Coordinates (elementary mathematics)

Coordinates (elementary mathematics)

This article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian 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.

Table of contents

1 Cartesian coordinates
2 Polar coordinates
3 Circular coordinates
4 Cylindrical coordinates
5 Spherical coordinates
6 Conversion between coordinate systems

6.1 Cartesian and circular
6.2 Cartesian and cylindrical
6.3 Cartesian and spherical
6.4 Cylindrical and spherical

7 See also

Cartesian coordinates

In the two-dimentional Cartesian coordinate system, a point P in the xy-plane is represent by a tuple of two components .

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-dimentional Cartesian coordinate system, a point P in the xyz-space is represent by a tuple of three components .

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.

Basic concept of coordinates is hard to explain in words.

For advanced topics, please refer to Cartesian coordinate system.

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.

The term polar coordinates often refered to circular coordinates (two-dimentional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both three-dimentional).

Circular coordinates

The circular coordinate system, often called simply 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 represent by a tuple of two components . 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.

Cylindrical coordinates

The cylindrical coordinate system is a three-dimentional polar coordinate.

In the cylindrical coordinate system, a point P is represent by a tuple of three components . 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.

Note: some sources use for ; there is no "right" or "wrong" convention, but the convention being used must be awared of.

The cylindrical coordinates involves some redundancy; loses its significance if .

Cylindrical coordinates are useful in analyzing systems that are symmetrical about an axis, the infinitely long cylinder that has the Cartesian equation has the very simple equation in cylindrical coordinates. Hence the name of "cylindrical" coordinates.

Spherical coordinates

The spherical coordinate system is a three-dimentional polar coordinate.

In the spherical coordinate system, a point P is represent by a tuple of three components . Using terms of the Cartesian coordinate system,

(radius) is the distance between the point P and the origin,
(colatitude) is the angle between the z-axis and the line from the origin to the point P, and
(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.

Note: some sources interchange the symbols and relative to this article, or use for ; there is no widely accepted convention.

The spherical coordinate system involves some redundancy; loses its significance if , and loses its significance if or or .

To construct a point from its spherical coordinates: from the origin, go along the positive z-axis, rotate about y-axis toward the direction of the positive x-axis, and rotate about the z-axis toward the direction of the positive y-axis.

Spherical coordinates are useful in analyzing systems that are symmetrical about a point; a sphere that has the Cartesian equation has the very simple equation in spherical coordinates. Hence the name of "spherical" coordinates.

Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry. In such a situation, one can describe waves using spherical harmonics. Another application is ergodynamic design, where is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.

Conversion between coordinate systems