Polar coordinate system


In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are
  • the point's distance from a reference point called the pole, and
  • the point's direction from the pole relative to the direction of the polar axis, a ray drawn from the pole.
The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The pole is analogous to the origin in a Cartesian coordinate system.
Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates.
The polar coordinate system is extended to three dimensions in two ways: the cylindrical coordinate system adds a second distance coordinate, and the spherical coordinate system adds a second angular coordinate.
Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the system's concepts in the mid-17th century, though the actual term polar coordinates has been attributed to Gregorio Fontana in the 18th century. The initial motivation for introducing the polar system was the study of circular and orbital motion.

History

The concepts of angle and radius were already used by ancient peoples of the first millennium BC. The Greek astronomer Hipparchus 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, Greek mathematician Archimedes describes his 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 —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 to its polar coordinates 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. Mathematicians from Jesuit, Grégoire de Saint-Vincent, and Italian 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. French mathematician Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.
In Method of Fluxions, English mathematician Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. He is credited with originating the polar coordinate system in its analytic form and for originating bipolar coordinates in a strict sense. In the journal Acta Eruditorum, Swiss mathematician 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 the calculation of the radius of curvature of curves expressed in these coordinates.
The term polar coordinates was attributed to Gregorio Fontana and 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 or . The angular coordinate is denoted by , specified by ISO standard 31-11, or in mathematical literature oftentimes.
Angles in polar notation are generally expressed in either degrees or radians. Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.
The angle is defined to start at 0° from a reference direction, and to increase for rotations in either clockwise or counterclockwise orientation. For example, in mathematics, the reference direction is usually drawn as a ray from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the respective opposite orientations.

Uniqueness of polar coordinates

Adding any number of full turns to the angular coordinate does not change the corresponding direction. Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates and, where is an arbitrary integer. Moreover, the pole itself can be expressed as for any angle.
Where a unique representation is needed for any point besides the pole, it is usual to limit to positive numbers and to either the interval or the interval, which in radians are or. Another convention, in reference to the usual codomain of the arctan function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to. In all cases, a unique azimuth for the pole must be chosen, e.g.,.

Converting between polar and Cartesian coordinates

The polar coordinates and can be converted to the Cartesian coordinates and by using the trigonometric functions of sine and cosine, respectively:
The Cartesian coordinates and can be converted to polar coordinates and, with and in the interval by:
where atan2 is a common variation on the arctangent function defined as
If r is calculated first as above, then this formula for φ may be stated more simply using the arccosine function:

Complex numbers

A complex number consists of real numbers and, as well as an imaginary number, which can be written as. Every complex number represents a point in the complex plane, thereby expressible by specifying either the point's Cartesian coordinates or the point's polar coordinates.
In polar form, the distance and angle coordinate are often referred to as the number's magnitude of modulus and argument, respectively. This can be obtained from a complex number, represented in rectangular form as, into a polar form, by substituting and :
The last expression is derived from Euler's formula, where is Euler's number approximately 2.718, and —expressed in radians—is the principal value of the complex number function arg applied to. To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. Equivalent are the cis—a function denotes —and angle notations:
For the operations of multiplication, division, exponentiation, and root extraction of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:
The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form and can be regarded as the graph of the polar function r. Note that, in contrast to Cartesian coordinates, the independent variable φ is the second entry in the ordered pair.
Different forms of symmetry can be deduced from the equation of a polar function r:
Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.
For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

Circle

The general equation for a circle with a center at and radius a is
This can be simplified in various ways, to conform to more specific cases, such as the equation
for a circle with a center at the pole and radius a.
When or the origin lies on the circle, the equation becomes
In the general case, the equation can be solved for, giving
The solution with a minus sign in front of the square root gives the same curve.