Green Eggs and Polar Equations

A look at r = a + b cos (nθ)

Assignment 11

By Amber Candela

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What is a polar equation?

A typical polar question is in the form r = f(θ) where f is some function ofθ . The independent variable is θ and the dependent variable is r. The graph is a collection of all points that have at least one set of polar coordinates that satisfy the equation. A point has more than one set of polar coordinates. Polar equations can be symmetrical. They can be symmetric to either the polar axis, the pole or the line θ= π/2.

Let's take a look at one!

When looking at the following polar equation, "a" and "b" were set equal to one and "n" varied from 1 to 10.

When n is equal to one there is a kidney bean shaped figure. When n is equal to 2 there are two "loops" that create a shape that looks like the infinity symbol. As "n" gets larger it starts to create petals or loops. At n =3 there are 3 loops, at n = 4 there are 4 loops and this continues in such a way that when n = 20 there are 20 loops. It then gives that with the equation, when a =1 and b=1 there are "n" number of loops around the origin. The animation below shows what happens to the equation as "n" goes from 1 to 20.

**r = a + b cos (nθ)**

When a = b, as "a" and "b" continue to get larger, the petals appear to get longer. The following video shows the following equation where "a" and "b" have been changed to "n" to see the effect when "n" goes from 1 to 3.

Where the number multiplied by theta gives the number of loops, "a" and "b" determine the length of the petals. Below shows a picture where a=b and "a" and "b" go from 1 to 2.75 in increments of .25. The effect is quite beautiful.