Polar coordinates provide a method of rendering graphs and indicating the positions of points on a two-dimensional (2D) surface. The polar coordinate system is employed in mathematics, physics, engineering, navigation, robotics, and other sciences.

The polar plane consists of a reference axis, or ray, that emanates from a point called the origin. Positions or coordinates are determined according to the distance or radius, from the origin, symbolized *r* , and the angle relative to the reference axis, symbolized by the lowercase Greek theta ( ). In the most common polar system, the reference ray points off toward the right, and angles are measured counterclockwise from it (illustration at left). This scheme is preferred, and is used by mathematicians, physicists, and engineers. In a less common scheme, the reference ray points upward, and angles are measured clockwise from it (illustration at right). This method is sometimes used by astronomers, navigators, military personnel, meteorologists, and robotics engineers.

Points or coordinates in either system are indicated by writing an opening parenthesis, the *r* value, a comma, the value, and a closing parenthisis in that order. The radius coordinates are, by convention, always nonnegative. Angles can be specified in degree s from 0 to 360, or in radian s from 0 to 2 pi , where pi is approximately 3.14159. An example using degrees is ( *r* , ) = (2,30). The origin is assigned *r* = 0.

If the scheme in the left-hand illustration is used, it is possible to convert a coordinate in the Cartesian *xy* -plane to polar values using these formulas:

*x* = *r* cos

*y* = *r* sin

Conversely, to convert a coordinate in the polar plane (as depicted in the left-hand illustration) to Cartesian values, use these formulas:

*r* = ( *x* ^{2} + *y* ^{2} ) ^{1/2}

= arctan ( *y* / *x* )

Compare Cartesian coordinates .

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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