Janet Kaplan



We are most familiar with functions as they appear when graphed on the rectangular coordinate plane. But often it is easier to work with the polar coordinate system of representation, particularly when describing motion using a function of one variable.  The polar coordinate system facilitates the study of circular motion.  It is extremely useful when describing the motion of electrons around the nucleus of an atom or the circular paths of orbiting spacecraft.

In this investigation we will look closely at several well-known polar equations and determine what gives them their characteristic shape. 

Before we begin our exploration, however, it may be useful to review how polar equations are graphed. CLICK HERE for a review of polar coordinates from Math Forum.

Let’s take a look, first of all, at the function r = a + b cos (kt).

   where a, b and k are positive integers, b is the coefficient of the trigonometric function,  t is the angle measurement, and k is its coefficient.

Polar equations of this form are known as limacons (for the French word for snail). When a = b, we get a special case of the limacon. The resulting graph is called a cardioid because of its heart like shape.

r = 2 + 2 cos (t)

But if we vary the values of a and b, the respective limacons take on different looks, as in the graph below.

r = 1 + 2 cos (t)     r = 2 + 2 cos (t)     r = 3 + 2 cos (t)     r = 5 + 2 cos (t)

As you might expect, there are reasons for the changing shape.  Limacons can be characterized according to the relative sizes of a and b.

r = 1 + 2 cos (t) is an example in which a < b

r = 2 + 2 cos (t) is the cardioid, in which a = b

r = 3 + 2 cos (t) is a case in which b < a < 2b       and

r = 5 + 2 cos (t) is a case in which a >= 2b


Now let’s look at a polar equation which is seemingly more basic, but has some beautiful designs in store for us.

r = b cos (kt)

Getting rid of the additional term, but varying the value of the coefficient k, we come across the lovely n-leaf rose.

r = 3 cos (1t)     r = 3 cos (3t)


r = 3 cos (2t)     r = 3 cos (4t)


 Notice how the number of petals corresponds to the value of k. When k is 1, there’s just the lone petal hanging to the right of the y-axis stalk. When k = 3, we get 3 petals. But when k = 2 or 4, we get 4 and 8 petals, respectively.  The pattern seems to be that when k is odd, the number of petals is equal to k.  But when k is even, the number of petals is equal to 2k. Now why would this be?

Well in fact, it’s really just an illusion. When k is odd, the points of each petal are traced twice as t ranges from 0 to 2Pi.  If k is even, the trace of each petal is drawn once, changing its orientation slightly around the origin, as t ranges from 0 to 2Pi.


What is the significance of the value of b? As the graph below indicates, the value of b determines the length of the leaf.

r = 1 cos (5t)     r = 2 cos (5t)     r = 3 cos (5t)


How does cos (t) compare with sin (t)?

r = 3 cos (2t)     r = 3 sin (2t)

Notice how the sine function is rotated 45 degrees from the cosine function. Just like in the rectangular coordinate system shown below, the two trigonometric functions are “out of phase” from each other by 45 degrees or (pi / 4).



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