Polar Equations

By Pei-Chun Shih

The **polar
coordinate system** is a
two-dimensional coordinate system in which the position of an object is
recorded using the **distance** from a
fixed point and an **angle** made with a
fixed ray from that point. Since distances and angles are the main components
of the polar coordinate system, this system is especially useful under the
circumstances where the relationship between two points is most easily
expressed in terms of angles and distances.

In
the polar coordinate system above, a fixed point O is called the **pole** (origin). A horizontal ray directed from the pole
toward the right is called the **polar axis**. If a point P has polar
coordinates (r, ), then the
distance from the pole to point P is | r | and the angle formed by OP and the
polar axis is . Note that the
angle, , can be
measured in degrees or radians.

An
equation expressed in terms of polar coordinates is called a **polar equation**. As mentioned earlier, distances and angles are two
main elements of the polar coordinate system and therefore a polar equation can
be specified by defining r (distances) as a function of (angles) in many cases. For example, r =
3 cos () is a polar
equation. A plot of all points whose coordinates (r, ) satisfy a
given polar equation is called a **polar graph**.

I am
going to explore some polar equations and their polar graphs in the following
sections.

LetÕs
start with the simplest polar equation **r = b
cos (**

[r = b cos ()]

Now
we know that we can alter the position and the size of a polar graph by
multiplying the function by a number. Next, we are going to explore the polar
graph of r = b cos () by adding the
function by a number, say a. The polar graph of

**r
= a + b cos (**

[r = a + b cos()]

How
about multiply by a number ** k**
to alter the graphs of the two polar equations above? LetÕs exam the polar
equation r = b cos (

[r = cos (k)]

From
the two graphs above, we can see that multiply by ** k** form petal shape curves. This type of curve is called
a

There
is an interesting finding here, the number of petals of a rose is determined by
whether ** k** is an odd number
or an even number. When

[r = cos (k)]

Now,
itÕs time to explore r = a + b cos (k). First, let a
= 1, 2, and 5 with b = 1 (again, b only affects the size of the curves), and
explore the curves when k = 2, 3, 4, and 5.

[r = a + cos (k) where a = 1, 2, and 5]

Recall
the shape of r = a + b cos () is a limacon.
From the two graphs above, we can see that if is multiplied by ** k** then the shape of y = a + b cos () changes
completely. It changes from a limacon to a Òflower-petalÓ shape. Here