The graphs generated by polar equations are
some of the most beautiful in all of mathematics. Before exploring
the graphs lets briefly define what polar equations are. Polar
equations are derived from polar coordinates, ** p**,
of the form (

Lets take a look at a basic curve.

What we see as the coefficient of cos changes the "inner" circle either gets smaller or non existent or increases in size. When the inner circle increase the outer circle increase in relation.

Now lets have some fun with the beauty of these equations.

1.)

2.)

3.)

In this series of curves, called rose petals, we will explore the general equation

where we will keep **a** and **b** constant
and vary ** k**. When

1.) (one petal)

2.) (two petals)

3.) (three petals)

4.) (ten petals)

5.) (fifty petals)

6.) (two thousand petals!)

So, what can we say about the change of ** k**?
Well, we know that as