Module talk:Location map/data/NATO

I find it difficult to understand the vertical stretch factor 575/324 in the formula for y introduced in [https://en.wikipedia.org/w/index.php?title=Module:Location_map/data/NATO&diff=prev&oldid=1212692713 this edit].

If I am not mistaken :File:NATO Territory Map.svg is a derivative of :File:North Atlantic Ocean laea location map.svg by cropping the lower border at 324/575 = 56.3478260869565%.

Imho this operation on a map with Lambert azimuthal equal-area projection cannot be compensated for by adding some vertical stretch factor but it requires the correction of the parameters $px\_0$, $py\_0$, $dx$ and $dy$.

The upper right corner of the map remains at lat = 71.02192702284165 N; lon = 38.61651360858501 E and the projection center remains at lat = 35 N; lon = 35 W. The projection of the upper right corner remains at

:$X\_\{UR\}\; =0.97716044\; \backslash text\{;\; \}\; Y\_\{UR\}\; =0.82614817$

However, the lower left corner moves to approximately lat = 14.35 N; lon = 96.55 W. Conversion of the lower left corner with the proj tool:

:

proj +proj=laea +lat_0=35 +lon_0=-35 +a=1 -f %.8f

-96.55 14.35

-0.97701214 -0.07076885

:$X\_\{LL\}\; =-0.97701214\; \backslash text\{;\; \}\; Y\_\{LL\}\; =-0.07076885$

From these values the projection parameters can be computed as:

:$px\_0\; =\; -100\; \backslash frac\{X\_\{LL\}\}\{X\_\{UR\}-X\_\{LL\}\}\; =\; -100\; \backslash frac\{-0.97701214\}\{0.97716044\; -\; -0.97701214\}\; =\; 49.9962$

:$py\_0\; =\; 100\; \backslash times\; \backslash left(1+\; \backslash frac\{Y\_\{LL\}\}\{Y\_\{UR\}-Y\_\{LL\}\}\backslash right)\; =\; 100\; \backslash times\; \backslash left(1+\; \backslash frac\{-0.07076885\}\{0.82614817\; -\; -0.07076885\}\backslash right)\; =\; 92.1098$

:$dx\; =\; \backslash frac\; 1\; \{X\_\{UR\}-X\_\{LL\}\}\; =\; \backslash frac\; 1\; \{0.97716044\; -\; -0.97701214\}\; =\; 0.5117$

:$dy\; =\; \backslash frac\; 1\; \{Y\_\{UR\}-Y\_\{LL\}\}\; =\; \backslash frac\; 1\; \{0.82614817\; -\; -0.07076885\}\; =\; 1.1149$

Spischot (talk) 12:14, 2 October 2024 (UTC)

:After spending some more time thinking I do now consider the vertical stretching by factor 575/324 as correct. All positions in the upper 324/575 part of the original map (y between 0% and 56.3...%) are strechteched out to extend vertically over all (0% - 100%) of the new, cropped map.

:How does this relate to my computation of the new factors above?

:The formula for

::$y\; =\; \backslash frac\{575\}\{324\}\; \backslash times\; ((100\; -\; 48.0905)\; -\; 62.8332\; \backslash times\; \backslash dotsb)$

:can be simplified as

::$y\; =\; \backslash frac\{575\}\{324\}\; \backslash times\; (51.9095\; -\; 62.8332\; \backslash times\; \backslash dotsb)$

::$=\; \backslash frac\{575\; \backslash times\; 51.9095\}\{324\}\; -\; \backslash frac\{575\; \backslash times\; 62.8332\}\{324\}\; \backslash times\; \backslash dotsb$

::$=\; 92.1234\; -\; 111.5096\; \backslash times\; \backslash dotsb$

:so the projection parameters are actually

::vertical projection center $py\_0\; =\; 92.1234$

::vertical scaling $dy\; =\; 1.115096$

:which are pretty close to my computation above. The deviation of 0.015% was caused by my approximation of the new lower left corner. Spischot (talk) 09:40, 3 October 2024 (UTC)