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.





Let’s start with the simplest polar equation r = b cos (). The graph is shown below. Notice that this graph are circles with diameters of b unit and pass through the origin. Also, the positive and negative signs of b determine the locations of the circles. When b is positive, the circles are on the right of the y-axis and the circles are on the left of the y-axis when b is negative.




[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 () below are curves with different shapes. When a is smaller than 1, the polar equation r = a + b cos () forms curves that have inner loops like the green and the pink ones below. When 1  a  2, it forms curves that have dimples like the red, the dark blue, and the light blue ones.  When a is larger than 2, the curves just curve outward without inner loops or dimples. All the curves form by the polar equation r = a + cos () are called limacon (pronounced lee muh SOHN).




[r = a + b cos()]



How about multiply  by a number k to alter the graphs of the two polar equations above? Let’s exam the polar equation r = b cos (k) first. Since b only determines the location and the size of r = b cos (k), let b = 1 to simplify our exploration.




[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 rose. It seems like whether k is positive or negative, it has nothing to do with the shape of the rose curves. Therefore, we will use only positive k in the rest of our exploration.


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 k is an odd number, there are k petals in a rose curve. When k is an even number, there are 2k petals in a rose. Please look at the graphs below for the differences.




[r = cos (k)]



Now, it’s 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 “flower-petal” shape. Here k determines the number of “petal”. In the polar equation r = a + b cos (k), there are exactly k “petals”. Besides, when a = 1, the petals “tie” at the origin. When a is larger than 1, the “ties” disappear. Also, the larger the value of a, the smoother the curve becomes.