Coordinates (mathematics)
From Wikipedia, the free encyclopedia
|
For a more advanced and general treatment, see Coordinate system.
Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level is a coordinate which is useful for describing points near the surface of the earth. A coordinate system, in a plane or in space, is a systematic method of assigning a pair or a triple of numbers to each point in the plane or in space (respectively) which describe its position uniquely. For example, the triple consisting of latitude, longitude and altitude (height above sea level) define a coordinate system near to the surface of the earth. Image:Cartesian-coordinate-system.svg
The Cartesian coordinate system.
Coordinates may be defined in more general contexts. For example, if one is not interested in height, then latitude and longitude form a coordinate system on the surface of the earth, which is (approximately) a sphere. Coordinates such as these are also important in astronomy for describing the location of objects in the (night) sky: see Celestial coordinate systems for further examples. For simplicity, however, this article will restrict attention to coordinate systems in a plane and in space.
Cartesian coordinatesIn the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represented by a pair of numbers Failed to parse (Missing texvc executable; please see math/README to configure.): (x, y) .
is the signed distance from the y-axis to the point P, and
is the signed distance from the x-axis to the point P. In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represented by a triple of numbers Failed to parse (Missing texvc executable; please see math/README to configure.): (x, y, z) .
is the signed distance from the yz-plane to the point P,
is the signed distance from the xz-plane to the point P, and
is the signed distance from the xy-plane to the point P. Image:Rectangular coordinates.svg Polar coordinatesThe polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles. They are the most common systems of curvilinear coordinates. The term polar coordinates often refers to circular coordinates (two-dimensional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both three-dimensional). Circular coordinatesThe circular coordinate system, commonly referred to as the polar coordinate system, is a two-dimensional polar coordinate system, defined by an origin, O, and a ray (or semi-infinite line) L leading from this point. L is also called the polar axis. In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis). Image:Coordonnees polaires plan.svg
In this picture x is L and the Cartesian coordinate axis is used for the purpose of illustration
In the circular coordinate system, a point P is represented by a pair (r, θ). Using terms of the Cartesian coordinate system,
(radius) is the distance from the origin to the point P, and
(azimuth) is the angle between the positive x-axis and the line from the origin to the point P. Possible coordinate transformations from one circular coordinate system to another include:
and combinations. More generally, transformations of the corresponding Cartesian coordinates can be translated into transformations from one circular coordinate system to another by basically transforming to Cartesian coordinates, transforming those, and transforming back to circular coordinates. This is e.g needed for:
A minor change is changing the range Failed to parse (Missing texvc executable; please see math/README to configure.): 0\leq\theta<360^\circ to e.g. Failed to parse (Missing texvc executable; please see math/README to configure.): -180^\circ<\theta\leq180^\circ
A complex number can be viewed as a point or a position vector on a plane, the so-called complex plane or Argand diagram. Here the circular coordinates are r = |z|, called the absolute value or modulus of z, and φ = arg(z), called the complex argument of z. These coordinates (mod-arg form) are especially convenient for complex multiplication and powers. Cylindrical coordinatesThe cylindrical coordinate system is a three-dimensional polar coordinate system. Image:Cylindrical coordinates2.svg In the cylindrical coordinate system, a point P is represented by a triple (r, θ, h). Using terms of the Cartesian coordinate system,
(radius) is the distance between the z-axis and the point P,
(azimuth or longitude) is the angle between the positive x-axis and the line from the origin to the point P projected onto the xy-plane, and
(height) is the signed distance from xy-plane to the point P.
for Failed to parse (Missing texvc executable; please see math/README to configure.): h
Cylindrical coordinates involve some redundancy; θ loses its significance if r = 0. Cylindrical coordinates are useful in analyzing systems that are symmetrical about an axis. For example the infinitely long cylinder that has the Cartesian equation Failed to parse (Missing texvc executable; please see math/README to configure.): x^2+y^2=c^2 has the very simple equation Failed to parse (Missing texvc executable; please see math/README to configure.): r=c in cylindrical coordinates. Spherical coordinatesThe spherical coordinate system is a three-dimensional polar coordinate system. Image:Spherical Coordinates.svg In the spherical coordinate system, a point Failed to parse (Missing texvc executable; please see math/README to configure.): P is represented by a triple (ρ,θ,φ). Using terms of the Cartesian coordinate system,
(radius) is the distance between the point Failed to parse (Missing texvc executable; please see math/README to configure.): P and the origin,
(zenith, colatitude or polar angle) is the angle between the Failed to parse (Missing texvc executable; please see math/README to configure.): z -axis and the line from the origin to the point P, and
(azimuth or longitude) is the angle between the positive Failed to parse (Missing texvc executable; please see math/README to configure.): x -axis and the line from the origin to the point P projected onto the Failed to parse (Missing texvc executable; please see math/README to configure.): xy -plane. There are different conventions for the exact letters used for the angles. The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates. Transformations between coordinate systems
Because there are many different possible coordinate systems for describing points in the plane or in space, it is important to understand how they are related. Such relations are described by coordinate transformations which give formulae for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x,y) and polar coordinates (r,θ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x = r cos θ and y = r sin θ. See also
Spherical coordinatesExternal links
gd:Co-chomharran ko:극좌표 nl:Coördinaat ja:極座標系 pl:Sferyczny układ współrzędnych ro:Coordonate polare ru:Координаты fi:Koordinaatisto sv:Koordinater th:พิกัด tr:Küresel koordinat sistemi |
