THE LIMACONS AND OTHER POLAR CURVES

By Dario Gonzalez Martinez

This exploration consists of analyzing the curves associated to the following polar expressions:

Where a, b and k are constants.

CARDIOD AND LOOPED LIMACONS

Let’s start fixing k = 1 and varying parameters a and b. Consider the simplest case when a = b = 1 to draw our initial graph of the expressions. Figure 1 below shows the graphs:

Figure 1(a)	Figure 1(b)

These curves are called Cardiods and, according what the figure 1 suggests, when the polar expression consists of a cosine, the curve will be symmetric with respect to x-axis. On the other hand, if the polar expression consists of a sine, the curve will be symmetric with respect to y-axis.

We should observe the shape of the curve when we make vary one of the parameters a or b. Let b = 1, and let’s vary a for the expressions. Animation 1 below shows the effects of parameter a on the graphs:

Animation 1(a)	Animation 1(b)

On the other hand, we could also consider the effects of parameter b when it varies, and a is fixed a = 1. Animation 2 shows this idea below:

Animation 2(a)	Animation 2(b)

Still another interesting observation is to see what happens when a = b. So, we can make them vary at the same time for the same values. Animation 3 below shows the effects on the graphs:

Animation 3(a)	Animation 3(b)

The latter animation gave us a good starting point to analyze these polar curves. We saw that when a = b the shape of the curve is always a cardiod; the only difference is its size, which is obviously related to the value of a. Actually, it is possible to generalize that the curve is always a cardiod while |a| = |b|.

The above explanation allows us to separate our analysis for two more cases. In other words, considering the above explanation, it just remains to explore and conclude what happens when |a| > |b| and |a| < |b|.

To determine the effect when |a| > |b|, consider the figure 2 and figure 3 below:

Figure 2

Figure 3

Thus, we can appreciate that when |a| > |b|, the cardiod not only increases its size but also the “heart” indentation is taken away from the coordinate system’s origin while |a| increases.

Similarly, we could analyze what happens with the graph when |a| < |b|. Consider figures 4 and 5 below:

Figure 4

Figure 5

From the figures above we can conclude that when |a| < |b|, the curve make a loop through the coordinate system’s origin. These looped curves are also known as Looped Limacons.

N-LEAVED ROSE

Now we will fix a and b, and we will vary parameter k. First, we will consider a = b = 1 which make the polar expressions as follow:

Also, we should observe animation 4 below which shows what happen when k varies from -4 to 4:

Animation 4(a)	Animation 4(b)

According to what the animation showed, we distinguished five curves that seem to be well-defined curves. These five curves are a circle, a cardiod, a 2-leafed “flower”, a 3-leafed “flower”, and a 4-leafed “flower.” We will refer the other curves as intermediate stages for now. Here is a summary of the curves before mentioned for the expression with sine:

SUMMARY FOR
Circle	k = 0
Cardiod	k = 1 and -1
2-leafed “flower”	k = 2 and -2
3-leafed “flower”	k = 3 and -3
4-leafed “flower”	k = 4 and -4

The last three curves are known as n-Leaved Roses.

The canonic polar expressions for n-leaved roses are:

As you probably noted, the value of k seems to be related to the number of leaves when k is an integer larger than 1. Let’s fortify our conjecture by observing the following sequence of graphs for the expression with sine:

SEQUENCE FOR
k = 2	4 leaves
k = 3	3 leaves
k = 4	8 leaves
k = 5	5 leaves

So, we can easily conclude that k indeed is related with the number of rose’s leaves. Actually, if k is odd, then number of leaves is equal to k. On the other hand, if k is even, then the number of leaves is 2k. for the expression with cosine the conclusion is the same.

Finally, to understand the effect of b on the rose, we will start considering the polar expression with cosine. Observe the animations below:

Animation 5(a)	Animation 5(b)

Similarly, we can see the effects on the polar expressions with sine:

Animation 6(a)	Animation 6(b)

While |b| increases the graph enlarges its size. An interesting effect occurs when the value of b changes from a positive number to a negative one (and vice versa) since the curve seems to be reflected around the coordinate system’s origin.

THE LIMACONS CURVES

The curves associated to the general polar expressions

are the Limacons Curves. The curves that we have analyzed above are examples of well-known Limacons Curves. In general, a Limacon can be formed when a circle rolls around the outside of another circle. The radius of each circle does not need to be equal each other. Figure 6 below shows an example of a 3-leaved rose:

Figure 6

Here are some animations in GSP to explore:

1) Click for a cardiod.

2) Click for looped limacon.

3) Click for a looped limacon and a 3-leaved rose.