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I am using the NAD 1983 2011 UTM Zone 13N system for my data in ArcGIS. I really want to know how the equations ArcGIS used to calculate the X and Y.

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    There is an open source library, proj4.org/about.html - written in C and C++ for doing coordinate transforms if you really want to know how this can be done - ArcGIS is proprietary software and so beyond trusting the documentation you can't be sure how they do it. – Spacedman Jul 20 '18 at 7:06
  • Related: gis.stackexchange.com/questions/1237/… Particularly look at Snyder's Map Projections. Esri's algorithm is close to Snyder's for normal zone/data. – mkennedy Jul 20 '18 at 18:15
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I suggest you have a look at Esri's page on Geographic Transformation Methods. In it, they explain that

A geographic transformation always converts geographic (latitude–longitude) coordinates. Some methods convert the geographic coordinates to geocentric (X,Y,Z) coordinates, transform the X,Y,Z coordinates, and convert the new values back to geographic coordinates.

Equation Based Methods describes four methods for converting coordinates:

This page on grid based methods is also likely relevant.

Also have a look at this technical article How To: Determine which NAD_1983_To_WGS_1984 transformation to use.

I'm unsure which method above was used for your specific calculations.

  • OP asked a UTM projection aka how does transverse Mercator work, not about converting between geographic coordinate reference systems. Sorry, but downvote. – mkennedy Jul 20 '18 at 18:06

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