Polar Graphs Using Geogebra

On November 24th, we had a worksheet where we investigated the consequences of changing the parameters in a few basic polar equations (such as r = a sin Θ). Mr. Honner showed us these polar graphs in Geogebra and I'm going to show the "evolution" of Limacons when the parameters are changed in Geogebra also.


Limacons (r = a + b cos Θ):

As I played around with Geogebra, I found out a few things:

As the ratio of a/b increases, the polar graph looks more circular.

• First polar graph: the polar graph has a loop inside itself. The ratio of a/b is 1/2.
• Second polar graph: the polar graph makes a heart shape; there is no loop. The ratio of a/b here is 1.
• Third polar graph: the graph here is getting more circular. The ratio of a/b is 1.33.
• Fourth polar graph: the graph is getting even more circular. The ratio of a/b is 4.

Perhaps we can state it in more general terms:

When a/b is between 0 and 1, the graph will have a loop.

When a/b is 1, the graph will make a "heart."

When a/b is between 1 and 2, the graph will be slightly more circular and have a little "dent."

When a/b is greater than or equal to 2, the graph will be nearly circular.