Investigate
Note:
* When a and b are equal, and k is an integer, this is one textbook version of the " n-leaf rose."
* Compare with
for various k. What if . . . cos( ) is replaced with sin( )?
At first, I set a=b=1 and k be an variable integer.
If k=0 , the shape is a circle whose center is origin and radius is 2.
When k=1 or k=-1, as you see above , they have the same shape(figure 1, 3)
When k=2 or k=-2, the shapes are also equal, which are 2-leaves rose. Since cos function is even, the shape would be equal regardless sign of k ( either k is positive or negative integer)
According to value k, we can expect the number of leaves. In other words, if k=n integer, then the shape will have n-leaves.
In addition, if n=odd number, then the shape is symetric to x-axis.
If n=even number, the the shape is symetric to both x- and y- axes.
In this time, when a=b=2, the results are like these.
If a=b=3, we can see the shapes like these.
As the a and b are greater, the shape dose not change but the size is getting bigger.
Now, for various k, we can see the variation.
Nect, what if . . . cos( ) is replaced with sin( )?
The number of leaves depends on k.
If we rotate the red one by some degrees, then we can get blue shape.