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.[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 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 restricts attention to the functions which are compatible with this structure. Examples include:
- Continuous functions on topological spaces;
- Smooth functions on smooth manifolds;
- Measurable functions on measure spaces;
- Rational functions on algebraic varieties;
- Linear functionals on vector spaces.
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-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.
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.
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.
- 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.
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:
- Whewell equation relates arc length and tangential angle.
- Cesàro equation relates arc length and curvature.
