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In mathematics and its applications, a coordinate (or co-ordinate) 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.[1] "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.

Although a specific coordinate system is useful for numerical calculations in a given space, the space itself is considered to exist independently of any particular choice of coordinates. From this point of view, a coordinate on a space is simply a function from the space (or a subset of the space) to the scalars. When the space has additional structure, one (typically) restricts attention to the functions which are compatible with this structure. Examples include:

The coordinates on a space transform naturally (by pullback) under the group of automorphisms of the space, and the set of all coordinates is a commutative ring called the coordinate ring of the space.

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.

Contents

Examples

The Cartesian coordinate system in the plane.

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.

The system of assigning longitude and latitude to geographical locations is a coordinate system. In this case the parametrization fails to be unique at the north and south poles.

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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.

Transformations

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 1D, 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:

There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length. These include:

A list of orthogonal coordinate systems

In mathematics, two vectors are orthogonal if they are perpendicular. The following coordinate systems all have the properties of being orthogonal coordinate systems, that is the coordinate surfaces meet at right angles.

  • Three dimensional orthogonal coordinate systems

Geographical systems

Geography and cartography utilize various geographic coordinate systems to map positions on the 3-dimensional globe to a 2-dimensional document.

The Global Positioning System uses the WGS84 coordinate system.

The Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) coordinate systems both use a metric-based cartesian grid laid out on a conformally projected surface to locate positions on the surface of the Earth. The UTM system is not a single map projection but a series of map projections, one for each of sixty zones. The UPS system is used for the polar regions, which are not covered by the UTM system.

During medieval times, the stereographic coordinate system was used for navigation purposes.[citation needed] The stereographic coordinate system was superseded by the latitude-longitude system, and more recently, the Global Positioning System.[citation needed]

Although no longer used in navigation, the stereographic coordinate system is still used in modern times to describe crystallographic orientations in the field of materials science.[citation needed]

The modern Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is commonly defined by two axes, at right angles to each other, forming a plane (an xy-plane). The horizontal axis is labeled x, and the vertical axis is labeled y. In a three dimensional coordinate system, another axis, normally labeled z, is added, providing a sense of a third dimension of space measurement. The axes are commonly defined as mutually orthogonal to each other (each at a right angle to the other). (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles.) All the points in a Cartesian coordinate system taken together form a so-called Cartesian plane.

The point of intersection, where the axes meet, is called the origin normally labeled O. With the origin labeled O, we can name the x axis Ox and the y axis Oy. The x and y axes define a plane that can be referred to as the xy plane. Given each axis, choose a unit length, and mark off each unit along the axis, forming a grid. To specify a particular point on a two dimensional coordinate system, you indicate the x unit first (abscissa), followed by the y unit (ordinate) in the form (x,y), an ordered pair. In three dimensions, a third z unit is added, (x,y,z).

The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.

Astronomical systems

Coordinate systems on the sphere are particularly important in astronomy: see astronomical coordinate systems.

See also

References and notes

  1. ^ Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore. p. 12. ISBN 0821810456. http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3Vi7xsLiYiWCK0erF6X2gczHOkJA#PPA12,M1. 

Further reading

External links


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