Projections

US Land Grid currently provides specific GIS datasets in Lat/Lang NAD83 and Lat/Long NAD27. Downloads are provided in both Datums automatically, and at no charge.

Definition

A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position. A common choice of coordinates is latitude, longitude and elevation.

Latitude and longitude in practice

Say you set up your Wild T4 next to the water tank north of the airport at Hilo, Hawaii, intending to determine its latitude and longitude by the stars. National Geodetic Survey (NGS) data predicts you will find the tank to be at 19.7323 deg North, 155.0412 deg West.

You cross the island and set the T4 next to the Keahole Point lighthouse; the NGS estimates that by the stars the lighthouse will turn out to be 19.7244 N 156.0787 W. [5] Calculating the distance from the water tank to the lighthouse using those lat-lons we get about 108.8 km, but if we measure the actual distance it turns out to be 105.5 km. What went wrong?

Hawaii is an extreme case of a problem that exists everywhere: when trying to measure latitude and longitude by the stars we can only orient our measuring device by gravity. We'd like the T4's axis to point to the center of the Earth, but the T4's level vials don't know where that is — all they know is the direction of gravity, which is much affected by that 4000-meter mountain 50 km away. So when we measure the lat-lons for two points the relationship between those two points can be distorted, which renders their lat-lons fairly useless for most people. When we measure the lat-lons of two points we want to be able to use those lat-lons to calculate the distance and direction from one to the other; we want to be able to draw a scale map and plot points on it by their lat-lons, and the distance between any pair of points on the map is supposed to closely match the actual distance we would measure on the ground.

So we need a different plan — a different definition of latitude and longitude. What they did in Hawaii circa 1930 was call the marker "Oahu West Base"[6] 21 deg 18 min 13.889 sec North, 157 deg 50 min 55.796 sec West, and define the lat-lon of every other point by its distance and direction from there.[7] The NGS now says that in 1993 that point was 21-18-02.54891 N 157-50-45.90280 W in the present NAD83 system. Was the old lat-lon off by 300+ meters? Well, yes, but the relationships between points in the islands were much more accurate than that. C&GS triangulated from island to island, calculating each successive point's lat-lon by its distance and direction from the previous points in the chain. Eventually they deemed the Hilo water tank to be at 19-43-54.526 N 155-03-26.463 W, which would make it 339191.7 meters from Oahu West Base on the Clarke 1866 spheroid. The NGS now figures those two points are 339192.8 meters apart.

Similarly in North America. If in 1980 you had asked the NGS for the lat-lons for the Empire State Building and a certain water tank in Anchorage, the NAD27 lat-lons they would have given you would be different from the current ones, but the distance you would have calculated then is 8.2 meters different from now. A transcontinental triangulation cannot do better than that.

UTM and UPS systems

The Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) coordinate systems both use a metric-based cartesian grid laid out on a conformally projected surface to locate positions on the surface of the Earth. The UTM system is not a single map projection but a series of map projections, one for each of sixty 6-degree bands of longitude. The UPS system is used for the polar regions, which are not covered by the UTM system.

Geodetic height

To completely specify a location of a topographical feature on, in, or above the Earth, one has to also specify the vertical distance from the centre of the Earth, or from the surface of the Earth. Because of the ambiguity of "surface" and "vertical", it is more commonly expressed relative to a precisely defined vertical datum which holds fixed some known point. Each country has defined its own datum. For example, in the United Kingdom the reference point is Newlyn, while in Canada, Mexico and the United States, the point is near Rimouski, Quebec, Canada. The distance to Earth's centre can be used both for very deep positions and for positions in space.

Cartesian coordinates

Every point that is expressed in ellipsoidal coordinates can be expressed as an x y z (Cartesian) coordinate. Cartesian coordinates simplify many mathematical calculations. The origin is usually the center of mass of the earth, a point close to the Earth's center of figure.

With the origin at the center of the ellipsoid, the conventional setup is the expected right-hand:

Z-axis along the axis of the ellipsoid, positive northward

X- and Y-axis in the plane of the equator, X-axis positive toward 0 degrees longitude and Y-axis positive toward 90 degrees east longitude

An example is the NGS data for a brass disk near Donner Summit, in California. Given the dimensions of the ellipsoid, the conversion from lat/lon/height-above-ellipsoid coordinates to X-Y-Z is straightforward—calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid. The reverse conversion is harder: given X-Y-Z we can immediately get longitude, but no closed formula for latitude and height exists. However, using Bowring's formula in 1976 Survey Review the first iteration gives latitude correct within degree as long as the point is within 10000 meters above or 5000 meters below the ellipsoid.

Shape of the Earth

The Earth is not a sphere, but an irregular shape approximating a biaxial ellipsoid. It is nearly spherical, but has an equatorial bulge making the radius at the equator about 0.3% larger than the radius measured through the poles. The shorter axis approximately coincides with axis of rotation. Map-makers choose the true ellipsoid that best fits their need for the area they are mapping. They then choose the most appropriate mapping of the spherical coordinate system onto that ellipsoid. In the United Kingdom there are three common latitude, longitude, height systems in use. The system used by GPS, WGS84, differs at Greenwich from the one used on published maps OSGB36 by approximately 112m. The military system ED50, used by NATO, differs by about 120m to 180m.[1]

Though early navigators thought of the sea as a flat surface that could be used as a vertical datum, this is far from reality. The Earth has a series of layers of equal potential energy within its gravitational field. Height is a measurement at right angles to this surface, roughly toward the centre of the Earth, but local variations make the equipotential layers irregular (though roughly ellipsoidal). The choice of which layer to use for defining height is arbitrary. The reference height we have chosen is the one closest to the average height of the world's oceans. This is called the geoid.[1][8]

The Earth is not static as points move relative to each other due to continental plate motion, subsidence, and diurnal movement caused by the Moon and the tides. The daily movement can be as much as a metre. Continental movement can be up to 10 cm a year, or 10 m in a century. A weather system high-pressure area can cause a sinking of 5 mm. Scandinavia is rising by 1 cm a year as a result of the melting of the ice sheets of the last ice age, but neighbouring Scotland is rising by only 0.2 cm. These changes are insignificant if a local datum is used, but are statistically significant if the global GPS datum is used.[1]

Expressing latitude and longitude as linear units

On the GRS80 or WGS84 spheroid at sea level at the equator, one latitudinal second measures 30.715 metres, one latitudinal minute is 1843 metres and one latitudinal degree is 110.6 kilometres. The circles of longitude, meridians, meet at the geographical poles, with the west-east width of a second naturally decreasing as latitude increases. On the equator at sea level, one longitudinal second measures 30.92 metres, a longitudinal minute is 1855 metres and a longitudinal degree is 111.3 kilometres. At 30° a longitudinal second is 26.76 metres, at Greenwich (51° 28' 38" N) 19.22 metres, and at 60° it is 15.42 metres.

On the WGS84 spheroid, the length in meters of a degree of latitude at latitude φ (that is, the distance along a north-south line from latitude (φ - 0.5) degrees to (φ + 0.5) degrees) is about 111132.954 - 559.822(cos 2φ) + 1.175(cos 4φ) (Those coefficients can be improved, but as they stand the distance they give is correct within a centimeter.)

To estimate the length of a longitudinal degree at latitude we can assume a spherical Earth (to get the width per minute and second, divide by 60 and 3600, respectively):

where Earth's average meridional radius is 6,367,449 m. Since the Earth isn't spherical that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude is where Earth's equatorial radius equals 6,378,137 m and ; for the GRS80 and WGS84 spheroids, b/a calculates to be 0.99664719. ( is known as the parametric or reduced latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 meter of each other if the two points are one degree of longitude apart.

Datums often encountered

Latitude and longitude values can be based on different geodetic systems or datums, the most common being WGS 84, a global datum used by all GPS equipment.[n 4] Other datums are significant because they were chosen by a national cartographical organisation as the best method for representing their region, and these are the datums used on printed maps. The latitude and longitude on a map may not be the same as on a GPS receiver. Coordinates from the mapping system can sometimes be roughly changed into another datum using a simple translation. For example, to convert from ETRF89 (GPS) to the Irish Grid add 49 metres to the east, and subtract 23.4 metres from the north.[9] More generally one datum is changed into any other datum using a process called Helmert transformations. This involves converting the spherical coordinates into Cartesian coordinates and applying a seven parameter transformation (translation, three-dimensional rotation), and converting back.

State Plane Coordinate System (SPCS)

In popular GIS software, data projected in latitude/longitude is often represented as a 'Geographic Coordinate System'. For example, data in latitude/longitude if the datum is the North American Datum of 1983 is denoted by 'GCS North American 1983'.