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## Cartesian coordinates

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

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.

## Polar coordinates

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

• (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

• (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 "right" or "wrong" convention, but the convention being used must be awared of.

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

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### Cartesian and cylindrical

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### Cartesian and spherical

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### cylindrical and spherical

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