Editing Geodesy (section)

== Coordinate systems in space ==
{{main|Geodetic system}}
{{further|World Geodetic System}}
{{unsourced section|date=February 2024}}
[[File:Datum Shift Between NAD27 and NAD83.png|220px|thumb|right|Datum shift between [[NAD27]] and [[NAD83]], in metres]]

The locations of points in 3D space most conveniently are described by three [[cartesian coordinate system|cartesian]] or rectangular coordinates, ''X'', ''Y'', and ''Z''. Since the advent of satellite positioning, such coordinate systems are typically [[geocentric]], with the Z-axis aligned to Earth's (conventional or instantaneous) rotation axis.

Before the era of [[satellite geodesy]], the coordinate systems associated with a geodetic [[datum (geodesy)|datum]] attempted to be [[geocentric]], but with the origin differing from the geocenter by hundreds of meters due to regional deviations in the direction of the [[plumbline]] (vertical). These regional geodetic datums, such as [[ED50|ED 50]] (European Datum 1950) or [[North American Datum#North American Datum of 1927|NAD 27]] (North American Datum 1927), have ellipsoids associated with them that are regional "best fits" to the [[geoid]]s within their areas of validity, minimizing the deflections of the vertical over these areas.

It is only because [[Global Positioning System|GPS]] satellites orbit about the geocenter that this point becomes naturally the origin of a coordinate system defined by satellite geodetic means, as the satellite positions in space themselves get computed within such a system.

Geocentric coordinate systems used in geodesy can be divided naturally into two classes:
# The [[inertial]] reference systems, where the coordinate axes retain their orientation relative to the [[fixed star]]s or, equivalently, to the rotation axes of ideal [[gyroscopes]].  The ''X''-axis points to the [[Equinox (celestial coordinates)|vernal equinox]].
# The co-rotating reference systems (also [[ECEF]] or "Earth Centred, Earth Fixed"), in which the axes are "attached" to the solid body of Earth. The ''X''-axis lies within the [[Greenwich meridian|Greenwich]] observatory's [[Meridian (geography)|meridian]] plane.

The coordinate transformation between these two systems to good approximation is described by (apparent) [[sidereal time]], which accounts for variations in Earth's axial rotation ([[day|length-of-day]] variations). A more accurate description also accounts for [[polar motion]] as a phenomenon closely monitored by geodesists.

=== Coordinate systems in the plane ===
{{main|Horizontal position}}
[[File:Elliptical coordinates grid.svg|225px|thumb|2D grid for elliptical coordinates]]
[[File:Litography archive of the Bayerisches Vermessungsamt.jpg|225px|thumb|A [[Munich]] archive with [[lithography]] plates of maps of [[Bavaria]]]]

In geodetic applications like [[surveying]] and [[map]]ping, two general types of coordinate systems in the plane are in use:

# '''Plano-polar''', with points in the plane defined by their distance, ''s'', from a specified point along a ray having a direction ''α'' from a baseline or axis.
# '''Rectangular''', with points defined by distances from two mutually perpendicular axes, ''x'' and ''y''. Contrary to the mathematical convention, in geodetic practice, the ''x''-axis points [[Northing|North]] and the ''y''-axis [[Easting|East]].

One can intuitively use rectangular coordinates in the plane for one's current location, in which case the ''x''-axis will point to the local north. More formally, such coordinates can be obtained from 3D coordinates using the artifice of a [[map projection]]. It is impossible to map the curved surface of Earth onto a flat map surface without deformation. The compromise most often chosen — called a [[conformal projection]] — preserves angles and length ratios so that small circles get mapped as small circles and small squares as squares.

An example of such a projection is UTM ([[Universal Transverse Mercator]]). Within the map plane, we have rectangular coordinates ''x'' and ''y''. In this case, the north direction used for reference is the ''map'' north, not the ''local'' north. The difference between the two is called [[Transverse Mercator projection#Convergence|meridian convergence]].

It is easy enough to "translate" between polar and rectangular coordinates in the plane: let, as above, direction and distance be ''α'' and ''s'' respectively; then we have:

:<math>\begin{align}
x &= s \cos \alpha\\
y &= s \sin \alpha
\end{align}</math>

The reverse transformation is given by:

:<math>\begin{align}
s &= \sqrt{x^2 + y^2}\\
\alpha &= \arctan\frac{y}{x}.
\end{align}</math>