A coordinate system is a system that uses numbers or coordinates to determine the position of a point or geometric element within a geographic framework. This allows geographic datasets to use common locations for integration. A coordinate system is also used as a reference system to represent locations of geographic features, observation points, GPS (Global Positioning System) points, and imagery within this framework.  The simplest coordinate system consists of coordinate axes oriented perpendicularly to each other, known as Cartesian coordinates. Depending on the type of problem under consideration, a specific type of coordinate systems may be better to use than others.
In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring. For complicated spaces, it is often not possible to provide one consistent practical coordinate system for the entire space. In this case, a collection of coordinate systems, called graphs, are put together to form an atlas covering the whole space. A simple example (which motivates the terminology) is the surface of the earth.
In informal usage, coordinate systems can have singularities: these are points where one or more of the coordinates is not well-defined. For example, the origin in the polar coordinate system (r,θ) on the plane is singular, because although the radial coordinate has a well-defined value (r = 0) at the origin, θ can be any angle, and so is not a well-defined function at the origin.
The prototypical example of a coordinate system is the Cartesian coordinate system, which describes the position of a point P in the Euclidean space Rn by an n-tuple
- P = (r1, ..., rn)
of real numbers
- r1, ..., rn.
These numbers r1, ..., rn are called the coordinates linear polynomials of the point P.
If a subset S of a Euclidean space is mapped continuously onto another topological space, this defines coordinates in the image of S. That can be called a parametrization of the image, since it assigns numbers to points. That correspondence is unique only if the mapping is bijective.
Defining a coordinate system based on another one
In geometry and kinematics, coordinate systems are used not only to describe the (linear) position of points, but also to describe the angular position of axes, planes, and rigid bodies. In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix, which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three unit vectors aligned with those axes.
To read the coordinate system you have to know what side is "n" (the bottom side with numbers) then you go from "n" to whatever your number is.
A coordinate transformation is a conversion from one system to another, to describe the same space.
With every bijection from the space to itself two coordinate transformations can be associated:
- such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
- such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)
For example, in one dimension, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more.
Systems commonly used
Some coordinate systems are the following:
- The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for three-dimensional flat space, uses three numbers representing distances. The three numbers represent first: the East or West direction, second: the North or South direction, and third: the Elevation.
- Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
- The polar coordinate systems:
- Circular coordinate system (commonly referred to as the polar coordinate system) represents a point in the plane by an angle and a distance from the origin.
- Cylindrical coordinate system represents a point in space by an angle, a distance from the origin and a height.
- Spherical coordinate system represents a point in space with two angles and a distance from the origin.
- Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates.
- Generalized coordinates are used in the Lagrangian treatment of mechanics.
- Canonical coordinates are used in the Hamiltonian treatment of mechanics.
- Parallel coordinates visualise a point in n-dimensional space as a polyline connecting points on n vertical lines.
Geographic Coordinate Systems
A geographic coordinate system (GCS) defines locations on the earth using a three-dimensional spherical surface , it is a reference system that uses latitude and longitude to identify locations on a spheroid or sphere. Oftentimes GCS are referred to solely as datums however, a datum is just one part of a GCS. A GCS includes a datum (based on a spheroid), prime meridian, and an angular unit of measure. .latitude and longitude values.  Latitude is the angle from a point on the Earth's surface to the equatorial plane, measured from the center of the sphere. Lines joining points of the same latitude are called parallels, which trace concentric circles on the surface of the Earth, parallel to the equator. Longitude is the angle east or west of a reference meridian, located at the Royal Observatory in Greenwich, England, to another meridian that passes through the observer's point. All meridians are halves of great circles, and are not parallel; they converge at the north and south poles.
Coordinates can be described using different frames of reference, or datums that are designed to be more accurate for different areas of the earth. The most common datum encountered is the World Geodetic System 1984, and is used in GPS applications.
Geographic coordinate systems vary from projected coordinate systems in that they reference the earth as a 3D object measured in degrees rather than using a 2D projection of the earth's surface to measure it using meters or feet. However, latitude and longitude degrees do not have a standard length. This makes it difficult to measure distance and represent data accurately on a computer screen or flat map.
An example of this would be showing the location of Salt Lake City, UT with the following coordinate pair: 40.758701 N and -111.876183 W. 
Projected Coordinate Systems (PCS)
Rather than using degrees to represent points on a spheroid, some coordinate systems are placed on 2-dimensional projections and function much like the previously mentioned Cartesian System. These coordinate systems often use measurements in feet or meters to express x and y coordinates for specific points. The advantages of these systems are in the easy to express and understand coordinates and simpler calculations of distance and area due to constant lengths and angles across the projection.  One of the most common PCS in the United States is the State Plane Coordinate System (SPCS). The SPCS is made of 125 zones which provides for high accuracy within each zone but low accuracy outside the zone. 
- Active and passive transformation
- Frame of reference
- Galilean transformation
- Coordinate-free approach
- Nomogram, graphical representations of different coordinate systems
References and notes
- Weisstein, Eric W. Coordinate System. From MathWorld--A Wolfram Web Resource. Accessed 11 May 2010
- Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu; Geometry of Differential Forms. American Mathematical Society Bookstore, 2001.
- Kennedy, M. Understanding Map Projections, 2000
- ESRI. "Support." GIS Dictionary. ESRI, n.d. Web. 21 Oct. 2013.
- "What are geographic coordinate systems?". ESRI. http://resources.arcgis.com/en/help/main/10.1/index.html#//003r00000006000000. Retrieved 2014-10-27.
- "Salt Lake City, UT". LatLong.net. http://www.latlong.net/place/salt-lake-city-uh-usa-728.html. Retrieved 2015-09-21.
- "Template:Citation error". http://resources.arcgis.com/en/help/main/10.1/index.html#//003r0000000p000000. Retrieved 2018-09-24.
- "Template:Citation error". https://www.ngs.noaa.gov/SPCS/index.shtml. Retrieved 2018-09-24.
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