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.
