Polar Coordinates

This article introduces polar coordinates. The article also coveres converting to and from polar coordinates and shows how to use them to add vectors.
Introduction The name polar coordinates comes from thinking of the origin of the plane at (0,0) as being the pole of the coordinate system. The standard way of finding points on the plane is do give the x and y coordinate so that, for example, (1,2) is one units to the right of the origin and two units above it. Polar coordinates give a direction and a distance from the origin instead of an x and y coordinate. The point (1,2) is in a direction about 63.4 degrees counterclockwise of the positive x axis and at a distance of the square root of five (about 2.236) units from the origin, as shown below.

Since angle repeat every 360 degrees, each point can be specified in many ways in polar coordinates. The point with radius 3 and angle 60 degrees is the same as the point with radius 3 and angle 420 degrees. To keep things simple we always keep the angle in the range from 0-360 degrees. Functions in polar coordinates. Consider a circle of radius 4 centered at the origin.

The set of points that make up this circle satisfy the equation

but this set of points isn't a function y=f(x) at all: it fails the vertical line test to be a function. In polar coordinates we make the radius of function of the angle and the equation of the circle of radius four is just r=4. The points at a radius of four in every direction from the origin do form a circle of radius four. In polar coordinates the vertical line test is for points moving out from the origin and the circle passes the test.

Polar coordinates are not better or worse than the standard x-y coordinates. Instead there are some problems that are easier to solve in polar coordinates, such as adding vectors. Converting back and forth between the two sorts of coordinates isn't too hard. Consider the following diagram:

Using trigonometry, this diagram implies that

Which we can solve to get equations for converting back and forth in both directions.

Some common sense is needed to use these equations. If, for example, x=0 then the angle is vertical - either 90 or 270 degrees. Likewise, the arctangent function returns results in the range from -90 to 90 degrees. This means that you may need to correct the angle by adding 180 degrees if the result is in the riegon from 90 to 270 degrees. To see an interesting application of polar coordinates, read the article on petal curves.

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