In the case k = 1, I draw the graphs
of the equation as a = b = 1 (purple),
a = b = 2 (red), and a = b = 3 (blue).
The inside area of the curve is getting
greater as a and b get greater together.
Here is a movie indicating this behavior
from a = b = 0 to a = b = 10.
In the case that k = 2, the graphs
are shown below, as a = b = 1 (purple), a = b = 2 (red), and a
= b = 3 (red).
The inside area is getting greater
as the same way.
Here is a movie telling this from a
= b = 0 to a = b = 10.
In the case k = 3, I draw three graphs
as a = b = 1 (purple), a = b = 2 (red), and a = b = 3 (red).
I give a movie showing the cases that
a = b = 1 and k is one integer of 1, 2, 3, ...9, and 10.
In summary, if a and b are equal and
are getting greater, the inside area of the curve gets greater.
In addition, the number of leaves is
to be k. For example, when k = 1 the number of leaves is 1 and
when k = 4 the number of leaves is 4.
I choose b = 1 and draw graphs for
k = 1 (purple), 2 (red), 3 (blue), 4 (green), and 5 (sky blue).
The number of leafs are same as k only
if k is an odd number. If k is an even number, the number of leaves
In order to show these facts, I made
a movie below.
You can click
While the number of step in this movie
has been 10, it would be more interesting to change the number
of step into 100.
See the graph below and click
All the graphs on the previous movie
are the cases that k is an integer. The latest movie is telling
us even the cases that k is not an integer.
Consider the what if cos is replaced
I show you the movie of ,
with k = 0, 1, 2, ...,9, 10.
It seems that the graphs (one
I showed as a movie) is rotated into the graphs at
Besides, here is a movie
with 100 steps.
All the graphs in this case go over
the point (0, 0), while the graps have
been going over (1, 0).