Polar
Coordinates and the Òn-Leaf RoseÓ
By
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 ÒleafsÓ 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 
?
LetÕs 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 numberÕs 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?
LetÕs 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 sortÉwhich gives rise to the name Òn-leaf rose.Ó
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?
