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 (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.