# 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 [math]\displaystyle{ r }[/math]) 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 [math]\displaystyle{ t }[/math]) 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).^{[1]}

## 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.^{[2]}
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.*^{[3]} 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.^{[4]} 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*.^{[5]}^{[6]} Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.^{[3]}

## 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:

- [math]\displaystyle{ \begin{align} x &= r \cos \varphi, \\ y &= r \sin \varphi. \end{align} }[/math]

The Cartesian coordinates *x* and *y* can be converted to polar coordinates *r* and *φ* with *r* ≥ 0 and *φ* in the interval (−π, π] by:^{[7]}

- [math]\displaystyle{ r = \sqrt{x^2 + y^2} \quad }[/math] (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 *[math]\displaystyle{ \phi }[/math]* (*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 [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ \theta }[/math] (*theta*).

## Polar Coordinate System Media

An illustration of a complex number plotted on the complex plane using Euler's formula

## Related pages

## References

- ↑ Brown, Richard G. (1997). Andrew M. Gleason (ed.).
*Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis*. Evanston, Illinois: McDougal Littell. ISBN 0-395-77114-5. - ↑ Friendly, Michael. "Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization". Archived from the original on 2011-03-20. Retrieved 2006-09-10.
- ↑
^{3.0}^{3.1}Coolidge, Julian (1952). "The Origin of Polar Coordinates".*American Mathematical Monthly*.**59**(2): 78–85. doi:10.1080/00029890.1952.11988074. - ↑ Boyer, C. B. (1949). "Newton as an Originator of Polar Coordinates".
*American Mathematical Monthly*.**56**(2): 73–78. doi:10.2307/2306162. JSTOR 2306162. - ↑ Miller, Jeff. "Earliest Known Uses of Some of the Words of Mathematics". Archived from the original on 2008-07-19. Retrieved 2006-09-10.
- ↑ Smith, David Eugene (1925).
*History of Mathematics, Vol II*. Boston: Ginn and Co. p. 324. - ↑ Torrence, Bruce Follett; Eve Torrence (1999).
*The Student's Introduction to Mathematica*. Cambridge University Press. ISBN 0-521-59461-8.

## Other websites

- FooPlot (online function plotter in polar coordinates) Archived 2011-10-13 at the Wayback Machine
- Online conversion tool between polar and Cartesian coordinates