GPS Coordinate Converter

The Washington Monument has an address in the United States National Grid standard: UTM zone 18, 323483.168 meters east and 4306479.498 meters north. The same document spells it 18SUJ2348306479. Type 18 north 323483.168 4306479.498 into the converter on this page and the obelisk also sits at 38.889467, -77.035240, or at 38 53.368 N, 77 02.114 W, or at 38 53 22.08 N, 77 02 06.86 W. Five spellings. One stone.

None of them is more correct than another. The U.S. Geological Survey says it flatly of UTM and latitude and longitude: one system is no more or less accurate than the other. What differs is the grammar. Each spelling has rules about order, sign, hemisphere, zone and rounding, and nearly every bad position in circulation is a grammar mistake, not an arithmetic one.

Degrees, written three ways

Latitude counts degrees north or south of the Equator, from 0 there to 90 at the poles. Longitude counts degrees east or west of the prime meridian, from 0 to 180. The National Geospatial-Intelligence Agency notes that degrees are divided the old sexagesimal way, into 60 minutes and each minute into 60 seconds, and that this yields three formats in common use: decimal degrees, degrees with decimal minutes, and degrees, minutes and seconds. They are one number written three ways. The monument’s latitude is 38.889467 degrees, which is 38 degrees and 53.368 minutes, which is 38 degrees, 53 minutes and 22.08 seconds.

What a digit is worth depends on where it sits. The Geological Survey puts one degree of latitude at about 69 miles, one minute at 1.15 miles and one second at 101 feet. Longitude is the awkward one, because the meridians converge toward the poles: at 38 degrees north a degree of longitude is down to 54.6 miles and a second to 80 feet. For decimal degrees, the GeoJSON standard observes that six decimal places amount to about 10 centimeters.

That is why minutes mistaken for decimals hurt so much. Read 38 53.368 as 38.53368 degrees, treat the longitude the same way, and the converter puts the result about 39.5 kilometers from the monument.

Sign and order are the other two traps. The agency’s grid manual is strict about sign: east longitude is positive, west is negative, and a value never carries both a sign and a letter. Order has no such referee. A geo: link, defined in RFC 5870, puts latitude first. GeoJSON puts longitude first, and so does a KML placemark, while GPX avoids the question by naming its two attributes lat and lon. Two bare numbers therefore say nothing about which came first. Swap the monument’s pair and it lands at 77 degrees south. A careful tool does not guess here. It states the order it assumed and offers the other.

Why a grid of meters exists

Angles are clumsy for the things people do with maps, which is measure and add. The agency’s manual says so directly: longitude and latitude are not suitable in all situations to report positions or to calculate distances or directions, and grids were invented to make those jobs convenient. The Universal Transverse Mercator grid cuts the world into 60 zones, each six degrees of longitude wide, numbered eastward from the 180 degree meridian. The conterminous United States takes ten of them, from zone 10 on the west coast to zone 19 in New England.

Inside a zone everything is meters. The central meridian is given an easting of 500,000, so grid values west of it are simply smaller, and northings count up from zero at the Equator. South of it, to avoid negative numbers, the Equator is assigned a false northing of 10,000,000 meters instead. Strictly there are 120 projections, two for each zone, one per hemisphere. The scale on the central meridian is fixed at 0.9996 exactly, and the grids standard says grid distance inside a zone may differ from ground distance by as much as 1 part in 1,000. At the monument the converter reports a point scale factor of 0.999984.

The price of that convenience is that two numbers no longer identify a place. The same easting and northing occur in every zone, and twice in each, once on either side of the Equator. Leave out the hemisphere and the monument’s numbers are just as valid near 51 degrees south, about 10,000 kilometers away. The usual shorthand makes it worse. One convention writes a latitude band letter after the zone number and another writes N or S for the hemisphere, and S is both: band S runs from 32 to 40 degrees north. The agency warns about exactly this confusion, and its manual forbids N and S as abbreviations for north and south.

Norway, Svalbard and the edges of the system

A six degree zone is a simple rule, and the rule has exceptions written into the standard. Between 56 and 64 degrees north, zone 32 is stretched west to 3 degrees east and zone 31 is squeezed into the strip from 0 to 3 degrees. Between 72 and 84 degrees north, zones 32, 34 and 36 do not exist at all. Zones 31, 33, 35 and 37 are widened there to cover 0 to 9, 9 to 21, 21 to 33 and 33 to 42 degrees east. A converter that finds the zone by dividing the longitude by six will file a point at 60.4 north, 5.3 east under zone 31. It belongs to 32V.

The edges are specified just as tightly. A zone owns its southern and western boundaries, so a point exactly on a zone meridian belongs to the zone east of it. UTM itself runs from 84 N to 80 S. Beyond those parallels the Universal Polar Stereographic grid takes over, with each pole placed at 2,000,000 meters east and north and a scale factor of 0.994 there.

The manual calls all of these administrative rules and admits that the mathematics does not require them. That is a useful reminder. A zone is a filing convention laid over a projection that keeps working some way past the line.

What the letters in MGRS add

The Military Grid Reference System is, in the manual’s own words, merely a respelling of UTM or UPS coordinates truncated to the desired precision. The zone number comes first, then a letter for a band of latitude: C through X starting from 80 degrees south, omitting I and O, each band eight degrees tall except X, which is twelve. Zone and band together form the grid zone designator, such as 18S. Two more letters name a square 100,000 meters on a side, and the digits that follow are what is left of the easting and the northing. Near the poles the zone number is dropped and A, B, Y or Z takes its place.

Reading one is a matter of counting digits. The National Grid standard walks the monument through every size: 18SUJ is the 100 kilometer square, 18SUJ20 narrows it to 10 kilometers, 18SUJ2306 to one kilometer, 18SUJ234064 to 100 meters, 18SUJ23480647 to 10 meters and 18SUJ2348306479 to one meter. The digits always split evenly, easting first and then northing, so at full length there are 10 digits, five for each. The standard expects a precision of one hundred or ten meters to be typical in the field, and that is the form worth learning to say aloud.

The rule people get wrong is the shortening. A grid reference is cut, never rounded. The grids standard says refinement is reduced not by rounding values but by truncation, because a reference names the square whose southwest corner it is, and rounding can name the neighbor instead. The monument’s northing digits are 06479. At ten meters they become 0647, not 0648. The slip is easy enough that the agency’s own overview page says references may be rounded and shortens its example, 15SWC8081751205, to 15SWC80825121. Its standard, applied to the same point, gives 15SWC80815120.

The manual also says where each form belongs. UTM is for calculation, analysis and storage. MGRS is for map margins, device displays and one person reading a position to another. A database full of grid letters is a database that someone will have to convert back.

The question the numbers cannot answer

Every spelling above assumes something it does not state: the datum. NOAA describes a geodetic datum as an abstract coordinate system with a reference surface that provides known locations from which to begin surveys and create maps. The one behind satellite positioning is WGS 84, a three-dimensional reference frame realized through the tracking stations of the Global Positioning System and built on an ellipsoid with a semi-major axis of 6,378,137 meters and a reciprocal flattening of 298.257223563. GPX, GeoJSON, KML and the geo: link all name WGS 84 as their frame.

Change the datum and the same stone gets different numbers. The National Grid standard gives the monument as 18SUJ23480647 on the North American Datum of 1983, which it treats as the equivalent of WGS 84, and as 18SUJ23450626 on the 1927 datum found on older maps. That is three squares west and twenty-one south, about 210 m. The agency’s own example moves 212 meters between the two datums and even changes its square letters, from WC to WN, because older ellipsoids take a different lettering scheme. Nothing in the digits warns you.

So there are things a converter cannot do, and an honest one says so. The tool on this page works on WGS 84 only and applies no datum shifts, because it cannot know which datum a paper map or an old survey used. It cannot add precision: a one kilometer reference converts to the corner of a one kilometer square, however many decimals follow. Its projection series is good to a few nanometers within 3,900 kilometers of a central meridian. Its distance between two points uses Vincenty’s formula of 1975, and that paper warns that the method may give no solution for two nearly antipodal points. Then the tool shows nothing, not a guess.

Which spelling to use follows from the job. Mariners, the Geological Survey notes, almost always use latitude and longitude because charts are built for it, while many experienced users of topographic maps prefer UTM, whose ticks are printed every 1,000 meters along the sheet. Files want decimal degrees, since that is what GPX, KML and GeoJSON store. A voice on a radio wants a grid reference at the precision the moment needs. And any position handed from one person to another should travel with three things stated: the format, the hemisphere or band, and the datum.

The arithmetic in all of this was settled long ago and can be checked to the millimeter. What moves a point by kilometers is a missing letter, a swapped pair, a rounded digit. Spell the place completely and it stays where it is.