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.