Polar Coordinates



In this exploration, we want to look at various polar coordinates and investigate the kinds of graphs will see.

Let's start out by investigating the following function:

If we let a and b both equal each other and k be some constant integer, we get the following graph:

Ok, we see that this gives us the textbook version of the 'n-leaf rose' which is also known as a cardiod. This falls under a special family of graphs called Limacon. We can obtain a cardiod by tracing the path of a point on a circle rolling around the circumference of another circle with the same radius. If we let k be constant and change a and b, we will get the following graphs:

So, we see that when b is less than or equal to 2a, the limacon is convex and when b and a equal, it degenerates to a cardiod. If b is strictly less than a, then the limacon has an inner loop like the graph above.

Now let's change the value of k but keep a and b constant.

Ok, I think we see a pattern. It seems as though k determines the number of petals in the graph. So we started out with k at 1 and increased it to 4. So our next investigation is to compare this kind of graph to

Now, let's look at this with various values of k.

So we see yet another pattern here. If k is an integer, the curve will be rosed shaped with 2k petals if k is even and k petals if k is odd. What will happen if k is an irrational?

So our results shows us that when k is an irrational number, then it is not closed and has infinite length and when it is rational, it has a finite length and the curve is closed.

Now let's investigate the same function but instead of cosine, lets use sine.

Again we see that the value of k determines the number of petals and the graph actually rotates towards the y-axis. We want to compare this to .

How lovly...ok, clearly we can see that sine and cosine are identical except for a rotation of radians.

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