I have tried to make this as easy to understand as I can. If you don't get what I mean, PLEASE, post a comment; thank you!
I want to find parametric equations for a helix where the pitch grows in relation to a particular angle. (I'm going to describe what I want in 2-D, to make things simpler.) These equations should be of the form $x, y=\frac{\sin(t)}{f(t)}, f(t)$. This form is important because it creates a 'hyperbolic' curve.' ($f(t)$ is just some expression in terms of $t$, it's what you'll be looking for.)
Specifically, I want the 'pitch' to be powers of $φ$ when $t$ equals multiples of $\frac{2π}{φ}$. By 'pitch' I mean the height of a full loop whose top is one such multiple. ($φ=1.618...$)
To provide an example, at $4·\frac{2π}{φ}$, this 'pitch' should be $φ^{4}$. The pattern should go on as such both positively and negatively forever. I wrote out a table to clarify this subject further. (Also, see my fig. which depicts the relevant concepts for pitch equals $φ^{5}$.)
$\begin{array}{c|c} t & Pitch \\ \hline -1·\frac{2π}{φ} & φ^{-1} \\ 0·\frac{2π}{φ} & φ^{0} \\ 1·\frac{2π}{φ} & φ^{1} \\ 2·\frac{2π}{φ} & φ^{2} \\ 3·\frac{2π}{φ} & φ^{3} \\ \end{array}$
