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.
[ ] "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.
Examples
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-tupleP = (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.
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:
- The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for three-dimensional flat space, uses three numbers representing distances.
- Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
- * 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.
While not coordinate systems, there are ways of describing curves 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.
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. The stereographic coordinate system was superseded by the latitude-longitude system, and more recently, the Global Positioning System.
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.
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.
An example of a point P on the system is indicated in the picture below using the coordinate (5,2).
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.
An example of a point P on the system is indicated in the picture below using the coordinate (5,2).
Astronomical systems
Coordinate systems on the sphere are particularly important in astronomy: see
astronomical coordinate systems.
See also
- Nomogram, graphical representations of different coordinate systems
References and notes
