I've been trying to derive the parametric equations for a specific type of sine-wave for quite some time, and now I think I know how to do it in principle but lack the skill in practice. So, I'd be very thankful for some help!
The wave I want is essentially the same as the one graphed here
with one key difference. The sine-wave in my graph has the following properties:
1. It has points of tangency to $y\cdot x=\pm 1$ (See my Graph.).
2. The vertical distance between these points of tangency on the respective sides increase by powers of $\varphi$ (STARTING AT $\varphi^1$ WHEN ONLY POSITIVE NUMBERS ARE GRAPHED) on alternate sides. (See my fig: $B$ is $\varphi$ times $A$.) The Pattern goes on repeating forever up the graph. ($\varphi$ is the Golden Ratio Constant: $1.618\ldots$, or $0.618\ldots$).
3. The graph starts at $(0, 1)$ for all positive numbers graphed.
4. Leaving aside the "$\sin(t)$" The function for $x(t)$ is the inverse of the function for $y(t)$. That's why it has points of tangency to $y\cdot x=\pm 1$.
5. The vertical distance between the aforementioned points of tangency are always powers of $\varphi$ times a constant $\alpha$.
Ok, so, I want to retain properties 1. through 4. (This Is Very Important!!), while being able to change the value of $\alpha$. To be specific, I want to be able to set $\alpha$ equal to $\varphi^{-2}$ (that is $1.618^{-2}$). In essence, that's it.
A little note on the overall nature of the problem: At first you might think of dividing / multiplying the function(s) ($\varphi^t/PI$ and $\varphi^-t/PI$) by something to solve the problem. But, I found that this is the same as using $\cos(t)$, and the issue with it and the thing that makes the problem rather tricky is that this will make it so that the graph does not start at $(0, 1)$ (for positive numbers). This is where the Key difficulty lies.
My graph and image should provide any other information you might need. I'm very excited to find an answer and can't wait for a response. Thank you all so much!
NOTES:
A: This question is cross-posted here
B: Make sure to look carefully at my graph; it shows the definition of α and shows points of tangency and more…
C: Please give answers in terms of "$\sin(t)$", not "$\cos(t)$", thank you!".
D: A version of this question with $\alpha=1$ is asked and answered in the question "Deriving Parametric Equations For A Hyperbolic PHI Sine-Wave".
Best Answer
This function should meet all 5 of your criteria, with the desired $\alpha=\phi^{-2}$ and $\phi=(1+\sqrt 5)/2$ (but it can be adjusted to any value of $\alpha$).
$$\{x(t),y(t)\}=\{\phi ^{-t/\pi} \cos t,\phi ^{t/\pi } \exp \left(-(\pi/t)^2\tfrac{1}{8}\ln\phi \sin ^2 t \right)\}$$