Editing Projected coordinate system (section)

==System specification==
Because the purpose of any coordinate system is to accurately and unambiguously measure, communicate, and perform calculations on locations, it must be defined precisely. The [[EPSG Geodetic Parameter Dataset]] is the most common mechanism for publishing such definitions in a machine-readable form, and forms the basis for many GIS and other location-aware software programs.<ref name="epsg"/> A projected SRS specification consists of three parts:
* An abstract two-dimensional [[Cartesian coordinate system]] that allows for the measurement of each location as a tuple (''x'', ''y''), which are also called the ''easting'' and ''northing'' in many systems such as UTM. Any coordinate system definition must include a planar surface, an origin point, a set of orthogonal axes to define the direction of each measurement, and a unit of measure (usually the [[meter]] or  [[survey foot|US foot]]).
* A choice of [[map projection]] that creates a planar surface for the coordinate system that is connected to locations on the Earth. In addition to the general type of projection (e.g., [[Lambert conformal conic]], [[transverse Mercator]]), a coordinate system definition will specify the parameters to be used, such as a center point, standard parallels, scale factor, false origin, and such. With these parameters, the underlying formulas of the projection convert latitude and longitude directly into the (''x'', ''y'') coordinates of the system.
* A choice of [[geodetic datum]], which includes a choice of [[earth ellipsoid]]. This binds the coordinate system to actual locations on the [[Earth]] by controlling the measurement framework for latitude and longitude (GCS). Thus, there will be a significant difference between the coordinate of a location in "UTM NAD83 Zone 14N" and for the same location in "UTM NAD27 Zone 14N", even though the UTM formulas are identical, because the underlying latitude and longitude values are different. In some GIS software, this part of the definition is called the choice of a particular geographic coordinate system.

=== Projections ===
{{further|Map projection}}

To establish the position of a geographic location on a [[map]], a map projection is used to convert geodetic coordinates to plane coordinates on a map; it projects the datum ellipsoidal coordinates and height onto a flat surface of a map. The datum, along with a map projection applied to a grid of reference locations, establishes a ''grid system'' for plotting locations.  [[Conformal map projection|Conformal]] projections are generally preferred. Common map projections include the [[Transverse Mercator projection|transverse Mercator]] (used in [[Universal Transverse Mercator coordinate system|Universal Transverse Mercator]], the [[Ordnance Survey National Grid|British National Grid]], the [[State Plane Coordinate System]] for some states), [[Lambert conformal conic projection|Lambert conformal conic]] (some states in the [[State Plane Coordinate System|SPCS]]), and  [[Mercator projection|Mercator]] ([[Swiss coordinate system]]).

Map projection formulas depend on the geometry of the projection as well as parameters dependent on the particular location at which the map is projected. The set of parameters can vary based on the type of project and the conventions chosen for the projection. For the [[transverse Mercator projection]] used in UTM, the parameters associated are the latitude and longitude of the natural origin, the false northing and false easting, and an overall scale factor.<ref name=OGP7_2>{{cite web |title=Geomatics Guidance Note Number 7, part 2 Coordinate Conversions and Transformations including Formulas |url = http://info.ogp.org.uk/geodesy/guides/docs/G7-2.pdf |publisher=International Association of Oil and Gas Producers (OGP) |access-date=5 March 2014 |pages=9–10 |url-status=dead |archive-url = https://web.archive.org/web/20140306005736/http://info.ogp.org.uk/geodesy/guides/docs/G7-2.pdf |archive-date=6 March 2014 }}</ref> Given the parameters associated with particular location or grin, the projection formulas for the transverse Mercator are a complex mix of algebraic and trigonometric functions.{{r|OGP7_2|page1=45–54}}

===Easting and northing===
{{anchor|False origin}}{{anchor|Natural origin}}Every map projection has a ''natural origin'', e.g., at which the ellipsoid and flat map surfaces coincide, at which point the projection formulas generate a coordinate of (0,0).<ref name=OGP7_2/> To ensure that the northing and easting coordinates on a map are not negative (thus making measurement, communication, and computation easier), map projections may set up a ''false origin'', specified in terms of ''false northing'' and ''false easting'' values, that offset the true origin. For example, in UTM, the origin of each northern zone is a point on the equator 500&nbsp;km west of the central meridian of the zone (the edge of the zone itself is just under 400&nbsp;km to the west). This has the desirable effect of making all coordinates within the zone positive values, being east and north of the origin. Because of this, they are often referred to as the ''easting'' and ''northing''.  

===Grid north===
'''Grid north''' ('''GN''') is a [[navigation]]al term referring to the direction northwards along the grid lines of a [[map projection]]. It is contrasted with [[true north]] (the direction of the [[North Pole]]) and [[magnetic north]] (the direction in which a compass needle points). Many [[topographic map]]s, including those of the [[United States Geological Survey]] and Great Britain's [[Ordnance Survey]], indicate the difference between grid north, true north, and magnetic north.<ref>{{Cite book|last=Estopinal|first=Stephen V.|url=https://books.google.com/books?id=g_Jvc0zBVwgC&dq=Grid+north&pg=PA35|title=A Guide to Understanding Land Surveys|date=2009|publisher=John Wiley & Sons|isbn=978-0-470-23058-9|pages=35|language=en}}</ref>

The grid lines on Ordnance Survey maps divide the UK into one-kilometre squares, east of an imaginary zero point in the Atlantic Ocean, west of Cornwall. The grid lines point to a Grid North, varying slightly from True North. This variation is zero on the central meridian (north-south line) of the map, which is at two degrees west of the [[Prime Meridian]], and greatest at the map edges. The difference between grid north and true north is very small and can be ignored for most navigation purposes. The difference exists because the correspondence between a flat map and the round Earth is necessarily imperfect.

At the [[South Pole]], grid north conventionally points northwards along the [[Prime Meridian]].<ref>[http://quest.nasa.gov/antarctica/background/cara/movpole.html "Moving the South Pole"]. {{webarchive|url=https://web.archive.org/web/20110716140440/http://quest.nasa.gov/antarctica/background/cara/movpole.html |date=2011-07-16 }}, NASA Quest</ref>  Since the meridians converge at the poles, true east and west directions change rapidly in a condition similar to [[gimbal lock]]. Grid north solves this problem.