Thomas Earl Ricks

Mathematics Education

 

Assignment # 11

 

“Polar Equations”

 


 

Assignment #11

Investigation #3

 

Investigate with different values of p:

 

 

For k > 1, k =1, k < 1.

 

Note:  The parameter k is called the “eccentricity” of these conics.  It is usually called “e” but for many software programs e is a constant and cannot be set as a variable.

 


 

We will begin our investigations by looking at all four graphs in unison for a set value of p, and varying the eccentricity parameter k.  Then we will investigate one graph in more depth.

 


 

Setting p = 1 and k = 1 yields:

 

 

Notice that each appears to be a parabola, with the sine equations being up and down facing parabolas, and the cosine graphs left and right.

 

Dropping k < 1 yields ellipse shapes of various kinds. 

 

For p = 1 and k = .7 we get:

 

 

 

 

Dropping k even further has the ellipses approach a circle.

 

For p =1 and k = .3 we get:

 

 

 

Thus as k approaches zero, the graphs not only shrink in size, but approximate a circle more and more.  Let us zoom in to observe the more circular shape:

 

 

What would you predict to happen if we raised k above 1?

 

If you guessed a hyperbola, you are right!

 

We will observe just the first graph as it will be confusing to interpret four overlapping hyperbolas all at once.

 

If p = 1 and k = 1.3 we get:

 

 

As we increase k, the hyperbola becomes more and more “steep” as in the example below where k = 2

 

 

 

We could observe the change in the shape of the graphs as k changes from below 1 to above one in the graph below:

 

 

Or if you wish to see a movie as the value of k varies between .1 and 2, click here.  The graph with k =1 is given in purple as a reference graph.

 

 

 

Now let us investigate what happens to a specific graph as p is varied.  Can you create a rough guess of what you think will happen?  Will it shift, expand, contract?

 

Let us one polar equation and vary p (holding k constant at k = 1):

 

 

We will graph various values for p.  When we do this we observe:

 

 

It appears that the p values changes the “steepness” of the graph, at least for a parabola.  What about for an ellipse?  Thus for a value of k < 1, say k = .5, we vary the p value for several graphs and obtain:

It appears that the right side of the ellipse goes through the p value on the x-axis.

 

Can you explore hyperbolas and determine what varying p does to them?

 

Have fun!

 


 

For more information on eccentricity, click here.

 


 

For more information on how to derive these polar equations from the rectangular conic formulas, click here.

 


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