Assignment # 11

Explorations with Polar Equations

by Michael Ferra

Proposed Investigation

i. Investigate r = a + b cos(kθ).

When a and b are equal, and k is an integer, this is one textbook version of the "n-leaf rose."

ii. Compare r = a + b cos(kθ) with r = b cos(kθ) for various k.

iii. What if cos() is replaced with sin()?

i. r = a + b cos(kθ)

Let's set a = b = 1 and test different values for k. Let's observe k = 1, 3, 5, a set of odd numbers for k.

Notice the value of k here determines the number of petals on each graph. Let's now look at when k = 2, 4, 6, a set of even numbers for k.

Notice once again that the value of k here determines the number of petals on each graph.

ii. Compare r = a + b cos(kθ) with r = b cos(kθ) for various k.

Let b = 1 and let's start by looking at k = 1, 3, 5, the same set of odd numbers for k as before.

Notice, as with r = a + b cos(kθ), that the value of k here when odd determines the number of petals on each graph. Thus restated, when k is odd, the number of petals is the same as the k value.

Let's now observe when k = 2, 4, 6, the same set of even numbers for k as before.

When k is even, the number of petals are twice the value of k, thus restated, the number of petals is 2k.

iii. What if cos() is replaced with sin()?

r = a + b sin(kθ)

Let's set a = b = 1 and test different values for k. Let's observe k = 1, 2, 3, 4.

Notice the same convention still holds that the value of k here determines the number of petals on each graph. Compared to the graphs of r = a + b cos(kθ), each graph has been rotated counterclockwise by 90/k degrees.

Explore r = b cos(kθ) compared to r = b sin(kθ) and see the same convention holds true for the equation with sin() as with cos() except the graph with sin() has a counterclockwise rotation by 90/k degrees.