Polar maps locate points by radius and swing from a pole
Polar coordinates name a plane point by distance from a pole and angle from a polar axis—radius and azimuth, with the pole playing Cartesian origin. Spirals and central motion love this frame. Cylindrical and spherical systems lift it into three dimensions with an extra length or angle.
The distance from the pole is the radial coordinate and the angle is the polar angle or azimuth. Converting back to a grid uses x equal to r times the cosine of the angle and y equal to r times its sine. Degrees dominate navigation and surveying, radians mathematics and physics. Mathematicians usually measure from a ray pointing right and count counterclockwise, whereas navigators put zero straight up and turn clockwise. Adding whole turns leaves a direction unchanged, and a negative radius with the angle shifted by 180 degrees names the same spot, so every point has infinitely many labels unless ranges are restricted.
Hipparchus, who lived from 190 to 120 BC, built a table of chord lengths and reportedly used angle-and-distance positions for stars, and Archimedes described a spiral whose radius grows with angle, though neither built a full coordinate system. From the eighth century Islamic astronomers worked out the direction and distance to Mecca from any place, and from the ninth they used spherical trigonometry and map projections to do it accurately, in effect converting latitude and longitude into polar coordinates.
Grégoire de Saint-Vincent, writing privately in 1625 and publishing in 1647, and Bonaventura Cavalieri developed the concept independently, motivated by circular and orbital motion. Newton’s Method of Fluxions, written in 1671 but printed in 1736, called it the Seventh Manner, For Spirals, among ten systems, and he is credited with its analytic form. Jacob Bernoulli used a related scheme in Acta Eruditorum in 1691.
Gregorio Fontana is credited with the name, which entered English in George Peacock’s 1816 translation of Lacroix. Alexis Clairaut first imagined polar coordinates in three dimensions, and Leonhard Euler actually developed them.
Source: Polar coordinate system
