On inversions of van der Grinten projections

Tomáš Bayer; Milada Kočandrlová

Applications of Mathematics (2021)

  • Volume: 66, Issue: 6, page 887-927
  • ISSN: 0862-7940

Abstract

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Approximately 150 map projections are known, but the inverse forms have been published for only two-thirds of them. This paper focuses on finding the inverse forms of van der Grinten projections I--IV, both by non-linear partial differential equations and by the straightforward inverse of their projection equations. Taking into account the particular cases, new derivations of coordinate functions are also presented. Both the direct and inverse equations have the analytic form, are easy to implement and are applicable to the coordinate transformations.

How to cite

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Bayer, Tomáš, and Kočandrlová, Milada. "On inversions of van der Grinten projections." Applications of Mathematics 66.6 (2021): 887-927. <http://eudml.org/doc/297905>.

@article{Bayer2021,
abstract = {Approximately 150 map projections are known, but the inverse forms have been published for only two-thirds of them. This paper focuses on finding the inverse forms of van der Grinten projections I--IV, both by non-linear partial differential equations and by the straightforward inverse of their projection equations. Taking into account the particular cases, new derivations of coordinate functions are also presented. Both the direct and inverse equations have the analytic form, are easy to implement and are applicable to the coordinate transformations.},
author = {Bayer, Tomáš, Kočandrlová, Milada},
journal = {Applications of Mathematics},
keywords = {mathematical cartography; inverse form; map; projection; van der Grinten; GIS},
language = {eng},
number = {6},
pages = {887-927},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On inversions of van der Grinten projections},
url = {http://eudml.org/doc/297905},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Bayer, Tomáš
AU - Kočandrlová, Milada
TI - On inversions of van der Grinten projections
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 6
SP - 887
EP - 927
AB - Approximately 150 map projections are known, but the inverse forms have been published for only two-thirds of them. This paper focuses on finding the inverse forms of van der Grinten projections I--IV, both by non-linear partial differential equations and by the straightforward inverse of their projection equations. Taking into account the particular cases, new derivations of coordinate functions are also presented. Both the direct and inverse equations have the analytic form, are easy to implement and are applicable to the coordinate transformations.
LA - eng
KW - mathematical cartography; inverse form; map; projection; van der Grinten; GIS
UR - http://eudml.org/doc/297905
ER -

References

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  1. Bayer, T., 10.1007/s10707-013-0200-4, GeoInformatica 18 (2014), 621-669. (2014) DOI10.1007/s10707-013-0200-4
  2. Bayer, T., 10.1007/s10707-015-0234-x, GeoInformatica 20 (2016), 241-284. (2016) DOI10.1007/s10707-015-0234-x
  3. Bayer, T., Detectproj, software for the map projection analysis, Available athttp://sourceforge.net/projects/detectproj/ (2017). (2017) 
  4. Bayer, T., Kočandrlová, M., 10.21136/AM.2018.0096-18, Appl. Math., Praha 63 (2018), 455-481. (2018) Zbl06945742MR3842963DOI10.21136/AM.2018.0096-18
  5. W. D. Brooks, C. E. Roberts, Jr., 10.1559/152304076784080096, American Cartographer 3 (1976), 143-150. (1976) DOI10.1559/152304076784080096
  6. Evenden, G., Butler, H., PROJ, Available at https://proj.org/ (1983-2020). (1983) 
  7. Fenna, D., 10.1201/b15876, CRC Press/Taylor & Francis, Boca Raton (2007). (2007) DOI10.1201/b15876
  8. Grossman, S. I., 10.1016/c2013-0-07351-3, Saunders College Pub., Portland (1995). (1995) DOI10.1016/c2013-0-07351-3
  9. Grosvenor, G., National Geographic Map Of World, National Geographic Society, New York (1943). (1943) 
  10. Lowman, P. D., Frey, H. V., A Geophysical Atlas for Interpretation of Satellite-Derived Data, National Aeronautics and Space Administration, Washington (1979). (1979) 
  11. O'Keefe, J. A., Greenberg, A., 10.1559/152304077784080392, American Cartographer 4 (1977), 127-132. (1977) DOI10.1559/152304077784080392
  12. Reeves, E. A., 10.2307/1776259, Geographical Journal 24 (1904), 670-672. (1904) DOI10.2307/1776259
  13. Rubincam, D. P., Inverting x , y Grid Coordinates to Obtain Latitude and Longitude in the Van Der Grinten Projection, NASA Technical Memorandum 81998. National Aeronautics and Space Administration, Washington (1980). (1980) 
  14. Snyder, J. P., 10.1559/152304079784022709, American Cartographer 6 (1979), 81. (1979) DOI10.1559/152304079784022709
  15. Snyder, J. P., 10.3133/pp1395, U.S. Geological Survey Profesional Paper 1395. U.S. Government Printing Office, Washington (1987). (1987) DOI10.3133/pp1395
  16. Snyder, J. P., Flattening the Earth: Two Thousand Years of Map Projections, University of Chicago Press, Chicago (1997). (1997) 
  17. Grinten, A. J. van der, Darstellung der ganzen Erdoberfläche auf einer kreisförmigen Projektionsebene, Petermann's Mitteilungen 50 (1904), 155-159 German. (1904) 
  18. Grinten, A. J. van der, 10.2475/ajs.s4-19.113.357, Am. J. Sci. 19 (1905), 357-366. (1905) DOI10.2475/ajs.s4-19.113.357
  19. Grinten, A. J. van der, Zur Verebnung der ganzen Erdoberfläche: Nachtrag zu der Darstellung in Pet. Mitt. 1904. Heft VII, S. 155-159, Petermann's Mitteilungen 51 (1905), 237 German. (1905) 
  20. Zöppritz, K. J., Leitfaden der Kartenentwurfslehre. 1 Teil. Die Projektionslehre, B. G. Teubner, Leipzig (1912), German 9999JFM99999 43.1102.07. (1912) 

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