Latitude and Longitude, GPS Conversion
Date: 12/07/2001 at 15:12:47 From: M.S. Subject: GPS conversion Hello, What is the equation to convert latitude/longitude/altitude (LLA) data into earth-centered/earth-fixed (ECEF) data? Thanks.
Date: 12/08/2001 at 11:07:30 From: Doctor Fenton Subject: Re: GPS conversion Dear M.S., Thanks for writing to Dr. Math. I'm assuming that "altitude" in your question refers to "altitude above the reference ellipsoid." If you are obtaining your altitude from a GPS receiver, I believe that this would be the correct interpretation. Many altitude specifications are "altitude above mean sea level," however, and this can make a difference of up to nearly 100 meters. You would have to get information on the height of the mean sea level surface (called the "geoid") relative to the reference ellipsoid in the area of interest. The defining document for WGS84 (World Geodetic System 1984 - the Department of Defense geodetic coordinate system used by GPS) gives a geoid map showing contours of geoid height and a table of geoid heights on a latitude-longitude grid. The basic formulas for converting from latitude, longitude, altitude (above reference ellipsoid) to Cartesian ECEF are given in the Astronomical Almanac in Appendix K. They depend upon the following quantities: a the equatorial earth radius f the "flattening" parameter ( = (a-b)/a ,the ratio of the difference between the equatorial and polar radii to a; this is a measure of how "elliptical" a polar cross-section is). The eccentricity e of the figure of the earth is found from e^2 = 2f - f^2 , or e = sqrt(2f-f^2) . For WGS84, a = 6378137 meters (1/f) = 298.257224 (the reciprocal of f is usually given instead of f itself, because the reciprocal is so close to an integer) Given latitude (geodetic latitude, not geocentric latitude!), compute 1 C = --------------------------------------------------- sqrt( cos^2(latitude) + (1-f)^2 * sin^2(latitude) ) and S = (1-f)^2 * C . Then a point with (geodetic) latitude "lat," longitude "lon," and altitude h above the reference ellipsoid has ECEF coordinates x = (aC+h)cos(lat)cos(lon) y = (aC+h)cos(lat)sin(lon) z = (aS+h)sin(lat) The Almanac also gives an iterative procedure for the inverse conversion from ECEF to Lat/lon/alt, although there is a method due to Bowring that may be better, and the Almanac also gives a reference to a closed form (non-iterative) method due to Borkowski. If you need any further information, please write us again. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/
Date: 11/12/2002 at 19:41:14 From: George Thomas Subject: GPS conversion Dear Dr. Math, You stated the equation for C to be as follows: 1 C = --------------------------------------------------- sqrt( cos^2(latitude) + (1-f)^2 * sin^2(latitude) ) I just want to verify that longitude is not necessary in the above equation. Thanks, GPS challenged
Date: 11/12/2002 at 21:38:00 From: Doctor Fenton Subject: Re: GPS conversion Hi George, Thanks for writing to Dr. Math. The formula for C is indeed independent of longitude, because the Earth is modeled as an (oblate) spheroid, which is a surface of revolution. Every plane containing the polar axis cuts the spheroid in exactly the same elliptical shape, so the expressions that determine the radius at a given latitude depend only upon the latitude, not the longitude. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/
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