Polar Coordinates and the “n-Leaf Rose”

 Gayle Gilbert & Greg Schmidt


Consider , where  and


Most of us are well acquainted with the rectangular coordinate system where the two coordinates are distances from perpendicular axis, but in order to investigate the graph of the above function we will use polar coordinates.  In polar coordinates, the pair of numbers that determines a point are an angle through which to rotate from the  and a distance from the origin.


Fix  and let , which gives us the following graphs














Now for the case where  it appears as though the number of “leafs” on the graph are in direct correspondence to .

Checking our hypothesis by letting  and setting  we see that we may be correct, but it is left to the reader to prove that is in fact the case.





But will this remain to be the case is ?


Let’s fix ,  and .  So we have



, ,


So we see that this does not remain to be the case when .  But we should be careful to note that this does not necessarily tell us that if  then we will not have  leaves. Notice when ,  and  we have



, ,


Now these leaves do not intersect at the origin, but nonetheless are leaves.  But then just what is happening?  Let us consider a few more examples.


Take , . 






This gives us a circle of radius 1 and center at the origin, which appears to be somewhat different from our other graphs, but when we recall that our graph is determined by a number’s distance from the origin and the angle through which it rotates about the  we immediately see how this graph it related to the others.  We also note that we need not specify the value of  so long as  is and integer, since the only specifies the number of traces about the given circle. 


But how does cosine affect this graph?  Or perhaps how would these graphs change if we were to replace cosine with sine?


Let’s see!


Well, we will obviously have no change whatsoever from the last graph when we replace cosine with sine, so consider

the following:








So our graphs with cosine replaced by sine remain similar thus far, only differing by a rotation.  This also makes sense once one recalls the behavior/relation of sine and cosine. 


Now, we may have thought that when  our graph resembles a flower of some sort…which gives rise to the name “n-leaf rose.”


Nevertheless, polar graphs of this sort produce a wide and interesting family of graphs, which are of great interest, and very difficult to predict.  Can you guess the values of ,  and  for the following graph?