The polar coordinate system is an adaptation of the two-dimensional  coordinate system invented in 1637 by French mathematician René Descartes (1596–1650). Several decades after Descartes published his two dimensional coordinate system, Sir Isaac Newton (1640–1727) developed ten different coordinate systems. One of the ten systems was a polar coordinate system. Newton and others used the polar coordinate system to plot a complex curve known as a spiral. It was Swiss mathematician Jakob Bernoulli (1654–1705) who first used a polar coordinate system for a wider array of calculus problems and coined the terms “pole” and “polar axis” that are still used today in polar coordinate systems.

Understanding Polar Coordinates

As with the two-dimensional Cartesian coordinate system, one can describe the location of points in a polar coordinate system by means of coordinates.

Both systems involve an origin point and axis lines. In the polar coordinate system a single axis, or a polar axis, extends indefinitely from the origin,  known in the polar coordinate system as the pole. This polar axis is a fixed line and the location of all points and figures is based on this fixed, polar axis. Every point in the polar coordinate system is described in terms of the directed distance the point is from the pole, and the angle of rotation the directed distance line makes with the polar axis.

In the diagram below, Point P is on a directed distance line that is at an angle  from the polar axis. Point P is at a distance r from the pole along the directed distance line. The polar coordinates for Point P are represented as (r, ). In a polar coordinate system, the angle of the directed distance line follows the distance from the pole that the point is located along the di- rected distance line.

Although it is possible to locate a point by representing the angle of rotation with degrees, in a polar coordinate system the angle is usually represented by radians. Radians can conveniently express angle measures in terms of  π. For example, an angle  π/2 of radians is equal to an angle of 90° and an angle of  π radians is equal to an angle of 180°. The rotation of any positive angle is always in a counterclockwise direction. Thus Point Q (2,  π/3) is shown below.

An angle measured as  π/3 radians is equal to a 60° angle. Point Q is located on a directed distance line that makes a 60° angle with the polar axis, at a distance of 2 units from the pole (Point O).

The angle for the rotation of the directed distance may be generated in a clockwise direction by labeling the angle with a negative sign. In the following diagram, Point T (5,   π/2) is shown. Point T is located 5 units from the origin on a directed distance line that makes a 90° angle with the polar axis, but in a clockwise or negative rotation.

The polar coordinates for Point T may also be written as (5, 3 π/2) to indicate a positive or counterclockwise rotation for the directed distance line.

Likewise, every point in a polar coordinate system may be represented by positive or negative rotations by the directed distance line. Generally, positive rotations are used to represent the location of a point in polar coordinate systems.

When the location of a point is represented in polar coordinates, it may be converted into two-dimensional or Cartesian coordinates. The angle of rotation for the directed distance line and the distance form the pole may be used to find the x, y coordinates on a Cartesian coordinate system as pictured below.

The point (x, y) in the Cartesian coordinate system is the same as (r, ) in the polar coordinate system. Simple trigonometry procedures enable mathematicians to convert from one system to the other.

Using Polar Coordinates

Mathematicians find the polar coordinate system may be used more easily than the two-dimensional coordinate system to represent circles and other figures with curved lines. The polar coordinate system can also be more easily applied to certain real-life situations because it requires only a single axis line in order to represent the location of any point, as for example with navigation. Any location may be selected as the pole, and then the polar axis may be determined by sighting along any line from the pole. Once the pole and polar axis have been determined, any location may be represented by polar coordinates. In addition, the directed distance line can be expressed simply as an angle of rotation from the polar axis. Thus, an expression such as  π/4can represent the heading of a ship at sea.

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Reference

Cajori, Florian. The History of Mathematics, 5th Ed. New York: Chelsea Publishing Company, 1991.

National Council of Teachers of Mathematics. Historical Topics for the Mathematics Classroom. Reston, VA: NCTM, 1969.