Bill Tankersley's EMT 668 Page
Write-up 10
Polar Equations

In this write-up, we will concentrate on investigating the following polar equations and their graphs:

1. , 2. , and 3.


Looking at the graph of equation 1 above, we get something like the following:

Let's look at another equation of the same form, such as . Here is the graph of this equation:

After experimenting around with other graphs of this type equation, we noticed that different numbers a, b, and c have a different effects on the graph of equations of the type .

What we see here in both cases is that we get an ellipse whose major axis is the r axis and whose minor axis is the t axis.

Looking at the graph of equation 2 above, we get this:

Again, we get an ellipse, but it looks like the major axis is along the angle for pi/4, or 45 degrees. This must surely be related to the pi/4 in the polar equation above. Let's look at another example of a graph of this type, say . Here is the graph:

Notice that the major axis of the ellipse is the line Pi/2, or 90 degrees. So, we can definitely say that the angle controls the orientation of the ellipse. Also, the values of a,b, and c control the size of the ellipse for graphs of the form .

Next, for the graph of equation 3 above, we get the following:

Let's look at a graph of another function of the same type, say . Here is the graph:

Clearly, changing the value in the numerator from 2 to 3 has lengthened the major axis of the figure above. One can make some very interesting graphs by multiplying t by various values in this form also.

This concludes the investigation of polar graphs of the given forms for this particular problem. Polar graphs have always been interesting to me, and it is nice to have a program such as Theorist with which to explore various graphs.

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