EMAT6680 Assignment 11

Polar Equations

By Kevin Perry


Polar Equations


In this discussion we will explore a family of polar equations known as the n-leaf rose.




A polar equation is one of the form r = f(θ), where r is the distance from the origin and θ is the angle from the positive x axis..


Simple Polar Equations


To get the feel of polar equations, we can first explore a simple example.  Let’s first look at



this equation says that for every any, the distance from the origin is 1.  The graph of this equation is



Next look at



which says that the distance from the origin at every angle is equal to the angle itself (in radians).  This graph looks like



We can see that these graphs are different than normal (x,y) graphs, in that they seem to rotate around the origin instead of going from left to right.


“n-leaf rose” Polar Equations


The family of functions called the “n-leaf rose” has the form



and a typical graph looks like



which has a = 1, b = 2, and k = 8.  From the graph, you can clearly see the flower pattern.  The special case where |a| = |b| gives the condition where k is the number of “petals” on the flower.  By changing b to 1 for the above graph we get



For conditions where |b| > |a|, you will always get k large leaves with k smaller leaves such as the first graph.  But if |b| < |a|, the graph changes shape again, for example, a = 2, b = 1, and k = 8.



Also, if a and b are not the same sign (i.e. one is positive and the other is negative), the graph will rotate so that the valley (or smaller petal) is now on the x axis.  For example, a = -2, b = 1, and k = 8.



Or a = 1, b = -2, and k = 8.



Another special case is when a = 0.  What we see is that the petals are all the same size again, but now there are 2k of them.  For example, a = 0, b = 2, and k = 8



There are lots of beautiful shapes that appear with this family of equations.  If you would like to create your own shapes on graphing calculator, click here.



Conclusions and Extensions


If instead of using k=8, what if we picked a k that was odd.  Let’s look at a = 1, b = 2, and k = 5.



The smaller leaves are now inside the larger leaves.


What if a, b, and k, are not integers?  And how large of a θ do we need to graph to in order to get the graph to overlap itself?


In this discussion, we have looked at some simple polar equations and seen that the shapes that result are unique and interesting.  These types of equations are not discussed and investigated as often as Cartesian functions, but they are just as useful and sometimes more fun!



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