Editing Geodesy (section)

=== 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>