Polar Opposites

By: John Vereen

            We will be exploring graphs in polar coordinates of the form   where a, b, and k are integers. First letŐs explain the significance of the variables r and . The variable r stands for radius, and each circle going outward in the graph above represents an increase of r by 1. The variable  represents the angle that is subtended by r at a specific time. Theta is measured by the lines going outward from the origin from [0,2pi].

            Now letŐs explore how variances in a affect the graph. As we can see above, the petals of the graph seem to become less defined as a increases in magnitude. Why is this? We see that a represents a shift, just as it does in the Cartesian plane. The constant a shifts the variable r. Since the cosine function represents values on the unit circle, it will only cover r values that are, at most, to radii apart. So, with the first graph , , we see a shift of r by 1. So, instead of sweeping the standard [-1,1], r will cover values [0,2]. We start seeing a pattern when a=2 in the equation   because r covers values [1,3], which produces less pronounced petals in the graph. Now we see, mathematically, why the petals are less pronounced; because r is being shifted outward by the value of a. It is also interesting to point out that the graphs     and   map onto each other, so the petals are less pronounced for negative a as well as positive a.

            We will also explore how changes in b affect graphs of form  . In the graph above, I fixed the values a=0 and k=4. When we varied b, the overall shape of the graph stayed the same; however, as b increased, we see changes in the radius length of each graph. The b value affects the radius just as the a value does. However, instead of shifting r additively like the constant a, the constant b shifts r multiplicatively. This is why the shape of the graph/petals is maintained and why the constant b is always zero when the cosine is zero: a+0=a while b(0)=0. Thus, we can conclude that the value b affects the amplitude of the graph in the polar plane.

            Finally, we will see how different values of k affect the shape of graphs of  . From the graphs above, we can conclude that the value of k has a direct effect on the number of petals that the graph will have. The number of petals in the polar plane is synonymous to periodicity in the Cartesian plane.

            An interesting fact about k is that it differs between even and odd numbers. As we can see above, if k is even, then the graph will have a number of petals equivalent to 2k. If k is odd, then it appears the graph will have a number of petals equivalent to k. This is because every polar graph of form   actually has a number of petals equal to 2k, but with odd numbers the second half of the petals are mapped onto the first half. So for example, a function   actually maps 10 petals, but the second 5 petals are mapped is directly onto the locations of the first 5 petals.

            If we changed    to , then we would have graphs of the same shape, just at different locations on the polar axes. A method we can use for mapping the cosine graph onto the sine graph is inputting a value theta such that, when multiplied by k, produces a 90 degree (or pi/2) rotation counterclockwise. This works because, on a standard unit circle, cosine=1 at 0 and when we rotate by 90 degrees, sine=1. So in order that the cosine function equals the sine function, we must have an input   = 90.