(The Journey from Polar to Rectangular)


Dixie Williford

Polar equations are composed of the polar coordinateswhere r is the directed distance of a point P on the polar curve to the origin; and is the angle measure, in radians, of the point P from the x-axis. Therefore, the coordinates determine the location of the point P. It is possible to rewrite an equation that is in polar coordinates as rectangular coordinates using the following coversion identities:


The following is an exploration into three specific polar functions:





FIRST, Let's take a look at the graph of .


The graph appears to be an ellipse. But, how can we know for sure? If it is an ellipse, how can we determine, algebraically, information about the center and the major and minor axes? Merely, by observation, we can estimate the following information :

R Major Axis: 5

R Minor Axis: 4

R Center: (3,0)

We could, therefore, guess that the equation for this figure, in rectangular coordinates is: .This information, however, would be more difficult to observe for various figures.

Click here to see if we have indeed confirmed that the figure is the ellipse described above.
SECONDLY,let us consider the polar equation:

As you will notice, this equation is very similar to the previous one. The only difference between the two is the angular measure of which the cosine is being taken for each .

What is the difference between verses ? Let's take a look at the graphs of these two polar equations together :

in blue; in green.

In rectangular coordinates, cos(x-2) is the translation of the function cos(x) 2 units to the right. Similarly, in polar coordinates, is the rotation of about the orgin by 45 degrees in the positive direction.

Let's take this knowledge about the cosine function as we look at the polar equation of real interest: and how it compares with: .

Let's look at the graphs of these two functions togethter :

in red; in blue.

Again, it seems as though the blue graph is a rotation of the red about the origin. Does it hold that the rotation angle is by 45 degrees as we saw in the graphs of cosine?

The following table shows values of both as well as as increases .









 (, 5.5581)


By looking at this table of values, you can see that indeed is the a or 45 degrees rotation of . Click here for a Graphing Calculator 2.2 animation demonstrating how this fact.

THIRDLY, let's investigate the function .

The following is a graph of the function:

Why is it that we get an assymptote at x=2 and y=-2? It is much easier to answer these questions by looking at the equation in its rectangular coordinates.

is equivalent to in rectangular coordinates. Click here to verify this.

It is easy to see that has a vertical assymptote at x=(-2) because this value of x gives a zero in the denominator of the function.

The horizontal assymptote can be found by taking the limit, L, of as x approaches infinity. Since the highest power of x in the numerator is equivalent to the highest power of x in the denominator, L = (-2/1) = -2. Therefore, a horizontal assymptote occurs at y=-2.

Therfore, because the vertical assymptote is determined by the "2" in the original polar equation, we see that whatever constant, c, appears in the numerator x= -c is the vertical assymptote. To test this out, let's try graphing

As you can see, the vertical assymptote is at x= -5, as predicted.

Notice, however, that by altering the numerator, you also alter the location of the horizontal asssymptote as well. By reverting to our rectangular coordinates, we will find that the new horizontal assymptote lies as y = - 5.

IN SUMMARY, we find that it is fairly easy (dispite slightly messy Algebra) to converty from polar equations to rectangular equations. Through this fairly simple conversion we learn some things about our graphs that are not so obvious from the polar form of the equation. It is very useful to be able to do such conversions in order to show students the relationships between the polar and rectangular coordinate sysytem. This conversion process can help students understand that one graph can be represented in numerous ways, one of which is its polar equation.