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Cartesian geographic coordinate system

A projected coordinate systemalso called a projected coordinate reference system, planar coordinate system, or grid reference systemis a type of spatial reference system that represents locations on Earth using Cartesian coordinates (x, y) on a planar surface created by a particular map projection. Each projected coordinate system, such as "Universal Transverse Mercator WGS 84 Zone 26N," is defined by a choice of map projection (with specific parameters), a choice of geodetic datum to bind the coordinate system to real locations on the earth, an origin point, and a choice of unit of measure. Hundreds of projected coordinate systems have been specified for various purposes in various regions. When the first standardized coordinate systems were created during the 20th century, such as the Universal Transverse Mercator, State Plane Coordinate System, and British National Grid, they were commonly called grid systems; the term is still common in some domains such as the military that encode coordinates as alphanumeric grid references. However, the term projected coordinate system has recently become predominant to clearly differentiate it from other types of spatial reference system. It is used in international standards such as the EPSG and ISO 19111 (also published by the Open Geospatial Consortium as Abstract Specification 2), and in most geographic information system software. During World War II, modern warfare practices required soldiers to quickly and accurately measure and report their location, leading to the printing of grids on maps by the U. S. Army Map Service (AMS) and other combatants. Initially, each theater of war was mapped in a custom projection with its own grid and coding system, but this resulted in confusion. This led to the development of the Universal Transverse Mercator coordinate system, possibly adopted from a system originally developed by the German Wehrmacht. To facilitate unambiguous reporting, the alphanumeric Military Grid Reference System (MGRS) was then created as an encoding scheme for UTM coordinates to make them easier to communicate. 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.

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). 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. 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.

Locations in a projected coordinate system, like any cartesian coordinate system, are measured and reported as easting/northing or (x, y) pairs. The pair is usually represented conventionally with easting first, northing second. For example, the peak of Mount Assiniboine (at on the British Columbia/Alberta border in Canada) in UTM Zone 11 is at (0594934mE, 5636174mN), meaning that is almost 600km east of the false origin for Zone 11 (95km east of the true central meridian at 117°W) and 5.6 million meters north of the equator. While such precise numbers are easy to store and calculate in GIS and other computer databases, they can be difficult for humans to remember and communicate. Thus, since the mid 20th century, there have been alternative encodings that shorten the numbers or convert the numbers into some form of alphanumeric string. For example, a truncated grid reference may be used where the general location is already known to participants and may be assumed. Because the (leading) most significant digits specify the part of the world and the (trailing) least significant digits provide a precision that is not needed in most circumstances, they may be unnecessary for some uses. This permits users to shorten the example coordinates to 949 361 by concealing , assuming the significant digits (3,4, and 5 in this case) are known to both parties. Alphanumeric encodings typically use codes to replace the most significant digits by partitioning the world up into large grid squares. For example, in the Military Grid Reference System, the above coordinate is in grid 11U (representing UTM Zone 11 5xxxxxx mN), and grid cell NS within that (representing the second digit 5xxxxxmE x6xxxxxm N), and as many remaining digits as are needed are reported, yielding an MGRS grid reference of 11U NS 949 361 (or 11U NS 9493 3617 or 11U NS 94934 36174). The Ordnance Survey National Grid (United Kingdom) and other national grid systems use similar approaches. In Ordnance Survey maps, each Easting and Northing grid line is given a two digit code, based on the British national grid reference system with an origin point just off the southwest coast of the United Kingdom. The area is divided into 100 km squares, each of which is denoted by a two letter code. Within each 100 km square, a numerical grid reference is used. Since the Eastings and Northings are one kilometre apart, a combination of a Northing and an Easting will give a four digit grid reference describing a one kilometre square on the ground. The convention is the grid reference numbers call out the lower left corner of the desired square. In the example map above, the town Little Plumpton lies in the square 6901, even though the writing which labels the town is in 6802 and 6902, most of the buildings (the orange boxed symbols) are in square 6901.

The more digits added to a grid reference, the more precise the reference becomes. To locate a specific building in Little Plumpton, a further two digits are added to the four digit reference to create a six digit reference. The extra two digits describe a position within the 1 kilometre square. Imagine (or draw or superimpose a Romer) a further 10x10 grid within the current grid square. Any of the 100 squares in the superimposed 10×10 grid can be accurately described using a digit from 0 to 9 (with 0 0 being the bottom left square and 9 9 being the top right square). For the church in Little Plumpton, this gives the digits 6 and 7 (6 on the left to right axis (Eastings) and 7 on the bottom to top axis (Northings). These are added to the four figure grid reference after the two digits describing the same coordinate axis, and thus our six figure grid reference for the church becomes 696017. This reference describes a 100 metre by 100 metre square, and not a single point, but this precision is usually sufficient for navigation purposes. The symbols on the map are not precise in any case, for example the church in the example above would be approximately 100x200 metres if the symbol was to scale, so in fact, the middle of the black square represents the map position of the real church, independently of the actual size of the church. Grid references comprising larger numbers for greater precision could be determined using large scale maps and an accurate Romer. This might be used in surveying but is not generally used for land navigating for walkers or cyclists, etc. The growing availability and decreasing cost of handheld GPS receivers enables determination of accurate grid references without needing a map, but it is important to know how many digits the GPS displays to avoid reading off just the first six digits. A GPS unit commonly gives a ten digit grid reference, based on two groups of five numbers for the Easting and Northing values. Each successive increase in precision (from 6 digit to 8 digit to 10 digit) pinpoints the location more precisely by a factor of 10. Since, in the UK at least, a 6 figure grid reference identifies a square of 100 metre sides, an 8 figure reference would identify a 10 metre square, and a 10 digit reference a 1 metre square. In order to give a standard 6 figure grid reference from a 10 figure GPS readout, the 4th, 5th, 9th and 10th digits must be omitted, so it is important not to read just the first 6 digits.