Testwiki:Reference desk/Archives/Mathematics/2023 March 8

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March 8

Converting UTM coordinates to latitude and longitude

I figure I should start with the article I'm referencing. In particular, I'm looking at the last set of formulae in this article, to convert UTM to lat-long coordinates:

Universal_Transverse_Mercator_coordinate_system#From_UTM_coordinates_(E,_N,_Zone,_Hemi)_to_latitude,_longitude_(φ,_λ)

The formula has a few variables that are a mystery to me, particularly "j" and beta j. Can a math wizard walk me through this calculation? I'd like to create a script to generate lat-long coordinates from a pair of UTM coordinates, all within 17N. - Floydian ^τ _¢ 17:58, 8 March 2023 (UTC)

The variable $j$ in these formulas is a bound variable. It is bound by the summations, written with a capital Greek letter $\sum .$ The notation

$\sum _{j=1}^{3}2j{\beta }_j\cos (2j\xi )\cosh (2j\eta )$

is convenient shorthand for

$2{\beta }_1\cos (2\xi )\cosh (2\eta )+4{\beta }_2\cos (4\xi )\cosh (4\eta )+6{\beta }_3\cos (6\xi )\cosh (6\eta ).$

The three coefficients ${\beta }_1,$ ${\beta }_2$ and ${\beta }_3$ are defined in the beginning of the section Simplified formulae.  --Lambiam 18:28, 8 March 2023 (UTC)

In the simplified formulae, no info is provided for the value of f. I hope its not too much to ask, but say I had a Northing of 5,100,000m and an Easting of 500,000m in zone 17N... could you walk me through the calculations? - Floydian ^τ _¢ 14:56, 10 March 2023 (UTC)

What do you mean? Template:Slink directly quotes the value of Template:Mvar for WGS 84. Anyhow, you might be interested in Charles Karney's paper or software. –jacobolus (t) 17:20, 10 March 2023 (UTC)

It is already all laid out in elementary steps that is almost a script. I think it is not going to help you if we write out the steps like,

Step 1: Calculate $\xi =\frac{N-N_0}{k_{0}A}.$

Step 2: Calculate $\eta =\frac{E-E_0}{k_{0}A}.$

Step 3: Calculate ${\xi }^{\prime }=\xi -\sum _{j=1}^3{\beta }_j\sin \left(2j\xi \right)\cosh \left(2j\eta \right).$

...

A problem we have with "walking you through the calculations" is that we don't know how we are supposed to make it simpler.  --Lambiam 17:44, 10 March 2023 (UTC)