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| author | Charles Karney <charles@karney.com> | 2017-12-03 17:06:02 -0500 |
|---|---|---|
| committer | Charles Karney <charles@karney.com> | 2017-12-03 17:06:02 -0500 |
| commit | a4308d9c529fd913cd8d50813fc913bf80552a38 (patch) | |
| tree | ae20e1e83a59c9f82fbdf3223b1052f40e245180 /docs | |
| parent | a141977eb6a476b4dfc7e35aaa26531f01f2fff9 (diff) | |
| download | PROJ-a4308d9c529fd913cd8d50813fc913bf80552a38.tar.gz PROJ-a4308d9c529fd913cd8d50813fc913bf80552a38.zip | |
Fixes to geodesic documention:
* replace SVG figure by PNG version,
* rewrite maths using :math:,
* put references in the main reference section,
* reformat references for consistency,
* put references in alphabetical order,
* use USGS URL for Snyder (1987).
Diffstat (limited to 'docs')
| -rw-r--r-- | docs/source/geodesic.rst | 187 | ||||
| -rw-r--r-- | docs/source/references.rst | 134 |
2 files changed, 179 insertions, 142 deletions
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]_. diff --git a/docs/source/references.rst b/docs/source/references.rst index 31100ded..e80e8caf 100644 --- a/docs/source/references.rst +++ b/docs/source/references.rst @@ -5,30 +5,110 @@ References ================================================================================ -.. [Evenden1995] Evenden, G. I., 1995, `Cartograpic Projection Procedures for the UNIX Environment - A User's Manual <https://github.com/OSGeo/proj.4/blob/master/docs/old/proj_4_3_12.pdf>`_ - -.. [Evenden2005] Evenden, G. I., 2005, `libproj4: A Comprehensive Library of Cartographic Projection Functions (Preliminary Draft) <https://github.com/OSGeo/proj.4/blob/master/docs/old/libproj.pdf>`_ - -.. [Steers1970] Steers, J.A., 1970, An introduction to the study of map projections (15th ed.): London, Univ. London Press, p. 229 - -.. [Snyder1987] Snyder. John P. 1987. `Map Projections - A Working Manual <https://github.com/OSGeo/proj.4/blob/master/docs/old/USGS-Snyder-Map-Projections-A-Working-Manual-1987.pdf>`_. US. Geological Survey professional paper; 1395. - -.. [Snyder1993] Snyder, 1993, Flattening the Earth, Chicago and London, The university of Chicago press - -.. [EberHewitt1979] Eber, L.E., and R.P. Hewitt. 1979. `Conversion algorithms for the CALCOFI station grid <http://www.calcofi.org/publications/calcofireports/v20/Vol_20_Eber___Hewitt.pdf>`__. California Cooperative Oceanic Fisheries Investigations Reports 20:135-137. - -.. [WeberMoore2013] Weber, E.D., and T.J. Moore. 2013. `Corrected Conversion Algorithms For The Calcofi Station Grid And Their Implementation In Several Computer Languages <http://calcofi.org/publications/calcofireports/v54/Vol_54_Weber.pdf>`__. California Cooperative Oceanic Fisheries Investigations Reports 54. - -.. [CalabrettaGreisen2002] M. Calabretta and E. Greisen, 2002, "Representations of celestial coordinates in FITS". Astronomy & Astrophysics 395, 3, 1077–1122. - -.. [ChanONeil1975] F. Chan and E.M.O'Neill, 1975, "Feasibility Study of a Quadrilateralized Spherical Cube Earth Data Base". Tech. Rep. EPRF 2-75 (CSC), Environmental Prediction Research Facility. - -.. [ONeilLaubscher1976] E.M. O'Neill and R.E. Laubscher, 1976, "Extended Studies of a Quadrilateralized Spherical Cube Earth Data Base". Tech. Rep. NEPRF 3-76 (CSC), Naval Environmental Prediction Research Facility. - -.. [LambersKolb2012] M. Lambers and A. Kolb, 2012, "Ellipsoidal Cube Maps for Accurate Rendering of Planetary-Scale Terrain Data", Proc. Pacfic Graphics (Short Papers). - -.. [Zajac1978] A. Zajac, 1978, "Atlas of distribution of vascular plants in Poland (ATPOL)". Taxon 27(5/6), 481–484. - -.. [Komsta2016] L. Komsta, 2016, `ATPOL geobotanical grid revisited – a proposal of coordinate conversion algorithms <http://wydawnictwo.up.lublin.pl/annales/Agricultura/2016/1/03.pdf>`__. Annales UMCS Sectio E Agricultura 71(1), 31-37. - -.. [Verey2017] M. Verey, 2017, `Theoretical analysis and practical consequences of adopting an ATPOL grid model as a conical projection, defining the conversion of plane coordinates to the WGS - 84 ellipsoid <http://www.botany.pl/atpol/Siatka%20ATPOL%20w%20analitycznym%20ujeciu.pdf>`__. Fragmenta Floristica et Geobotanica Polonica (preprint submitted). +.. [Bessel1825] F. W. Bessel, 1825, + `The calculation of longitude and latitude from geodesic measurements + <https://arxiv.org/abs/0908.1824>`_, + Astron. Nachr. **331**\ (8), 852–861 (2010), + translated by C. F. F. Karney and R. E. Deakin. + +.. [CalabrettaGreisen2002] M. Calabretta and E. Greisen, 2002, + "Representations of celestial coordinates in FITS". + Astronomy & Astrophysics 395, 3, 1077–1122. + +.. [ChanONeil1975] F. Chan and E. M. O'Neill, 1975, + "Feasibility Study of a Quadrilateralized Spherical Cube Earth Data Base", + Tech. Rep. EPRF 2-75 (CSC), Environmental Prediction Research Facility. + +.. [Danielsen1989] J. Danielsen, 1989, + `The area under the geodesic + <https://doi.org/10.1179/003962689791474267>`_, + Survey Review **30**\ (232), 61–66. + +.. [EberHewitt1979] L. E. Eber and R.P. Hewitt, 1979, + `Conversion algorithms for the CALCOFI station grid + <http://www.calcofi.org/publications/calcofireports/v20/Vol_20_Eber___Hewitt.pdf>`_, + California Cooperative Oceanic Fisheries Investigations Reports 20:135-137. + +.. [Evenden1995] G. I. Evenden, 1995, + `Cartograpic Projection Procedures for the UNIX Environment - + A User's Manual + <https://github.com/OSGeo/proj.4/blob/master/docs/old/proj_4_3_12.pdf>`_. + +.. [Evenden2005] G. I. Evenden, 2005, + `libproj4: A Comprehensive Library of Cartographic Projection Functions + (Preliminary Draft) + <https://github.com/OSGeo/proj.4/blob/master/docs/old/libproj.pdf>`_. + +.. [GeodesicBib] `A geodesic bibliography + <https://geographiclib.sourceforge.io/geodesic-papers/biblio.html>`_. + +.. [GeodesicWiki] The wikipedia page, + `Geodesics on an ellipsoid + <https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid>`_. + +.. [Helmert1880] F. R. Helmert, 1880, + `Mathematical and Physical Theories of Higher Geodesy, Vol 1 + <https://doi.org/10.5281/zenodo.32050>`_, + (Teubner, Leipzig), Chaps. 5–7. + +.. [Karney2011] C. F. F. Karney, 2011, + `Geodesics on an ellipsoid of revolution + <https://arxiv.org/abs/1102.1215v1>`_; + `errata + <https://geographiclib.sourceforge.io/geod-addenda.html#geod-errata>`_. + +.. [Karney2013] C. F. F. Karney, 2013, + `Algorithms for geodesics + <https://doi.org/10.1007/s00190-012-0578-z>`_, + J. Geodesy **87**\ (1) 43–55; + `addenda <https://geographiclib.sourceforge.io/geod-addenda.html>`_. + +.. [Komsta2016] L. Komsta, 2016, + `ATPOL geobotanical grid revisited - a proposal of coordinate conversion + algorithms + <http://wydawnictwo.up.lublin.pl/annales/Agricultura/2016/1/03.pdf>`_, + Annales UMCS Sectio E Agricultura 71(1), 31–37. + +.. [LambersKolb2012] M. Lambers and A. Kolb, 2012, + "Ellipsoidal Cube Maps for Accurate Rendering of Planetary-Scale + Terrain Data", Proc. Pacfic Graphics (Short Papers). + +.. [ONeilLaubscher1976] E. M. O'Neill and R. E. Laubscher, 1976, + "Extended Studies of a Quadrilateralized Spherical Cube Earth Data Base", + Tech. Rep. NEPRF 3-76 (CSC), + Naval Environmental Prediction Research Facility. + +.. [Snyder1987] J. P. Snyder, 1987, + `Map Projections - A Working Manual + <https://pubs.er.usgs.gov/publication/pp1395>`_. + U.S. Geological Survey professional paper 1395. + +.. [Snyder1993] J. P. Snyder, 1993, + Flattening the Earth, Chicago and London, The University of Chicago press. + +.. [Steers1970] J. A. Steers, 1970, + An introduction to the study of map projections (15th ed.), + London Univ. Press, p. 229. + +.. [Verey2017] M. Verey, 2017, + `Theoretical analysis and practical consequences of adopting an ATPOL + grid model as a conical projection, defining the conversion of plane + coordinates to the WGS-84 ellipsoid + <http://www.botany.pl/atpol/Siatka%20ATPOL%20w%20analitycznym%20ujeciu.pdf>`_, + Fragmenta Floristica et Geobotanica Polonica (preprint submitted). + +.. [Vincenty1975] T. Vincenty, 1975, + `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. + +.. [WeberMoore2013] E. D. Weber and T.J. Moore, 2013, + `Corrected Conversion Algorithms For The Calcofi Station Grid And Their + Implementation In Several Computer Languages + <http://calcofi.org/publications/calcofireports/v54/Vol_54_Weber.pdf>`_, + California Cooperative Oceanic Fisheries Investigations Reports 54. + +.. [Zajac1978] A. Zajac, 1978, + "Atlas of distribution of vascular plants in Poland (ATPOL)", + Taxon 27(5/6), 481–484. |
