| Title:
|
On inversions of van der Grinten projections (English) |
| Author:
|
Bayer, Tomáš |
| Author:
|
Kočandrlová, Milada |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
66 |
| Issue:
|
6 |
| Year:
|
2021 |
| Pages:
|
887-927 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
mathematical cartography |
| Keyword:
|
inverse form |
| Keyword:
|
map |
| Keyword:
|
projection |
| Keyword:
|
van der Grinten |
| Keyword:
|
GIS |
| MSC:
|
53A20 |
| MSC:
|
86A30 |
| idZBL:
|
Zbl 07442411 |
| idMR:
|
MR4342613 |
| DOI:
|
10.21136/AM.2021.0026-20 |
| . |
| Date available:
|
2021-11-18T15:28:55Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149268 |
| . |
| Reference:
|
[1] Bayer, T.: Estimation of an unknown cartographic projection and its parameters from the map.GeoInformatica 18 (2014), 621-669. 10.1007/s10707-013-0200-4 |
| Reference:
|
[2] Bayer, T.: Advanced methods for the estimation of an unknown projection from a map.GeoInformatica 20 (2016), 241-284. 10.1007/s10707-015-0234-x |
| Reference:
|
[3] Bayer, T.: Detectproj, software for the map projection analysis.Available athttp://sourceforge.net/projects/detectproj/ (2017). |
| Reference:
|
[4] Bayer, T., Kočandrlová, M.: Reconstruction of map projection, its inverse and re-projection.Appl. Math., Praha 63 (2018), 455-481. Zbl 06945742, MR 3842963, 10.21136/AM.2018.0096-18 |
| Reference:
|
[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. 10.1559/152304076784080096 |
| Reference:
|
[6] Evenden, G., Butler, H.: PROJ.Available at https://proj.org/ (1983-2020). |
| Reference:
|
[7] Fenna, D.: Cartographic Science: A Compendium of Map Projections, with Derivations.CRC Press/Taylor & Francis, Boca Raton (2007). 10.1201/b15876 |
| Reference:
|
[8] Grossman, S. I.: Multivariable Calculus, Linear Algebra, and Differential Equations.Saunders College Pub., Portland (1995). 10.1016/c2013-0-07351-3 |
| Reference:
|
[9] Grosvenor, G.: National Geographic Map Of World.National Geographic Society, New York (1943). |
| Reference:
|
[10] Lowman, P. D., Frey, H. V.: A Geophysical Atlas for Interpretation of Satellite-Derived Data.National Aeronautics and Space Administration, Washington (1979). |
| Reference:
|
[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. 10.1559/152304077784080392 |
| Reference:
|
[12] Reeves, E. A.: Van der Grinten's projection.Geographical Journal 24 (1904), 670-672. 10.2307/1776259 |
| Reference:
|
[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). |
| Reference:
|
[14] Snyder, J. P.: Communications from readers: Projection notes.American Cartographer 6 (1979), 81. 10.1559/152304079784022709 |
| Reference:
|
[15] Snyder, J. P.: Map projections: A working manual.U.S. Geological Survey Profesional Paper 1395. U.S. Government Printing Office, Washington (1987). 10.3133/pp1395 |
| Reference:
|
[16] Snyder, J. P.: Flattening the Earth: Two Thousand Years of Map Projections.University of Chicago Press, Chicago (1997). |
| Reference:
|
[17] Grinten, A. J. van der: Darstellung der ganzen Erdoberfläche auf einer kreisförmigen Projektionsebene.Petermann's Mitteilungen 50 (1904), 155-159 German. |
| Reference:
|
[18] Grinten, A. J. van der: New circular projection of the whole Earth's surface.Am. J. Sci. 19 (1905), 357-366. 10.2475/ajs.s4-19.113.357 |
| Reference:
|
[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. |
| Reference:
|
[20] Zöppritz, K. J.: Leitfaden der Kartenentwurfslehre. 1 Teil. Die Projektionslehre.B. G. Teubner, Leipzig (1912), German \99999JFM99999 43.1102.07. |
| . |
