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.
Cartesian coordinates
- 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.
- 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.
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 two-dimentional polar coordinate system is the circular coordinate system.
The three-dimentional polar coordinate systems are cylindrical coordinate system and spherical coordinate system.
Circular coordinates
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
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.
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
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.
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
Cartesian and circular
Cartesian and cylindrical
- (NEED FIX)
- (NEED FACT-CHECK)
Cartesian and spherical
- (NEED FIX)
- (NEED FACT-CHECK)
cylindrical and spherical
- (NEED FACT-CHECK)
See also
For spherical coordinates:
- Gimbal lock
- Spherical harmonics
- Euler angles
- Yaw, pitch and roll