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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.


CARTESIAN COORDINATES

In the two-dimensional Cartesian coordinate system, a point P in the ''xy''-plane is represented by a tuple of two components (x, y).
  • x is the signed distance from the ''y''-axis to the point P, and

  • y 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 (x, y, z).
  • x is the signed distance from the ''yz''-plane to the point P,

  • y is the signed distance from the ''xz''-plane to the point P, and

  • z is the signed distance from the ''xy''-plane to the point P.


For advanced topics, please refer to Cartesian Coordinate System .


POLAR COORDINATES

The polar coordinate systems are Coordinate System s 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 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 (r, heta). Using terms of the Cartesian Coordinate System ,
  • 0\leq{r} ( Radius ) is the distance from the origin to the point P, and

  • 0\leq heta<360^\circ ( 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

  • changing from the angle increasing anticlockwise to increasing clockwise or conversely

  • 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 0\leq heta<360^\circ to e.g. -180^\circ< heta\leq180^\circ

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).