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.
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Description
The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth.
History
The concepts of angle and radius were already used by ancient peoples of the first millennium BC. The Greek astronomer and astrologer Hipparchus (190–120 BC) 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.
From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca (qibla)—and its distance—from any location on the Earth.
From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its antipodal point.
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.
Conventions
The radial coordinate is often denoted by r or ρ, and the angular coordinate by ϕ, θ, or t. The angular coordinate is specified as ϕ by ISO standard 31-11.
Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.
In many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis.
In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.
Connection to spherical and cylindrical coordinates
The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system.
Applications
Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane.
They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point.
For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate.
Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates.
The initial motivation for the introduction of the polar system was the study of circular and orbital motion.
Polar coordinates are used often in navigation as the destination or direction of travel can be given as an angle and distance from the object being considered.
For instance, aircraft use a slightly modified version of the polar coordinates for navigation.
In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system.
Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.
Modeling
Systems displaying radial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole.
A prime example of this usage is the groundwater flow equation when applied to radially symmetric wells. Systems with a radial force are also good candidates for the use of the polar coordinate system.
These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas.
Radially asymmetric systems
Radially asymmetric systems may also be modeled with polar coordinates.
Microphones
For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves.
The pattern shifts toward omnidirectionality at lower frequencies.
See also
- Coordinate system
- Curvilinear coordinates
- Cylindrical coordinate system
- List of canonical coordinate transformations
- Log-polar coordinates
- Mathematics
- Polar decomposition
- Spherical coordinate system
