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

= 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

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 $$\textstyle\sum.$$ The notation
 * $$\sum_{j=1}^32j\beta_j\cos(2j\xi)\cosh(2j\eta)$$
 * 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? directly quotes the value of $f$ 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 $$\xi'=\xi-\sum_{j=1}^{3}\beta_j\sin\left(2j\xi\right)\cosh\left(2j\eta\right).$$
 * 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)
 * 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)