ← [α\ :sub:`1`,α\ :sub:`2`] + [δ,δ], for arbitrary δ. .. _background: 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 then 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: 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 `_.