From Simple English Wikipedia, the free encyclopedia
Points in the polar coordinate system with poleO and polar axisL. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).
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 familiarCartesian or rectangular coordinate system, such a relationship can only be found throughtrigonometric 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 thepole (equivalent to theorigin in the Cartesian system). The angular coordinate (also known as the polar angle or theazimuth angle, and usually denoted by θ or) denotes thepositive oranticlockwise (counterclockwise) angle required to reach the point from the 0°ray orpolar axis (which is equivalent to the positivex-axis in the Cartesian coordinate plane).[1]
A polar grid with several angles labeled in degrees
The concepts of angle and radius were already used by ancient peoples of the 1st millenniumBCE.Hipparchus (190-120 BCE) created a table ofchord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.[2]InOn Spirals,Archimedes describes theArchimedean 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 inHarvard professorJulian Lowell Coolidge'sOrigin of Polar Coordinates.[3]Grégoire de Saint-Vincent andBonaventura 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 anArchimedean spiral.Blaise Pascal subsequently used polar coordinates to calculate the length ofparabolic arcs.
InMethod of Fluxions (written 1671, published 1736), SirIsaac 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 journalActa Eruditorum (1691),Jacob Bernoulli used a system with a point on a line, called thepole andpolar axis respectively. Coordinates were specified by the distance from the pole and the angle from thepolar axis. Bernoulli's work extended to finding theradius of curvature of curves expressed in these coordinates.
The actual termpolar coordinates has been attributed toGregorio Fontana and was used by 18th-century Italian writers. The term appeared inEnglish inGeorge Peacock's 1816 translation ofLacroix'sDifferential and Integral Calculus.[5][6]Alexis Clairaut was the first to think of polar coordinates in three dimensions, andLeonhard Euler was the first to actually develop them.[3]
Converting between polar and Cartesian 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 aheight above the original coordinate system. Each point is uniquely identified by a distance to the origin, calledr here, an angle, called (phi), and a height above the plane of the coordinate system, calledZ in the picture.
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).
