It is known that the spiral phyllotactic pattern is common in Nature, especially in Botany.
It consists of two group of clockwise and anticlockwise spirals, starting from the center. In most cases the number of those spirals are two consecutive Fibonacci numbers: $F_n, F_{n+1}$. Also there are patterns where number of spirals are doubled Fibonacci numbers, Lucas numbers, or Fibonacci ± 1.
The equations for generating some sort of a spiral phyllotaxis are:
$$ \phi = \pi(1+\sqrt{5}) \\ \forall n \in [0, N] \\ \theta = n \phi \\ r = \sqrt{n} \\ x = r \cos{\theta} \\ y = r \sin{\theta} $$
How to change that equations to take into account the predefined number of spirals? For example, I want to generate a phyllotactic pattern consisted of (11, 18) spirals: 11 clockwise and 18 anticlockwise. How to do that?
UPDATE.
It looks that the number of visible spirals is depended on the radius of a pattern.