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diff --git a/docs/source/geodesic.rst b/docs/source/geodesic.rst index 29327ae8..d54212ca 100644 --- a/docs/source/geodesic.rst +++ b/docs/source/geodesic.rst @@ -10,53 +10,52 @@ Geodesic calculations 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 +Consider a ellipsoid of revolution with equatorial radius :math:`a`, polar +semi-axis :math:`b`, and flattening :math:`f=(a−b)/a`. Points on +the surface of the ellipsoid are characterized by their latitude :math:`\phi` +and longitude :math:`\lambda`. (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`) +:math:`(\phi_1,\lambda_1)` and :math:`(\phi_2,\lambda_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`. +:math:`s_{12}` and the geodesic from point 1 to point 2 has forward +azimuths :math:`\alpha_1` and :math:`\alpha_2` at the two end +points. In this figure, we have :math:`\lambda_{12}=\lambda_2-\lambda_1`. .. raw:: html <center> - <img src="https://upload.wikimedia.org/wikipedia/commons/c/cb/Geodesic_problem_on_an_ellipsoid.svg" + <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Geodesic_problem_on_an_ellipsoid.svg/320px-Geodesic_problem_on_an_ellipsoid.svg.png" alt="Figure from wikipedia" - width="250"> + width="320"> </center> 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 programs +Solution of geodesic problems ----------------------------- 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`. +* the direct problem — given :math:`\phi_1`, + :math:`\lambda_1`, :math:`\alpha_1`, :math:`s_{12}`, + determine :math:`\phi_2`, :math:`\lambda_2`, :math:`\alpha_2`. -* 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`. +* the inverse problem — given :math:`\phi_1`, + :math:`\lambda_1`, :math:`\phi_2`, :math:`\lambda_2`, + determine :math:`s_{12}`, :math:`\alpha_1`, + :math:`\alpha_2`. PROJ incorporates `C library for Geodesics <https://geographiclib.sourceforge.io/1.49/C/>`_ from `GeographicLib <https://geographiclib.sourceforge.io>`_. This library provides routines to solve the direct and inverse geodesic problems. Full double -precision accuracy is maintained provided that −0.02 < *f* < 0.02. Refer +precision accuracy is maintained provided that +:math:`\lvert f\rvert<\frac1{50}`. Refer to the `application programming interface @@ -70,9 +69,9 @@ rest of PROJ: * angles (latitudes, longitudes, and azimuths) are in degrees (instead of in radians); -* the shape of ellipsoid is specified by the flattening *f*; this can - be negative to denote a prolate ellipsoid; setting *f* = 0 corresponds - to a sphere in which case the geodesic becomes a great circle. +* the shape of ellipsoid is specified by the flattening :math:`f`; this can + be negative to denote a prolate ellipsoid; setting :math:`f=0` corresponds + to a sphere, in which case the geodesic becomes a great circle. PROJ also includes a command line tool, :ref:`geod`\ (1), for performing simple geodesic calculations. @@ -82,47 +81,41 @@ Additional properties The routines also calculate several other quantities of interest -* *S*\ :sub:`12` is the area between the geodesic from point 1 to +* :math:`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 - (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 + :math:`(\phi_1,\lambda_1)`, :math:`(0,\lambda_1)`, + :math:`(0,\lambda_2)`, and + :math:`(\phi_2,\lambda_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 +* :math:`m_{12}`, the reduced length of the geodesic is defined such + that if the initial azimuth is perturbed by :math:`d\alpha_1` + (radians) then the second point is displaced by :math:`m_{12}\,d\alpha_1` + in the direction perpendicular to the + geodesic. :math:`m_{12}` is given in meters. On a curved surface + the reduced length obeys a symmetry relation, :math:`m_{12}+m_{21}=0`. + On a flat surface, we have :math:`m_{12}=s_{12}`. +* :math:`M_{12}` and :math:`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*\ :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. + :\math`dt`, then they are separated by a distance :math:`M_{12}\,dt` at + point 2. :math:`M_{21}` is defined similarly (with the geodesics + being parallel to one another at point 2). :math:`M_{12}` and + :math:`M_{21}` are dimensionless quantities. On a flat surface, + we have :math:`M_{12}=M_{21}=1`. +* :math:`\sigma_{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°. + the next is always :math:`180^\circ`. 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` +* :math:`s_{13}=s_{12}+s_{23}`, +* :math:`\sigma_{13}=\sigma_{12}+\sigma_{23}`, +* :math:`S_{13}=S_{12}+S_{23}`, +* :math:`m_{13}=m_{12}M_{23}+m_{23}M_{21}`, +* :math:`M_{13}=M_{12}M_{23}-(1-M_{12}M_{21})m_{23}/m_{12}`, +* :math:`M_{31}=M_{32}M_{21}-(1-M_{23}M_{32})m_{12}/m_{23}`. Multiple shortest geodesics --------------------------- @@ -132,42 +125,41 @@ The shortest distance found by solving the inverse problem is 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 +* :math:`\phi_1=-\phi_2` (with neither point at + a pole). If :math:`\alpha_1=\alpha_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 + :math:`[\alpha_1,\alpha_2]\leftarrow[\alpha_2,\alpha_1]`, + :math:`[M_{12},M_{21}]\leftarrow[M_{21},M_{12}]`, + :math:`S_{12}\leftarrow-S_{12}`. + (This occurs when the longitude difference is near :math:`\pm180^\circ` + for oblate ellipsoids.) +* :math:`\lambda_2=\lambda_1\pm180^\circ` (with + neither point at a pole). If :math:`\alpha_1=0^\circ` or + :math:`\pm180^\circ`, 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 + :math:`[\alpha_1,\alpha_2]\leftarrow[-\alpha_1,-\alpha_2]`, + :math:`S_{12}\leftarrow-S_{12}`. (This occurs when + :math:`\phi_2` is near :math:`-\phi_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. + :math:`[\alpha_1,\alpha_2]\leftarrow[\alpha_1,\alpha_2]+[\delta,-\delta]`, + for arbitrary :math:`\delta`. (For spheres, this prescription applies when points 1 and 2 are antipodal.) -* *s*\ :sub:`12` = 0 (coincident points). There are infinitely many +* :math:`s_{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. + :math:`[\alpha_1,\alpha_2]\leftarrow[\alpha_1,\alpha_2]+[\delta,\delta]`, + for arbitrary :math:`\delta`. 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 algorithms implemented by this package are given in [Karney2013]_ +and are based on [Bessel1825]_ and [Helmert1880]_; the algorithm for +areas is based on [Danielsen1989]_. These improve on the work of +[Vincenty1975]_ 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 @@ -178,40 +170,5 @@ Vincenty (1975) in the following respects: 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) - <https://arxiv.org/abs/0908.1824>`_, - 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 - <https://doi.org/10.5281/zenodo.32050>`_, - (Teubner, Leipzig, 1880), Chaps. 5–7. -* T. Vincenty, - `Direct and inverse solutions of geodesics on the ellipsoid with - application of nested equations - <http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf>`_, - Survey Review **23**\ (176), 88–93 (1975). -* J. Danielsen, - `The area under the geodesic - <https://doi.org/10.1179/003962689791474267>`_, - Survey Review **30**\ (232), 61–66 (1989). -* C. F. F. Karney, - `Algorithms for geodesics - <https://doi.org/10.1007/s00190-012-0578-z>`_, - J. Geodesy **87**\ (1) 43–55 (2013); - `addenda <https://geographiclib.sourceforge.io/geod-addenda.html>`_. -* C. F. F. Karney, - `Geodesics on an ellipsoid of revolution - <https://arxiv.org/abs/1102.1215v1>`_, - Feb. 2011; - `errata - <https://geographiclib.sourceforge.io/geod-addenda.html#geod-errata>`_. -* `A geodesic bibliography - <https://geographiclib.sourceforge.io/geodesic-papers/biblio.html>`_. -* The wikipedia page, - `Geodesics on an ellipsoid - <https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>`_. +Additional background material is provided in [GeodesicBib]_, +[GeodesicWiki]_, and [Karney2011]_. |