Polar coordinate system

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed with angles and distance; in the more familiar Cartesian or rectangular coordinate system, such a relationship can only be found through trigonometric formulae.

As the coordinate system is two-dimensional, each point is determined by two polar coordinates: the radial coordinate and the angular coordinate. The radial coordinate (usually denoted as ) denotes the point's distance from a central point known as the _pole_ (equivalent to the _origin_ in the Cartesian system). The angular coordinate (also known as the polar angle or the azimuth angle, and usually denoted by θ or ) denotes the positive or anticlockwise (counterclockwise) angle required to reach the point from the 0° ray or _polar axis_ (which is equivalent to the positive x-axis in the Cartesian coordinate plane).

## History

The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. Hipparchus (190-120 BCE) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.
In _On Spirals,_ Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.

There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's _Origin of Polar Coordinates._ Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.

In _Method of Fluxions_ (written 1671, published 1736), Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. In the journal _Acta Eruditorum_ (1691), Jacob Bernoulli used a system with a point on a line, called the _pole_ and _polar axis_ respectively. Coordinates were specified by the distance from the pole and the angle from the _polar axis_. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.

The actual term _polar coordinates_ has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's _Differential and Integral Calculus_. Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.

## Converting between polar and Cartesian coordinates

The polar coordinates _r_ and _φ_ can be converted to the Cartesian coordinates _x_ and _y_ by using the trigonometric functions sine and cosine:

The Cartesian coordinates _x_ and _y_ can be converted to polar coordinates _r_ and _φ_ with _r_ ≥ 0 and _φ_ in the interval (−, ] by:
- (as in the Pythagorean theorem)

## Cylindrical coordinates

Cylindrical coordinates take the same idea that polar coordinates use, but they extend it further. To get a third dimension, each point also has a _height_ above the original coordinate system. Each point is uniquely identified by a distance to the origin, called _r_ here, an angle, called __ (_phi_), and a height above the plane of the coordinate system, called _Z_ in the picture.

## Spherical coordinates

The same idea as is used by polar coordinates can also be extended in a different way. Instead of using two distances, and one angle only, it is possible to use one distance only, and two angles, called  and  (_theta_).

## Related pages

- Geographic coordinate system
- Spherical coordinate system
- Celestial coordinate system

## Other websites

- FooPlot (online function plotter in polar coordinates)&type0=1)
- Online conversion tool between polar and Cartesian coordinates