UTM, or Universal Transverse Mercator, is a popular method of map projection. Since the Earth is a sphere and maps are generally flat, there are inherent errors when cartographers project the Earth onto a flat map. In a UTM projection, there is a small angular difference between true North, i.e. the direction to the North pole, and grid North, the vertical lines on a particular gridded UTM map. That difference at any particular point is its convergence. UTM maps come in a series of 60 maps, spaced 6 degrees in longitude apart, and only one central grid line on each map runs true north-south.

Along a particular line of longitude (not a true-north grid line), UTM convergence is zero at the equator and a maximum at the poles.

Take the tangent of the longitude, using positive for longitudes East of the true North meridian for the map and negative for West of it. For example, the geographical coordinates of New York City are approximately 40.6 degrees North and 74 degrees West. The true North meridian is 75 degrees West there. Therefore, tan(1) is 0.0175.

Take the sine of your latitude, using positive for northerly latitudes and negative for southerly latitudes. For New York City, sin(40.6) is 0.6508.

Take the product of the first two steps. With these numbers, the product of 0.0175 and 0.6508 is 0.0114.

Take the inverse tangent, or arctan, of the previous result. The inverse tangent of 0.0114 is 0.65. This is the convergence, in degrees, of the UTM projection at New York City.

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