[The following is a highly technical note (i.e, odds are you don't want to read it at all), which I'm writing in the form of a blog post mainly because I'm too lazy to start another Web page—but it might get copied elsewhere eventually (if someone wants to dump it on Wikipedia, feel free to).]
I've long been intrigued by the oft-used (especially in conjunction with GPS units) UTM coordinates: explanations as to how UTM coordinates are defined exactly is very hard to find (the spherical case is simple enough, but the ellipsoidal one is a much tougher nut), and although there are many online tools (such as this one) to convert from latitude+longitude to UTM and back, they use black box formulæ, converging power series, and looking at the source will give you very little insight on what is going on. So here is an attempt at a mathematically precise definition (it took me a whole day of angry formula-crunching before I came up with something entirely correct, so I won't spare the details).
First of all, what is a transverse mercator projection? It can be defined by starting from a central meridian (UTM uses 60 possible central meridians, one for each UTM zone) and constructing the mathematically unique projection from a spheroid (= rotation of an ellipse about one of its axes) to a plane which is conformal (= preserves angles) and maps the central meridian to a straight line with constant scale. So here are the defining features of Universal Transverse Mercator:
- It divides the Earth in 60 longitude zones, each 6° wide: zone 1 ranges from 180° to 174°W, zone 2 from 174°W to 168°W and so on through zone 30 from 6°W to 0°, zone 31 from 0° to 6°E, up to zone 60 from 174°E to 180°. Each zone's central meridian is halfway between the limiting longitudes (e.g., zone 33's central meridian is 15°E). UTM coordinates are given relative to a longitude zone and a hemisphere (North or South).
- In each zone, UTM is strictly conformal (= preserves angles). In practice, this means that it does not distort shapes anywhere, or that the UTM grid is a square grid everywhere, and it appears square when displayed on any conformal map projection. (Note that this does not imply that the lines of the grid are aligned north-south and east-west! But they are always perpendicular.)
- Each zone's central meridian (which receives aneastingcoordinate of 500000, see below) is mapped to a vertical line along which the vertical (northing) coordinate is simply proportional to distance (on the ellipsoid) along that meridian; however,
- the scale along the central meridian is not 1:1, it is 9996:10000. In other words, two points lying at 10km's distance from one another on the central meridian will have vertical (northing) coordinates differing by only 9996. The reason for this is that the map scale increases on either side of the meridian (as is unavoidable when mapping a curve surface) and the 0.04% decrease at the center is chosen to make the average scale on a 6°-wide zone roughly 1:1.
- The two UTM coordinates, easting (given first) and northing (given second) are orthogonal and measured in meters (after the transverse mercator projection and the 9996:10000 scale are applied). Easting is measured horizontally, with the central meridian having value 500000 (in practice, it can take values from 166021 to 833979 at the equator, the range being more narrow at higher latitudes). Northing is measured vertically, with the equator having value 0 in the northern hemisphere and 10000000 in the southern hemisphere; in the northern hemisphere, it ranges from 0 at the equator to 9328094 at 84°N on the central meridian, or even 9997965 if used all the way to the pole (but this normally isn't supposed to happen: beyond 84°N, and beyond 80°S Universal Polar Stereographic coordinates should be used instead of UTM).
- UTM is (nowadays) almost exclusively used with the WGS84 ellipsoid: this has a semimajor axis of a = 6378137m (exactly) and a flattening of f = 1 − b/a = 1/298.257223563 (exactly).
This is a complete definition, however it isn't a very usable one
because it lacks a description of how to use the
of the definition to actually convert geodetic coordinates to
Universal Transverse Mercator. If we were to use a spherical Earth
model, computations would be easy enough: