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See Cartesian coordinate system or Coordinates (mathematics) for a more elementary introduction to this topic.
In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. "Numbers" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. If the space or manifold is curved, it may not be possible to provide one consistent coordinate system for the entire space. In this case, a set of coordinate systems, called charts, are cobbled together to form an atlas for the space.
When the space has some additional algebraic structure, then the co-ordinates will also transform under rings or groups; a particularly famous example in this case are the Lie groups.
Although any 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. By convention the origin of the coordinate system in Cartesian coordinates is the point (0, 0, ..., 0), which may be assigned to any given point of Euclidean space.
In physics, a scalar is a physical quantity which assumes a single value which is a "real" quantity independent of the coordinate system. In this sense coordinates are not scalars (although, of course, a scalar field can be defined which for one particular coordinate system corresponds to a particular coordinate).
In some coordinate systems some points are associated with multiple tuples of coordinates, e.g. the origin in polar coordinates: r = 0 but θ can be any angle.
Examples
An example of a coordinate system is to describe 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 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.
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, 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.
If the bijection is an involution, e.g. a reflection, then the two associated coordinate transformations are the same, e.g., in 1D, x becomes 7-x.
Examples of bijections include the invertible affine transformations. Of these, the similarity transformations preserve distance ratios, hence magnitude ratios, and angles, so that e.g. decomposition of a vector into perpendicular components is preserved. In the case that vector quantities are considered in relation to position and displacement, as in vector fields, a similarity transformation of space is normally accompanied by a corresponding linear transformation of the other vector quantities, to preserve angles between e.g. a force and a displacement, hence preserve e.g. dot products up to scaling. The transformation is linear because, as opposed to position, most vector quantities have a natural origin, e.g. zero force. However, velocity translation preserves the laws of motion, because an inertial frame of reference is preserved. (But if there is e.g. air-resistance, a velocity translation will affect tacitly assumed stationarity of air.)
In diagrams showing vectors of multiple physical dimensions, e.g. forces and displacements, scaling of one kind of vectors does not affect relevant properties: a force and a displacement having the same length in a diagram has no particular significance.
Singularities
Some choices of coordinate systems may lead to paradoxes, for example, close to a black hole, but can be understood by changing the choice of coordinate system. At an actual mathematical singularity the coordinate system breaks down.
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.
- For any finite-dimensional vector space and any basis, the coefficients of the basis vectors can be used as coordinates. Changing the basis is a coordinate transformation, a linear transformation that can be summarized by a matrix, and is computationally the same as a mapping of points to other points keeping the bases the same: e.g. in 2D:
- a clockwise rotation is a mapping of points to other points which changes the coordinates the same as keeping the points in place but rotating the coordinate axes anti-clockwise. The rotation of coordinate systems is covered in depth on wikibooks.
- an expansion by a factor two in the direction of one basis vector is a mapping of points to other points which changes the coordinates the same as keeping the points in place but halving the magnitude of that basis vector (in both cases the corresponding coordinate is doubled).
- a mapping of points to other points which distorts a rectangle to a parallelogram changes the coordinates the same as keeping the points in place but changing the basis vectors from being two sides of that parallelogram to perpendicular ones, two sides of that rectangle.
- Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
- The polar coordinate systems:
- Generalized coordinates are used in the Lagrangian treatment of mechanics.
- Canonical coordinates are used in the Hamiltonian treatment of mechanics.
- Intrinsic coordinates describe a point upon a curve by the length of the curve to that point and the angle the tangent to that point makes with the x-axis.
Astronomical systems
See also
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