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How can I update MGRS coordinates in the attribute table of a point feature, when the position of points has been added without using a tool that creates a new feature?

I am not using a special extension for ArcGIS 10.3.

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    The ArcGIS Convert Coordinate Notation Tool will output MGRS coordinates. It is located in the Projections and Transformations Toolbox of Data Management. This is available at any license level. Commented Jan 17, 2018 at 20:34

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First, make sure both the X and Y fields you already have are up to date, as moving features does not update these automatically. Re-calculate Geometry. In a model, I would make these updates a precondition of the MGRS update step.

I have several short Python field calculation scripts. They are designed for my area of interest, so you will have to generalize this sample:

def mgrse(easting):
   if (easting < 500000):
     return "M"
   else:
     return "N"

def mgrsn(northing):
   if (northing < 3900000):
       return "U"
   else:
       return "V"


__esri_field_calculator_splitter__
mgrse( !CoordinateX! ) + mgrsn( !CoordinateY! ) + " " + str( !CoordinateX! )[1:6]  +  " " + str(!CoordinateY!)[2:7] 

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