question: How does the gps-coordinate system (XYZ-plane) accounts for the earth’s curvature?
comment: GPS doesn't use XYZ coordinates, but ellipsoidal coordinates. Math is used to convert them to something easier for us to use on a map. URL_0
passages: Methods for computing the geodesic distance are widely available in
  
  geographical information systems, software libraries, standalone
  
  utilities, and online tools. The most widely used algorithm is by
  
  Vincenty,
  
  who uses a series which is accurate to third order in the flattening of
  
  the ellipsoid, i.e., about 0.5 mm; however, the algorithm fails to
  
  converge for points that are nearly antipodal. (For
  
  details, see Vincenty's formulae.) This defect is cured in the
  
  algorithm given by
  
  Karney,
  
  who employs series which are accurate to sixth order in the flattening.
  
  This results in an algorithm which is accurate to full double precision
   Section::::Algorithm description.
  
  Section::::Algorithm description.:Step 1: Curvilinear structure detections.
   It is possible to combine the principal radii of curvature above in a non-directional manner.
  
  The Earth's Gaussian radius of curvature at latitude is:
  
  The Earth's mean radius of curvature at latitude is:
  
  Section::::Global average radii.
  
  The Earth can be modeled as a sphere in many ways. This section describes the common ways. The various radii derived here use the notation and dimensions noted above for the Earth as derived from the WGS-84 ellipsoid; namely,
   The geoid surface is irregular, unlike the reference ellipsoid (which is a mathematical idealized representation of the physical Earth), but is considerably smoother than Earth's physical surface. Although the physical Earth has excursions of +8,848 m (Mount Everest) and −11,034 m (Marianas Trench), the geoid's deviation from an ellipsoid ranges from +85 m (Iceland) to −106 m (southern India), less than 200 m total.
   While the mean Earth ellipsoid is the ideal basis of global geodesy, for regional networks a so-called "reference ellipsoid" may be the better choice. When geodetic measurements have to be computed on a mathematical reference surface, this surface should have a similar curvature as the regional geoid - otherwise, reduction of the measurements will get small distortions.
   Section::::Effects and empirical evidence.:Observing the Sun.:Determining the shape of the Earth.
  
  On a given day, if many different cities measure the angle of the Sun at local noon, the resulting data, when combined with the known distances between cities, shows that the Earth has 180 degrees of north-south curvature. (A full range of angles will be observed if the north and south poles are included, and the day chosen is either the autumnal or spring equinox.) This is consistent with many rounded shapes, including a sphere, and is inconsistent with a flat shape.
   Biruni's method was intended to avoid "walking across hot, dusty deserts," and the idea came to him when he was on top of a tall mountain in India. From the top of the mountain, he sighted the angle to the horizon which, along with the mountain's height (which he calculated beforehand), allowed him to calculate the curvature of the Earth.
  
  He also made use of algebra to formulate trigonometric equations and used the astrolabe to measure angles.
  
  According to John J. O'Connor and Edmund F. Robertson,
  
  BULLET::::- Applications
   BULLET::::- Maximum: The summit of Chimborazo is from the Earth's center.
  
  BULLET::::- Minimum: The floor of the Arctic Ocean is approximately from the Earth's center.
  
  Section::::Location-dependent radii.:Radii of curvature.
  
  Section::::Location-dependent radii.:Radii of curvature.:Principal sections.
  
  There are two principal radii of curvature: along the meridional and prime-vertical normal sections.
  
  Section::::Location-dependent radii.:Radii of curvature.:Principal sections.:Meridional.
  
  In particular, the Earth's "radius of curvature in the (north–south) meridian" at is:
  
  This is the radius that Eratosthenes measured.
  
  Section::::Location-dependent radii.:Radii of curvature.:Principal sections.:Prime vertical.
  
  If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction.
   BULLET::::- EGM96: The NASA GSFC and NIMA Joint Geopotential Model
  
  BULLET::::- Earth Gravitational Model 2008 (EGM2008)
  
  BULLET::::- GeographicLib provides a utility GeoidEval (with source code) to evaluate the geoid height for the EGM84, EGM96, and EGM2008 Earth gravity models. Here is an online version of GeoidEval.
   measures, and describe the "direction" of the local normal to the reference ellipsoid of revolution. This is "approximately" the same as the direction of the plumbline, i.e., local gravity, which is also the normal to the geoid surface. For this reason, astronomical position determination – measuring the direction of the plumbline by astronomical means – works fairly well provided an ellipsoidal model of the figure of Earth is used.
   However, for international networks, GPS positioning, or astronautics, these regional reasons are less relevant. As knowledge of the Earth's figure is increasingly accurate, the International Geoscientific Union IUGG usually adapts the axes of the Earth ellipsoid to the best available data.
  
  Section::::Historical method of determining the ellipsoid.
   In practice, this turns out to be an unreliable method of measurement, due to variations in atmospheric refraction. This additional effect can give the impression that the Earth's surface is flat, curved more convexly than it is, or even that it is concave, by bending light travelling near the surface of the Earth (as happened in various trials of the famous Bedford Level experiment).
   The Earth is not a perfect sphere nor is mass evenly distributed within the Earth. This results in the point-mass gravity model being inaccurate for orbits around the Earth, particularly Low Earth orbits. To account for variations in gravitational potential around the surface of the Earth, the gravitational field of the Earth is modeled with spherical harmonics which are expressed through the equation:
  
  where
  
  where:
   about , can also be used.
  
  Section::::Global average radii.:Mean curvature.
  
  The mean curvature in all directions at all points on the surface is given by the weighted mean Gaussian curvature:
  
  For the WGS 84 ellipsoid, the mean curvature equals .
  
  Section::::Global average radii.:Average distance from center to surface.
   The mathematics of navigation by GPS assume that satellites are moving in known orbits around an approximately spherical surface. The accuracy of GPS navigation in determining latitude and longitude and the way these numbers map onto locations on the ground show that these assumptions are correct. The same is true for the operational GLONASS system run by Russia, and the in-development European Galileo, Chinese BeiDou, and Indian IRNSS.
   Section::::Data Products.:Regional Moment Tensors.
   BULLET::::- EGM96 NASA GSFC Earth gravity model
  
  BULLET::::- Earth Gravitational Model 2008 (EGM2008, Released in July 2008)
  
  BULLET::::- NOAA Geoid webpage
  
  BULLET::::- International Centre for Global Earth Models (ICGEM)
  
  BULLET::::- GeographicLib provides a utility GeoidEval (with source code) to evaluate the geoid height for the EGM84, EGM96, and EGM2008 earth gravity models. Here is an online version of GeoidEval.
  
  BULLET::::- Kiamehr's Geoid Home Page
  
  BULLET::::- A free windows calculator which yields, among other calculation, the height difference between EGM96 geoid and mean sea level at every point on earth
  
  BULLET::::- Geoid tutorial from Li and Gotze (964KB pdf file)
   BULLET::::- Figure of the Earth
  
  BULLET::::- History of the metre
  
  BULLET::::- Spherical Earth#History
  
  Section::::References.
  
  BULLET::::- An early version of this article was taken from the public domain source at http://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003A.HTM#ZZ4.
  
  BULLET::::- J. L. Greenberg: "The problem of the Earth's shape from Newton to Clairaut: the rise of mathematical science in eighteenth-century Paris and the fall of "normal" science." Cambridge : Cambridge University Press, 1995
  
  BULLET::::- M .R. Hoare: "Quest for the true figure of the Earth: ideas and expeditions in four centuries of geodesy". Burlington, VT: Ashgate, 2004
   GeoTrellis started as a research project at Azavea, a geospatial software company based in Philadelphia. A precursor software component, DecisionTree, was developed beginning in 2006 with support from a Small Business Innovation Research grant from the U.S. Department of Agriculture. In 2009, with financial support from the William Penn Foundation and Stroud Water Research Center, Azavea embarked on early development of GeoTrellis.
  
  GeoTrellis was released as an open source project in 2011 with the goal of supporting fast processing of geospatial raster data at scale.
   It was stated earlier that measurements are made on the apparent or topographic surface of the Earth and it has just been explained that computations are performed on an ellipsoid. One other surface is involved in geodetic measurement: the geoid. In geodetic surveying, the computation of the geodetic coordinates of points is commonly performed on a reference ellipsoid closely approximating the size and shape of the Earth in the area of the survey. The actual measurements made on the surface of the Earth with certain instruments are however referred to the geoid. The ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the other hand, coincides with that surface to which the oceans would conform over the entire Earth if free to adjust to the combined effect of the Earth's mass attraction (gravitation) and the centrifugal force of the Earth's rotation. As a result of the uneven distribution of the Earth's mass, the geoidal surface is irregular and, since the ellipsoid is a regular surface, the separations between the two, referred to as geoid undulations, geoid heights, or geoid separations, will be irregular as well.
   Some claim that this experiment assumes a very distant Sun, such that the incoming rays are essentially parallel, and if a flat Earth is assumed, that the measured angles can allow one to calculate the distance to the Sun, which must be small enough that its incoming rays are not very parallel. However, if more than two relatively well-separated cities are included in the experiment, the calculation will make clear whether the Sun is distant or nearby. For example, on the equinox, the 0 degree angle from the North Pole and the 90 degree angle from the equator predict a Sun which would have to be located essentially next to the surface of a flat Earth, but the difference in angle between the equator and New York City would predict a Sun much further away if the Earth is flat. Because these results are contradictory, the surface of the Earth cannot be flat; the data "is" consistent with a nearly spherical Earth and a Sun which is very far away compared with the diameter of the Earth.
   Geodesy is primarily concerned with positioning and the gravity field and geometrical aspects of their temporal variations, although it can also include the study of Earth's magnetic field. Especially in the German speaking world, geodesy is divided into geomensuration ("Erdmessung" or "höhere Geodäsie"), which is concerned with measuring the Earth on a global scale, and surveying ("Ingenieurgeodäsie"), which is concerned with measuring parts of the surface.
  
  The Earth's shape can be thought of in at least two ways;
  
  BULLET::::- as the shape of the geoid, the mean sea level of the world ocean; or
   In accomplishing WGS 60, a combination of available surface gravity data, astro-geodetic data and results from HIRAN and Canadian SHORAN surveys were used to define a best-fitting ellipsoid and an earth-centered orientation for each of initially selected datum. (Every datum is relatively oriented with respect to different portions of the geoid by the astro-geodetic methods already described.) The sole contribution of satellite data to the development of WGS 60 was a value for the ellipsoid flattening which was obtained from the nodal motion of a satellite.
   spheroid is also used) was a well-accepted approximation to the
  
  figure of the Earth. The adjustment of triangulation networks
  
  entailed reducing all the measurements to a reference ellipsoid and
  
  solving the resulting two-dimensional problem as an exercise in
  
  spheroidal trigonometry 
  
  It is possible to reduce the various geodesic problems into one of two
  
  types. Consider two points: at latitude
  
  (from to ) is , of length
  
  , which has azimuths and
  
  considered are:
  
  BULLET::::1. the "direct geodesic problem" or "first geodesic problem", given , , and , determine and ;
   Section::::Projection description.
  
  The space-oblique Mercator projection provides continual conformal mapping of the swath sensed by a satellite. Scale is true along the ground track, varying 0.01 percent within the normal sensing range of the satellite. Conformality is correct within a few parts per million for the sensing range. Distortion is essentially constant along lines of constant distance parallel to the ground track. SOM is the only projection presented that takes the rotation of Earth into account.
  
  Section::::Equations.
  
  The forward equations for the Space Oblique Mercator projection for the sphere are as follows:
  
  Section::::References.
  
presuppositions: GPS system uses XYZ coordinates.  GPS coordinates are in XYZ plane.
corrections: The GPS system does not use XYZ coordinates but ellipsodial coordinated instead.  GPS doesn't use XYZ coordinates.
raw_presuppositions: GPS system uses XYZ coordinates.  GPS coordinates are in XYZ plane.
raw_corrections: The GPS system does not use XYZ coordinates but ellipsodial coordinated instead.  GPS doesn't use XYZ coordinates.