This is a graphical exploration of the curve:
Will consider for now that k=17, which will produce 17 leafs, and will explore the curve for different aís.
Apparently the curve has two values of r for each theta? Is that weird? Will see try to see what happens latter.













































If a is between ñ5 and 5 the size of the larger leaf is equal to a+b and the size of the small leaf (only when K is odd) is ba.
When a is larger than 5, the graph of the function will be exterior to a circle centered at (0,0) with radius ab. Any of the curves is always included in the circle centered at (0,0) and radius equal to a+b.
We will try now to see why is that, apparently, there are two values of r for the same angle. If we walk with the grid lines, one way, counterclockwise, on the graph of:
we can see in what order the leafs are ran through. If, as in this case, k (=7) is odd and there are leafs of different size they are ran in the following order:
What happens with cos(x) in the interval from 0 to 2Pi will happen with cos(7x) in the interval from 0 to 2Pi/7. Thus:
























and the same is repeated until the interval from 0 to 2Pi is exhausted by jumps of Pi/14 and the 14 leafs are run through.
Considering a and b fixed, how does the position of the leafs depends on k, considering k integer?
If k is even, in this case k=8 and a=1 and b=5, then we have 16 leafs:
If k is odd, for example, k=9, and equally a=1 and b=5, we have 18 leafs but half of then nested in the other half:
Graphical relation between leafs of curve and derivative
The number of leafs in:
and the number of leafs in itís derivative function:
varies in the following way as a dependent on k (integer): If k is even, the number of leafs in the derivative is double than the number of leafs in the function which is exactly equal to k:
If the k is odd the number of leafs in the function and in the derivative is the same:
Note: If in each of the above curves we draw the exterior and the interior circles (when possible) we get stain glass motives.