These formulas are slightly modified from Army (1973). They are accurate to within less than a meter within a given grid zone. The original formulas include a now obsolete term that can be handled more simply – it merely converts radians to seconds of arc. That term is omitted here but discussed below.

Symbols

• lat = latitude of point
• long = longitude of point
• long₀ = central meridian of zone
• k₀ = scale along long₀ = 0.9996. Even though it’s a constant, we retain it as a separate symbol to keep the numerical coefficients simpler, also to allow for systems that might use a different Mercator projection.
• e = SQRT(1-b²/a²) = .08 approximately. This is the eccentricity of the earth’s elliptical cross-section.
• e’² = (ea/b)² = e²/(1-e²) = .007 approximately. The quantity e’ only occurs in even powers so it need only be calculated as e’².
• n = (a-b)/(a+b)
• rho = a(1-e²)/(1-e²sin²(lat))^3/2. This is the radius of curvature of the earth in the meridian plane.
• nu = a/(1-e²sin²(lat))^1/2. This is the radius of curvature of the earth perpendicular to the meridian plane. It is also the distance from the point in question to the polar axis, measured perpendicular to the earth’s surface.
• p = (long-long₀) in radians (This differs from the treatment in the Army reference)

Calculate the Meridional Arc
S is the meridional arc through the point in question (the distance along the earth’s surface from the equator). All angles are in radians.

• S = A’lat – B’sin(2lat) + C’sin(4lat) – D’sin(6lat) + E’sin(8lat), where lat is in radians and
• A’ = a[1 - n + (5/4)(n² - n³) + (81/64)(n⁴ - n⁵) ...]
• B’ = (3 tan/2)[1 - n + (7/8)(n² - n³) + (55/64)(n⁴ - n⁵) ...]
• C’ = (15 tan²/16)[1 - n + (3/4)(n² - n³)  ...]
• D’ = (35 tan³/48)[1 - n + (11/16)(n² - n³)  ...]
• E’ = (315 tan⁴/512)[1 - n  ...]

The USGS gives this form, which may be more appealing to some. (They use M where the Army uses S)

• M = a[(1 - e²/4 - 3e⁴/64 - 5e⁶/256 ....)lat
  - (3e²/8 + 3e⁴/32 + 45e⁶/1024...)sin(2lat)
  + (15e⁴/256 + 45e⁶/1024 + ....)sin(4lat)
  - (35e⁶/3072 + ....) sin(6lat) + ....)] where lat is in radians

This is the hard part. Calculating the arc length of an ellipse involves functions called elliptic integrals, which don’t reduce to neat closed formulas. So they have to be represented as series.


Converting Latitude and Longitude to UTM
All angles are in radians.

y = northing = K1 + K2p² + K3p⁴, where

• K1 = Sk₀,
• K2 = k₀ nu sin(lat)cos(lat)/2 = k₀ nu sin(2 lat)/4
• K3 = [k₀ nu sin(lat)cos³(lat)/24][(5 - tan²(lat) + 9e'²cos²(lat) + 4e'⁴cos⁴(lat)]

x = easting = K4p + K5p³, where

• K4 = k₀ nu cos(lat)
• K5 = (k₀ nu cos³(lat)/6)[1 - tan²(lat) + e'²cos²(lat)]

Easting x is relative to the central meridian. For conventional UTM easting add 500,000 meters to x.

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