Assignment # 11

Explorations with Polar Equations

by Michael Ferra

Proposed Investigation

Investigatei..r = a + b cos(kθ)When

andaare equal, andbis an integer, this is one textbook version of the "n-leaf rose."k

Compareii.withr = a + b cos(kθ)for variousr = b cos(kθ).k

What ifiii.is replaced withcos()?sin()

i. r = a + b cos(kθ)Let's set

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

Notice the value of

here determines the number of petals on each graph. Let's now look at whenk= 2, 4, 6, a set of even numbers fork.kNotice once again that the value of

here determines the number of petals on each graph.k

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

= 1, 3, 5, the same set of odd numbers forkas before.kNotice, as with

, that the value ofr = a + b cos(kθ)here when odd determines the number of petals on each graph. Thus restated, whenkis odd, the number of petals is the same as thekvalue.kLet's now observe when

= 2, 4, 6, the same set of even numbers forkas before.kWhen

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

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

r = a + b sin(kθ)Let's set

and test different values fora = b = 1. Let's observek= 1, 2, 3, 4.kNotice the same convention still holds that the value of

here determines the number of petals on each graph. Compared to the graphs ofk), each graph has been rotated counterclockwise by 90/k degrees.r = a + b cos(kθExplore

r = b cos(kθ)compared tor = 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.