Reconstruction of map projection, its inverse and re-projection

Tomáš Bayer; Milada Kočandrlová

Applications of Mathematics (2018)

  • Volume: 63, Issue: 4, page 455-481
  • ISSN: 0862-7940

Abstract

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This paper focuses on the automatic recognition of map projection, its inverse and re-projection. Our analysis leads to the unconstrained optimization solved by the hybrid BFGS nonlinear least squares technique. The objective function is represented by the squared sum of the residuals. For the map re-projection the partial differential equations of the inverse transformation are derived. They can be applied to any map projection. Illustrative examples of the stereographic and globular Nicolosi projections frequently used in early maps are involved and their inverse formulas are presented.

How to cite

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Bayer, Tomáš, and Kočandrlová, Milada. "Reconstruction of map projection, its inverse and re-projection." Applications of Mathematics 63.4 (2018): 455-481. <http://eudml.org/doc/294576>.

@article{Bayer2018,
abstract = {This paper focuses on the automatic recognition of map projection, its inverse and re-projection. Our analysis leads to the unconstrained optimization solved by the hybrid BFGS nonlinear least squares technique. The objective function is represented by the squared sum of the residuals. For the map re-projection the partial differential equations of the inverse transformation are derived. They can be applied to any map projection. Illustrative examples of the stereographic and globular Nicolosi projections frequently used in early maps are involved and their inverse formulas are presented.},
author = {Bayer, Tomáš, Kočandrlová, Milada},
journal = {Applications of Mathematics},
keywords = {mathematical cartography; inverse projection; analysis; nonlinear least squares; partial differential equation; optimization; hybrid BFGS; early map; re-projection},
language = {eng},
number = {4},
pages = {455-481},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Reconstruction of map projection, its inverse and re-projection},
url = {http://eudml.org/doc/294576},
volume = {63},
year = {2018},
}

TY - JOUR
AU - Bayer, Tomáš
AU - Kočandrlová, Milada
TI - Reconstruction of map projection, its inverse and re-projection
JO - Applications of Mathematics
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 455
EP - 481
AB - This paper focuses on the automatic recognition of map projection, its inverse and re-projection. Our analysis leads to the unconstrained optimization solved by the hybrid BFGS nonlinear least squares technique. The objective function is represented by the squared sum of the residuals. For the map re-projection the partial differential equations of the inverse transformation are derived. They can be applied to any map projection. Illustrative examples of the stereographic and globular Nicolosi projections frequently used in early maps are involved and their inverse formulas are presented.
LA - eng
KW - mathematical cartography; inverse projection; analysis; nonlinear least squares; partial differential equation; optimization; hybrid BFGS; early map; re-projection
UR - http://eudml.org/doc/294576
ER -

References

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  1. Al-Baali, M., Fletcher, R., 10.2307/2582880, J. Oper. Res. Soc. 36 (1985), 405-421. (1985) Zbl0578.65064DOI10.2307/2582880
  2. Barancsuk, Á., 10.1007/978-3-319-19602-2_17, Progress in Cartography. EuroCarto 2015 G. Gartner, M. Jobst, H. Huang Springer, Cham 267-288 (2016). (2016) DOI10.1007/978-3-319-19602-2_17
  3. Bayer, T., 10.1007/s10707-013-0200-4, GeoInformatica 18 (2014), 621-669. (2014) DOI10.1007/s10707-013-0200-4
  4. Bayer, T., 10.1007/s10707-015-0234-x, GeoInformatica 20 (2016), 241-284. (2016) DOI10.1007/s10707-015-0234-x
  5. Bayer, T., Detectproj---software for the projection analysis, Available athttp://sourceforge.net/projects/detectproj/ (2017). (2017) 
  6. Bayer, T., 10.14311/gi.17.2.3, Geoinformatics FCE CTU 17 (2018), 31-64. (2018) DOI10.14311/gi.17.2.3
  7. Bildirici, I. O., 10.1016/S0098-3004(03)00090-6, Computers & Geosciences 29 (2003), 1003-1011. (2003) DOI10.1016/S0098-3004(03)00090-6
  8. Bildirici, I. O., 10.1080/15230406.2016.1200492, Cartography and Geographic Information Science 44 (2017), 463-471. (2017) DOI10.1080/15230406.2016.1200492
  9. Evenden, G. I., libproj4: A comprehensive library of cartographic projection functions, Falmouth, Massachusetts (2005). (2005) 
  10. Flacke, W., Kraus, B., Working with Projections and Datum Transformations in ArcGIS: Theory and Practical Examples, Points Verlag, Norden (2005). (2005) 
  11. Fletcher, R., Xu, C., 10.1093/imanum/7.3.371, IMA J. Numer. Anal. 7 (1987), 371-389. (1987) Zbl0648.65051MR0968531DOI10.1093/imanum/7.3.371
  12. Harrison, E., Mahdavi-Amiri, A., Samavati, F., 10.1109/CW.2011.36, Cyberworlds (CW), 2011 International Conference on Cyberworlds IEEE (2011), 136-143. (2011) DOI10.1109/CW.2011.36
  13. Huschens, J., 10.1137/0804005, SIAM J. Optim. 4 (1994), 108-129. (1994) Zbl0798.65064MR1260409DOI10.1137/0804005
  14. Ipbüker, C., 10.1007/978-3-642-02454-2_40, International Conference on Computational Science and Its Applications O. Gervasi et al. Springer, Berlin (2009), 553-564. (2009) DOI10.1007/978-3-642-02454-2_40
  15. Ipbüker, C., Bildirici, I., A general algorithm for the inverse transformation of map projections using jacobian matrices, Proceedings of the Third International Symposium Mathematical & Computational Applications 2002 Konya, Turkey (2002), 175-182. (2002) 
  16. Jenny, B., Map Analyst, Available at http://mapanalyst.org (2011). (2011) 
  17. Lapaine, M., Mollweide map projection, KoG 15 (2011), 7-16. (2011) Zbl1261.51016MR2951618
  18. Lukšan, L., 10.1007/BF02191760, J. Optimization Theory Appl. 83 (1994), 27-47. (1994) Zbl0819.90097MR1298855DOI10.1007/BF02191760
  19. Lukšan, L., 10.1007/BF02275350, J. Optimization Theory Appl. 89 (1996), 575-595. (1996) Zbl0851.90118MR1393364DOI10.1007/BF02275350
  20. Lukšan, L., Spedicato, E., 10.1016/S0377-0427(00)00420-9, J. Comput. Appl. Math. 124 (2000), 61-95. (2000) Zbl0985.65066MR1803294DOI10.1016/S0377-0427(00)00420-9
  21. Ratner, D. A., 10.1559/152304091783805536, Cartography and Geographic Information Systems 18 (1991), 104-108. (1991) DOI10.1559/152304091783805536
  22. Šavrič, B., Jenny, B., 10.1080/13658816.2014.924628, International Journal of Geographical Information Science 28 (2014), 2373-2389. (2014) DOI10.1080/13658816.2014.924628
  23. Smart, W. M., Green, R. M., 10.1017/cbo9781139167574, Cambridge University Press, Cambridge (1977). (1977) MR0181431DOI10.1017/cbo9781139167574
  24. Snyder, J. P., Map Projections Used by the US Geological Survey, Technical report, US Government Printing Office, Washington (1982). (1982) 
  25. Snyder, J. P., Map projections: A working manual, US Government Printing Office, Washington (1987). (1987) 
  26. Tobler, W. R., 10.1111/j.1467-8306.1966.tb00562.x, Annals of the Association of American Geographers 56 (1966), 351-360. (1966) DOI10.1111/j.1467-8306.1966.tb00562.x
  27. Tobler, W. R., Numerical approaches to map projections, Beitrage zur theoretischen Kartographie, Festschrift f{ü}r Erik Amberger, hg. Ingrid Kretschmer Franz Deuticke, Wien (1977), 51-64. (1977) 
  28. Tobler, W. R., 10.1559/152304086783900103, The American Cartographer 13 (1986), 135-139. (1986) DOI10.1559/152304086783900103
  29. Yabe, H., Takahashi, T., 10.1007/BF01586927, Math. Program., Ser. A 51 (1991), 75-100. (1991) Zbl0737.90064MR1119246DOI10.1007/BF01586927
  30. Yang, Q. H., Snyder, J. P., Tobler, W. R., Map Projection Transformation. Principles and Applications, Taylor & Francis, London (2000). (2000) Zbl0980.86006MR1832180
  31. Zhou, W., Chen, X., 10.1137/090748470, SIAM J. Optim. 20 (2010), 2422-2441. (2010) Zbl1211.90131MR2678399DOI10.1137/090748470

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