1、 Unlike latitude and longitude, there is no physical frame of reference for UTM grids. Latitude is determined by the earths polar axis. Longitude is determined by the earths rotation. If you can see the stars and have a sextant and a good clock set to Greenwich time, you can find your latitude and l
2、ongitude. But there is no way to determine your UTM coordinates except by calculation.UTM grids, on the other hand, are created by laying a square grid on the earth. This means that different maps will have different grids depending on the datum used （model of the shape of the earth）. I saw US milit
3、ary maps of Germany shift their UTM grids by about 300 meters when a more modern datum was used for the maps. Also, old World War II era maps of Europe apparently used a single grid for all of Europe and grids in some areas are wildly tilted with respect to latitude and longitude.The two basic refer
4、ences for converting UTM and geographic coordinates are U.S. Geological Survey Professional Paper 1395 and U. S. Army Technical Manual TM 5-241-8 （complete citations below）. Each has advantages and disadvantages. For converting latitude and longitude to UTM, the Army publication is better. Its notat
5、ion is more consistent and the formulas more clearly laid out, making code easier to debug. In defense of the USGS, their notation is constrained by space, and the need to be consistent with cartographic literature and descriptions of several dozen other map projections in the book.For converting UT
6、M to latitude and longitude, however, the Army publication has formulas that involve latitude （the quantity to be found） and which require reverse interpolation from tables. Here the USGS publication is the only game in town.Some extremely tiny terms that will not seriously affect meter-scale accura
7、cy have been omitted.Converting Between Decimal Degrees, Degrees, Minutes and Seconds, and Radians（dd + mm/60 +ss/3600） to Decimal degrees （dd.ff）dd = whole degrees, mm = minutes, ss = secondsdd.ff = dd + mm/60 + ss/3600Example: 30 degrees 15 minutes 22 seconds = 30 + 15/60 + 22/3600 = 30.2561Decima
8、l degrees （dd.ff） to （dd + mm/60 +ss/3600）For the reverse conversion, we want to convert dd.ff to dd mm ss. Here ff = the fractional part of a decimal degree.mm = 60*ffss = 60*（fractional part of mm）Use only the whole number part of mm in the final result.30.2561 degrees = 30 degrees.2561*60 = 15.36
9、6 minutes.366 minutes = 22 seconds, so the final result is 30 degrees 15 minutes 22 secondsDecimal degrees （dd.ff） to RadiansRadians = （dd.ff）*pi/180Radians to Decimal degrees （dd.ff）（dd.ff） = Radians*180/piDegrees, Minutes and Seconds to DistanceA degree of longitude at the equator is 111.2 kilomet
10、ers. A minute is 1853 meters. A second is 30.9 meters. For other latitudes multiply by cos（lat）. Distances for degrees, minutes and seconds in latitude are very similar and differ very slightly with latitude. （Before satellites, observing those differences was a principal method for determining the
11、exact shape of the earth.）Converting Latitude and Longitude to UTMOkay, take a deep breath. This will get very complicated, but the math, although tedious, is only algebra and trigonometry. It would sure be nice if someone wrote aspreadsheet to do this.P = point under considerationF = foot of perpen
12、dicular from P to the central meridian. The latitude of F is called thefootprint latitude.O = origin （on equator）OZ = central meridianLP = parallel of latitude of PZP = meridian of POL = k0S = meridional arc from equatorLF = ordinate of curvatureOF = N = grid northingFP = E = grid distance from cent
13、ral meridianGN = grid northC = convergence of meridians = angle between true and grid northAnother thing you need to know is the datum being used:DatumEquatorial Radius, meters （a）Polar Radius, meters （b）Flattening （a-b）/aUseNAD83/WGS846,378,1376,356,752.31421/298.257223563GlobalGRS 806,356,752.3141
14、1/298.257222101USWGS726,378,1356,356,750.51/298.26NASA, DODAustralian 19656,378,1606,356,774.71/298.25AustraliaKrasovsky 19406,378,2456,356,863.01/298.3Soviet UnionInternational （1924） -Hayford （1909）6,378,3886,356,911.91/297Global except as listedClake 18806,378,249.16,356,514.91/293.46France, Afri
15、caClarke 18666,378,206.46,356,583.81/294.98North AmericaAiry 18306,377,563.46,356,256.91/299.32Great BritainBessel 18416,377,397.26,356,079.01/299.15Central Europe, Chile, IndonesiaEverest 18306,377,276.36,356,075.41/300.80South AsiaDont interpret the chart to mean there is radical disagreement abou
16、t the shape of the earth. The earth isnt perfectly round, it isnt even a perfect ellipsoid, and slightly different shapes work better for some regions than for the earth as a whole. The top three are based on worldwide data and are truly global. Also, you are very unlikely to find UTM grids based on
17、 any of the earlier projections.The most modern datums （jars my Latinist sensibilities since the plural of datum in Latin is data, but that has a different meaning to us） are NAD83 and WGS84. These are based in turn on GRS80. Differences between the three systems derive mostly from redetermination o
18、f station locations rather than differences in the datum. Unless you are locating a first-order station to sub-millimeter accuracy （in which case you are way beyond the scope of this page） you can probably regard them as essentially identical.I have no information on the NAD83 and WGS84 datums, nor
19、can my spreadsheet calculate differences between those datums and WGS84.Formulas For Converting Latitude and Longitude to UTMThese 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 ter
20、m that can be handled more simply - it merely converts radians to seconds of arc. That term is omitted here but discussed below.Symbolslat = latitude of point long = longitude of point long0 = central meridian of zone k0 = scale along long0 = 0.9996. Even though its a constant, we retain it as a sep
21、arate 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 earths elliptical cross-section. e2 = （ea/b）2 = e2/（1-e2） = .007 approximately. The quantity e onl
22、y occurs in even powers so it need only be calculated as e2. 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
23、 distance from the point in question to the polar axis, measured perpendicular to the earths surface. p = （long-long0） in radians （This differs from the treatment in the Army reference） Calculate the Meridional ArcS is the meridional arc through the point in question （the distance along the earths s
24、urface from the equator）. All angles are in radians.S = Alat - Bsin（2lat） + Csin（4lat） - Dsin（6lat） + Esin（8lat）, where lat is in radians and A = a1 - 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 tan
25、3/48）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
26、） + .） where lat is in radians This is the hard part. Calculating the arc length of an ellipse involves functions called elliptic integrals, which dont reduce to neat closed formulas. So they have to be represented as series.All angles are in radians.y = northing = K1 + K2p2 + K3p4, whereK1 = Sk0, K
27、2 = k0 nu sin（lat）cos（lat）/2 = k0 nu sin（2 lat）/4 K3 = k0 nu sin（lat）cos3（lat）/24（5 - tan2（lat） + 9e2cos2（lat） + 4e4cos4（lat） x = easting = K4p + K5p3, whereK4 = k0 nu cos（lat） K5 = （k0 nu cos3（lat）/6）1 - tan2（lat） + e2cos2（lat） Easting x is relative to the central meridian. For conventional UTM eas
28、ting add 500,000 meters to x.What the Formulas MeanThe hard part, allowing for the oblateness of the Earth, is taken care of in calculating S （or M）. So K1 is simply the arc length along the central meridian of the zone corrected by the scale factor. Remember, the scale is a hair less than 1 in the
29、middle of the zone, and a hair more on the outside.All the higher K terms involve nu, the local radius of curvature （roughly equal to the radius of the earth or roughly 6,400,000 m）, trig functions, and powers of e2 （ = .007 ）. So basically they are never much larger than nu. Actually the maximum va
30、lue of K2 is about nu/4 （1,600,000）, K3 is about nu/24 （267,000） and K5 is about nu/6 （1,070,000）. Expanding the expressions will show that the tangent terms dont affect anything.If we were just to stop with the K2 term in the northing, wed have a quadratic in p. In other words, wed approximate the parallel of latitude as a parabola. The r
