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Keywords:
mathematical cartography; inverse form; map; projection; van der Grinten; GIS
mathematical cartography; inverse form; map; projection; van der Grinten; GIS
Summary:
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.
References:
[1] Bayer, T.: Estimation of an unknown cartographic projection and its parameters from the map. GeoInformatica 18 (2014), 621-669. DOI 10.1007/s10707-013-0200-4
[2] Bayer, T.: Advanced methods for the estimation of an unknown projection from a map. GeoInformatica 20 (2016), 241-284. DOI 10.1007/s10707-015-0234-x
[3] Bayer, T.: Detectproj, software for the map projection analysis. Available at http://sourceforge.net/projects/detectproj/ (2017).
[4] Bayer, T., Kočandrlová, M.: Reconstruction of map projection, its inverse and re-projection. Appl. Math., Praha 63 (2018), 455-481. DOI 10.21136/AM.2018.0096-18 | MR 3842963 | Zbl 06945742
[5] W. D. Brooks, C. E. Roberts, Jr.: An analysis of a modified van der Grinten equatorial arbitrary oval projection. American Cartographer 3 (1976), 143-150. DOI 10.1559/152304076784080096
[6] Evenden, G., Butler, H.: PROJ. Available at https://proj.org/ (1983-2020).
[7] Fenna, D.: Cartographic Science: A Compendium of Map Projections, with Derivations. CRC Press/Taylor & Francis, Boca Raton (2007). DOI 10.1201/b15876
[8] Grossman, S. I.: Multivariable Calculus, Linear Algebra, and Differential Equations. Saunders College Pub., Portland (1995). DOI 10.1016/c2013-0-07351-3
[9] Grosvenor, G.: National Geographic Map Of World. National Geographic Society, New York (1943).
[10] Lowman, P. D., Frey, H. V.: A Geophysical Atlas for Interpretation of Satellite-Derived Data. National Aeronautics and Space Administration, Washington (1979).
[11] O'Keefe, J. A., Greenberg, A.: A note on the van der Grinten projection of the whole earth onto a circular disk. American Cartographer 4 (1977), 127-132. DOI 10.1559/152304077784080392
[12] Reeves, E. A.: Van der Grinten's projection. Geographical Journal 24 (1904), 670-672. DOI 10.2307/1776259
[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).
[14] Snyder, J. P.: Communications from readers: Projection notes. American Cartographer 6 (1979), 81. DOI 10.1559/152304079784022709
[15] Snyder, J. P.: Map projections: A working manual. U.S. Geological Survey Profesional Paper 1395. U.S. Government Printing Office, Washington (1987). DOI 10.3133/pp1395
[16] Snyder, J. P.: Flattening the Earth: Two Thousand Years of Map Projections. University of Chicago Press, Chicago (1997).
[17] Grinten, A. J. van der: Darstellung der ganzen Erdoberfläche auf einer kreisförmigen Projektionsebene. Petermann's Mitteilungen 50 (1904), 155-159 German.
[18] Grinten, A. J. van der: New circular projection of the whole Earth's surface. Am. J. Sci. 19 (1905), 357-366. DOI 10.2475/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.
[20] Zöppritz, K. J.: Leitfaden der Kartenentwurfslehre. 1 Teil. Die Projektionslehre. B. G. Teubner, Leipzig (1912), German \99999JFM99999 43.1102.07.
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