Polar Coordinates

One of the several systems for addressing points in the plane is the polar-coordinate system. In this system a point P is identified with an ordered pair (r,θ) where r is a distance and θ an angle. The angle is measured counter-clockwise from a fixed ray OA called the "polar axis." The distance to P is measured from the end point O of the ray. This point is called the "pole." Thus each pair determines the location of a point precisely.

When a point P is given coordinates by this scheme, both r and θ will be positive. In working with polar coordinates, however, it occasionally happens that r,θ , or both take on negative values. To handle this one can either convert the negative values to positive ones by appropriate rules, or one can broaden the system to allow such possibilities. To do the latter, instead of a ray through O and P one can imagine a number line with θ the angle formed by OA and the positive end of the number line, as shown here. One can also say that an angle measured in a clockwise direction is negative. For example, the point (5, 30°) could also be represented by (-5, -150°).

To convert r and θ to positive values, one can use these rules:

(Notice that θ can be measured in radians, degrees, or any other measure as long as one does it consistently.) Thus one can convert (-5, -150°) to (5, 30°) by rule I alone. To convert (-7, -200°) would require two steps. Rule I would take it to (7, -20°). Rule II would convert it to (7, 340°).

Rule II can also be used to reduce or increase θ by any multiple of 2π or 360°. The point (6.5, 600°) is the same as (6.5, 240°), (6.5, 960°), (6.5, -120°), or countless others.

It often happens that one wants to convert polar coordinates to rectangular coordinates, or vice versa. Here one assumes that the polar axis coincides with the positive x-axis and the same scale is used for both. The equations for doing this are

For example, the point (3, 3) in rectangular coordinates becomes (√18, 45°) in polar coordinates. The polar point (7, 30°) becomes (6.0622, 3.5). Some scientific calculators have built-in functions for making these conversions.

These formulas can also be used in converting equations from one form to the other. The equation r = 10 is the polar equation of a circle with it center at the origin and a radius of 10. Substituting for r and simplifying the result gives x² + y² = 100. Similarly, 3x - 2y = 7 is the equation of a line in rectangular coordinates. Substituting and simplifying gives r = 7/(3 cos θ - 2 sin θ) as its polar equation.

As these examples show, the two systems differ in the ease with which they describe various curves. The Archimedean spiral r = kθ is simply described in polar coordinates. In rectangular coordinates, it is a mess. The parabola y = x² is simple. In polar form it is r = sin θ/(1 - sin²θ ). (This comparison is a little unfair. The polar forms of the conic sections are more simple if one puts the focus at the pole.) One particularly interesting way in which polar coordinates are used is in the design of radar systems. In such systems, a rotating antenna sends out a pulsed radio beam. If that beam strikes a reflective object the antenna will pick up the reflection. By measuring the time it takes for the reflection to return, the system can compute how far away the reflective object is. The system, therefore, has the two pieces of information it needs in order determine the position of the object. It has the angular position,θ , of the antenna, and the distance r, which it has measured. It has the object's position (r,θ ) in polar coordinates.

For coordinating points in space a system known as cylindrical coordinates can be used. In this system, the first two coordinates are polar and the third is rectangular, Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.


Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group.


representing the point's distance above or below the polar plane. Another system, called a spherical coordinate system, uses a radius and two angles, analogous to the latitude and longitude of points on earth.

Polar coordinates were first used by Isaac Newton and Jacob (Jacques) Bernoulli in the seventeenth century, and have been used ever since. Although they are not as widely used as rectangular coordinates, they are important enough that nearly every book on calculus or analytic geometry will include sections on them and their use; and makers of professional quality graph paper will supply paper printed with polar-coordinate grids.