Isn't Santa at the North Polar?

by

Mike Rosonet

This exercise will investigate polar equations of the form

and how the value of k affects the graph. To begin with, what is a polar equation?

A polar equation is a function expressed in polar coordinates with the radius r expressed as a function of the angle .

Consider the equation r = a. This is a transformation of the original equation of this investigation with b = 0. If this equation is graphed in the polar coordinate plane, it looks like this:

Graphs of polar equations with differing values of a

Thus, the polar equation r = a is a circle with a radius of a. The reason for this is because every point in the polar coordinate system is determined by its distance r from the origin (0, 0) and the measure of the angle constructed by the segment from the point to the origin and the initial side of the angle, the x-axis.

The parameter a.

What does changing the value of a do to the graph of the given polar equation?

graph of the inital equation with a = 1

Let b = 4 and k = 1. Thus, the equation is r = a + 4cos. Allowing the value of a to vary from 0 to 10, the graph changes like so:

changing the value of a from 0 to 10

When a = 0, the graph is of a circle with a diameter equal to 4 with endpoints at the points (0, 0) and (4, 0) and center at (2, 0). As the value of a increases from 0 to 4, the circle expands to reveal that it is actually two circles, and while one is expanding into a bud shape, the other is shrinking and looks like a tear drop. When a = 0, the tear drop can no longer be seen. As a increases from 4 to 10, the tear drop circle pushes out along the x-axis, increasing the circumference of the larger bud-shape as it expands to look more and more like a perfect circle.

The parameter b.

What does changing the parameter of b do to the graph of the given polar equation?

Here, let a = 4 and k = 1. Thus, the equation is r = 4 + bcos. Allowing the value of b increase from 0 to 8:

changing the value of b from 0 to 8

When b = 0, the graph is of a circle with center (0, 0) and a radius of 4. As b increases from 0 to 4, the center of the circle shifts to the right and the left side of the circle begins to indent while the circle's circumference expands. As b increases from 4 to 8, a loop forms inside the circle; its circumference increases while the circumference of the circle increases. The inner loop approaches the outer circle but never reaches it as b increases.

The relationship between a and b

The parameters a and b have inverse effects on the polar equation and its graph. When a = 0, it graphs as a perfect circle tangent to the y-axis. As a increases to infinity, the circle separates into two loops, one inside the other, until the inner loop is joined with the outer loop at a = b. From that point to infinity, the graph expands in all directions and approaches the shape of a perfect circle. As b increases to infinitiy, the circle indents on its left side as it shifts to the right. The dent forms a loop inside the growing circle and both the inner loop and the outer circle expand as b increases to infinity.

One thing worth noticing is that the graph when b = 0 is the same as the graph as a approaches infinity. Conversely, the graph as b approaches infinity grows very close to the graph when a = 0. The neutral case is when a = b. Each of these findings, however, depend on the parameter k = 1.

The parameter k.

The parameter k affects the graph of the polar equation in a totally different manner from those effects made by changing the values of parameters a and b. It is important, however, to look at three different cases, a < b, a = b, and a > b.

I. a < b.

For this case, the base equation will be r = a + bcos(k ). The general form, with k = 1, is below.

graph of the base equation with a = 1, b = 2 so that a < b, and k = 1

How will this graph change as the value of k increases?

graphs of the equation with different values for k, with a < b (a = 1, b = 2)

From these examples, it is clear to see that when k is an even integer, there will be a number of loops equal to k outside of the larger loops. Likewise, when k is an odd integer, there will be a nmber of loops equal to k inside the larger loops.

II. a = b.

For this case, the base equation will be r = 2 + 2cos(k ). The general form, with k = 1, is below.

How will this graph change as the value of k increases to 2, 3, 4, and 5, respectively?

graphs of the equation with different values of k, and a = b (a = b = 2)

From these examples, it is plain to see that, when a = b, the value of k equals the number of loops, or, what look like flower petals in these examples. Notice that each petal begins and ends at the origin and each petal is the same size as the other petals in each graph.

III. a > b.

For this case, the base equation will be r = a + bcos(k ). The general form, with k = 1, is below.

the graph of the equation with a > b (a = 2, b = 1), k = 1

How will this graph change as the value of k increases?

graph of the equation with increasing values of k, and a > b (a = 2, b = 1)

From these examples, it can be seen that the value of k corresponds to the number of petal-like divisions that occur. The center of each graph is the origin, and each petal is the same size as the other petals in each graph.

Conclusion

The behavior of the polar equation r = a + bcos(k ) is predictable, as shown by each of these investigations. By changing the values of the parameters a, b, or k, the graph of the equation can take on many different shapes. By letting k = 1, base equations and graphs can be used to determine the effects of changing a and b. Then, changing the value of k will divide the graph into k equal sections.

Extension 1

What happens to the graphs when cos ( ) is replaced with sin ( )?

* The theme of this page was inspired by my wife, who loves Christmas.

Return