# 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>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>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]}

## Contents

## 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]}

## 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>\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>\phi</math> and <math>\theta</math> (*theta*).

## Related pages

## References

- ↑ Template:Cite book
- ↑ Friendly, Michael. "Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization". Retrieved 2006-09-10.
- ↑
^{3.0}^{3.1}Coolidge, Julian (1952). "The Origin of Polar Coordinates".*American Mathematical Monthly***59**: 78-85. http://www-history.mcs.st-and.ac.uk/Extras/Coolidge_Polars.html. - ↑ Boyer, C. B. (1949). "Newton as an Originator of Polar Coordinates".
*American Mathematical Monthly***56**: 73-78. http://links.jstor.org/sici?sici=0002-9890%28194902%2956%3A2%3C73%3ANAAOOP%3E2.0.CO%3B2-V&size=LARGE. - ↑ Miller, Jeff. "Earliest Known Uses of Some of the Words of Mathematics". Retrieved 2006-09-10.
- ↑ Template:Cite book