Module talk:Location map/data/NATO

Formulae after cropping

I find it difficult to understand the vertical stretch factor 575/324 in the formula for y introduced in 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

$X_{UR}=0.97716044{\text{; }}Y_{UR}=0.82614817$

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

$X_{LL}=-0.97701214{\text{; }}Y_{LL}=-0.07076885$

From these values the projection parameters can be computed as:

$px_{0}=-100{\frac {X_{LL}}{X_{UR}-X_{LL}}}=-100{\frac {-0.97701214}{0.97716044--0.97701214}}=49.9962$

$py_{0}=100\times \left(1+{\frac {Y_{LL}}{Y_{UR}-Y_{LL}}}\right)=100\times \left(1+{\frac {-0.07076885}{0.82614817--0.07076885}}\right)=92.1098$

$dx={\frac {1}{X_{UR}-X_{LL}}}={\frac {1}{0.97716044--0.97701214}}=0.5117$

$dy={\frac {1}{Y_{UR}-Y_{LL}}}={\frac {1}{0.82614817--0.07076885}}=1.1149$

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

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

can be simplified as

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

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

$=92.1234-111.5096\times \dotsb$

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)