NOTE $0$ times the golden angle is a $G_1$ point, too, and should give a distance $φ^{-3}$.
Some of this may be hard to visualize so, see my figs., also see a graph here.
Also, see here for a similar, very nice derivation: https://math.stackexchange.com/a/3520568/708680 See Here Also for the answerer profile.
(Note. My figs. depict the concepts not an actual graph of the wave that I'm looking for, for obvious reasons.)
If $φ=\left(\frac{1+5^{1/2}}{2}\right), α=φ^{-2}, β=1$, then the parametric equations, $x, y=sin(t)·\left(α·φ^{t-{π/2}/π}+β-\frac{α}{φ^{1/2}}\right)^{-1}, \left(α·φ^{t-{π/2}/π}+β-\frac{α}{φ^{1/2}}\right)$ produce a graph where the vertical distances between points of tangency with $x·y=±1$ on alternate sides are powers of φ. (when only positive numbers are graphed, starting at $φ^{-1}$ and proceeding as follows: $ φ^{0}, φ^{1}, φ^{2}, φ^{3}$).
To learn more about the above, see the update to the answer at this link: https://math.stackexchange.com/a/3515756/708680
In the above $α$ is defined in terms of points given by multiples of $π/2$: The distance between any two such points is $α$ times powers of the golden ratio constant (1.618...) (see the above answer / update for more information). I would like to find similar parametric equations where $α$ is defined in terms of points given by multiples of the "Golden Angle" $2·π·φ^{-1}$. These shall be called 'golden points': There are two forms of such points, $g_1$ and $g_2$:
$g_1$ is given by setting $t$ (for an expression of the kind under discussion) to some whole number multiple of the golden angle, (i.e., the golden angle times -2, -1, 0, 1, 2 etc.).
A $g_1$ point's corresponding $g_2$ point has $x$ given by its $g_1$ point. Its $y$ is given by the first (Lower-down) point where $x=$ the $x$ coordinate of $g_1$ intersects the 'side' of the wave on which $g_1$ is situated. (By side the following is meant: For the aforementioned equations, $0≤t≤\left(\frac{π}{2}\right)$ is the back side $\left(\frac{π}{2}\right)≤t≤\left(\frac{π}{2}\right)+π$ is the front side, and $\left(\frac{π}{2}\right)+π≤t≤\left(\frac{π}{2}\right)+π+π$ is the back side etc... Momentarily thinking in terms of 3-space may help.)
I want to find equations where the distance between any two 'golden 'points' $g_1$ and $g_2$ is a power of $φ$ times $α$ such that the first such distance is $φ^{-1}$ for positive numbers; proceeding: $φ^{0}, φ^{1}, φ^{2}, φ^{3}$. These equations should have the following attributes, too:
0. Be of the form: $x,y=f(t)^{-1}·sin(t), f(t)$. (To add clarity, for the original equations, this $f(t)$ was $f(t)=\left(α·φ^{t-{π/2}/π}+β-\frac{α}{φ^{1/2}}\right)$.
1. Start at $(0, 1)$ for positive and negative numbers.
2. Have points of tangency to $x·y=±1$ (as a result of 0.).
3. Maintain a smooth, sinusoidal, 2-D spiral nature throughout.
4. Be written in terms of $sin(t)$.
Thanks so much for the help!
This has an error in it: we can't fully forget about x(t), because of slight de-phasing; however, what I said holds as a sort of approximation that moves us closer.
This isn't an answer, but rather a more thorough examination of the question. The answerer to a prior iteration of this question (and presumably other persons) seemed to find, in essence, that this problem is overly arbitrary. I hope to show why this is not the case, and to add greater insight.
As we all know by now (that is all who have been following these posts) $x, y=\left(\frac{\sin(t)}{f(t)}\right), f(t)$ meets and is tangent to the hyperbola(s) $x·y=±1$.
Consequently, we can really forget about $x(t)$ for this problem as it's given by $y(t)$. Indeed, because of this relation, we can forget about the hyperbola(s), too, and think about the problem in terms of a wave of the form $x, y=\sin(t), f(t)$. All that matters is $f(t)$.
Now, this problem is probably easier to solve in 2-D; but it's probably going to be easier to make sense of it in 3-D.
To visualize what's meant to be happening, lets imagine a helix given by $x, y, z=\sin(t), \cos(t), f(t)$, where $f(t)$ is the function I'm looking for. ($f(t)$ will presumably take the form $µ^{t·ζ}$ where $µ$ is some positive, real number and $ζ$ is some real number. Or some modification thereof, obviously keeping in mind that it should retain a starting point for positive and negative numbers at $(0, 1)$ in 2-D.)
Alright, the equations for this helix should allow us to do the following:
We'll enter $2·π·φ^{-1}$ ($φ$ being $1.618..$) into these equations to get a point. Next we'll draw an imaginary line down from this point until it touches the closest lower loop.
Aright, now we can depart from any kind of formal thought, imagine this situation in the real world. Imagine moving this imaginary 'line' up the helix: It will lengthen gradually until it is its original length times $φ$ at $2·(2·π·φ^{-1})$. And so on in the same way for every multiple.
So, returning to the actual equations I'm looking for; this $z(t)$ or in 2-D $y(t)$ will allow us to create a triple $φ$ curve: The distance between loops grows by $φ$ every time the angle is a multiple of the $φ$ angle, and the whole thing moves closer and closer to the $y$ axis in the same way.
Now, note that the original equations, the ones in the update linked above, do the same exact thing (when converted to 3-D), they grow the distance between loops, only they do this via $π/2$.
I hope this yielded some insight.