Coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
In this system, an arbitrary point O (the origin) is chosen on a given line.
[6] This can be generalized to create n coordinates for any point in n-dimensional Euclidean space.
There are two common methods for extending the polar coordinate system to three dimensions.
For example, Plücker coordinates are used to determine the position of a line in space.
It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis.
An example of this is the systems of homogeneous coordinates for points and lines in the projective plane.
However, one of the coordinate curves is reduced to a single point, the origin, which is often viewed as a circle of radius zero.
[15] It is often not possible to provide one consistent coordinate system for an entire space.
In this case, a collection of coordinate maps are put together to form an atlas covering the space.
[16] 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).
The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems.
Starting with the Greeks of the Hellenistic period, a variety of coordinate systems have been developed based on the types above, including:
