Polar Equations

Using Theorist

Let's investigate the polar equation:

Let a = b

What happens when a = b and our k varies from 1 to 3? The following equations are the ones that are graphed below:


From the graph we see that the value for k changes the number of "petals." The green graph has a k-value of 1 and the red graph's k-value is 3.

How would the graphs change if we left a = b and varied them?


In the illustration above, we have three cardioids with various "widths." The red cardioid was produced with the equation where a = b = 4.

We again vary a and b and we k=4. This creates a 4 petal flower with differing petal lengths. So the green flower is the smallest and is thus the result of the 1st equation below.



Let a > b

What happens when a > b and our k varies from 1 to 3?


When k = 1, we have a dimpled limcon. When k > 1, we have flowers with k number of petals, but the petals do not meet at the origin as they do when a = b.

Now let's vary our a's and b's and leave our k the same.


Again we have petals that do not join at the origin and the length of each petal varies with a and b.

Let b > a

When b > a, we have different graphs from the other situations. When k = 1, we have a limacon. When k = 2, we have a leminscate with two extra little petals. This could also be thought of as a flower with two large petals and two small petals. When k = 3, the graph has 3 petals with 3 smaller petals inside the other petals.


Let's vary our a's and b's again and leave our k the same.


As you can see, the petals are the same, but the length of each petal varies with a and b.

Click to see what happens when k is NOT an integer.

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