Coordinates (mathematics)
From Wikipedia, the free encyclopedia
- For an advanced treatment, see Coordinate system.
The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space. A coordinate system is a plane or space where the origin and axes are defined so that coordinates can be measured.
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[edit] Cartesian coordinates
In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represented by a tuple of two components <math>(x, y)</math>.
- <math>x</math> is the signed distance from the y-axis to the point P, and
- <math>y</math> 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 tuple of three components <math>(x, y, z)</math>.
- <math>x</math> is the signed distance from the yz-plane to the point P,
- <math>y</math> is the signed distance from the xz-plane to the point P, and
- <math>z</math> is the signed distance from the xy-plane to the point P.
[edit] Polar coordinates
The 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).
[edit] Circular coordinates
The 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 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).
In the circular coordinate system, a point P is represented by a tuple of two components <math>(r, \theta)</math>. Using terms of the Cartesian coordinate system,
- <math>0\leq{r}</math> (radius) is the distance from the origin to the point P, and
- <math>0\leq\theta<360^\circ</math> (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:
- change of zero direction (such as making north the zero direction)
- changing from the angle increasing anticlockwise to increasing clockwise or conversely (as in a compass)
- change of scale
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:
- change of origin
- change of scale in one direction
A minor change is changing the range <math>0\leq\theta<360^\circ</math> to e.g. <math>-180^\circ<\theta\leq180^\circ</math>
Circular coordinates can be convenient in situations where only the distance, or only the direction to a fixed point matters, rotations about a point, etc. (by taking the special point as the origin).
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.
[edit] Cylindrical coordinates
The cylindrical coordinate system is a three-dimensional polar coordinate system.
In the cylindrical coordinate system, a point P is represented by a tuple of three components <math>(r, \theta, h)</math>. Using terms of the Cartesian coordinate system,
- <math>0\leq{r}</math> (radius) is the distance between the z-axis and the point P,
- <math>0\leq\theta<360^\circ</math> (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
- <math>h</math> (height) is the signed distance from xy-plane to the point P.
- Note: some sources use <math>z</math> for <math>h</math>; there is no "right" or "wrong" convention, but it is necessary to be aware of the convention being used.
Cylindrical coordinates involve some redundancy; <math>\theta</math> loses its significance if <math>r=0</math>.
Cylindrical coordinates are useful in analyzing systems that are symmetrical about an axis. For example the infinitely long cylinder that has the Cartesian equation <math>x^2+y^2=c^2</math> has the very simple equation <math>r=c</math> in cylindrical coordinates.
[edit] Spherical coordinates
The spherical coordinate system is a three-dimensional polar coordinate system.
In the spherical coordinate system, a point <math>P</math> is represented by a tuple of three components <math>(\rho, \theta, \phi)</math>. Using terms of the Cartesian coordinate system,
- <math>0\leq\rho</math> (radius) is the distance between the point <math>P</math> and the origin,
- <math>0\leq\phi\leq 180^\circ</math> (zenith, colatitude or polar angle) is the angle between the <math>z</math>-axis and the line from the origin to the point P, and
- <math>0\leq\theta<360^\circ</math> (azimuth or longitude) is the angle between the positive <math>x</math>-axis and the line from the origin to the point P projected onto the <math>xy</math>-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.
[edit] Conversions
- main article: List of canonical coordinate transformations
[edit] See also
- coordinate rotation
- coordinate system
- curvilinear coordinates
- nabla in cylindrical and spherical coordinates
- parabolic coordinates
- vector (spatial)
- vector fields in cylindrical and spherical coordinates
[edit] Spherical coordinates
[edit] External links
- Frank Wattenberg has made some attractive animations illustrating spherical and cylindrical coordinate systems.
- http://www.physics.oregonstate.edu/bridge/papers/spherical.pdf is a description of the different conventions in use for naming components of spherical coordinates, along with a proposal for standardizing this.
- http://mathworld.wolfram.com/SphericalCoordinates.html is a very thorough review of all possible partial derivatives etc.de:Koordinaten
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