I'm struggling trying to convert this movement behavior into an equation.
For future reference, this is as a result of searching for a solution for a previous question of mine.
Thank you @eyeballfrog for helping me reformulate my question.
Setup
Starting with the most basic scenario for this problem, let $A$, $C$, $D$ & $F$ be unique arbitrary points, where $AC$ & $DF$, each represent a line segment with a moving point along them, being $B$ & $E$ respectively. So far, $\frac{AB}{BC} = \frac{DE}{EF} = r$.
Now, since $AC < DF$, we define a new point $G$, such that $AB = DG$. Hypothetically, if $AC > DF$, we'd have $DE = AG$.
At this point, by defining the $X_i$ sequence of points as $GX_{i+1}/X_{i+1}X_i = r; X_0 = E$, we could generate an arbitrary number of new points; for this simple case, $0 <= i <= 2$.
Last, but not least, $AC$ & $DF$ are parallel between each other, and $AC \ne DF$.
Question(s)
Given the above, the question is: Would it be possible to obtain a $f_i(x)$, which accepts as input a value in $[0, 1]$ and returns $x(X_i)$ -- The $x$ coordinate of $X_i$?
And moreover: Would there be a way to expand such reasoning to allow for $N > 2$ line segments, such that $h_{ij}(x) = x(X_{ij})$
Simple static diagram
Where the percentage slider represents $100r$.
Motivation
The main idea behind this, as stated in my previous question, it's to move two points relatively, along their own line segment, attempting to keep both points reasonably close/aligned each other, despite difference in lengths, which, due to the relative nature of the movement, ends up creating a difference in pace for every point, consequently shifting points away.
Additionally, all points must start & finish at the same time.
For the sake of completeness, and as a proof of concept, here's how it looks when using $4$ -- $j$ segments, at $9$ different «softness» -- $i$ levels, including «stock» level, $[0, 1, 2, 4, 8, 16, 32, 64, 101]$, from right to left (or from red moving point to yellow point).
Note how the shortest line segment doesn't have more than 3 points: That's because every other line segment pivot depends on its moving point; since it doesn't depend on any other segment, there's no need for more than 3 points on it.
After all, the ultimate objective of all this, its to pick a single «softness» level -- color, and discard the rest.
Extended dynamic behavior
Say that $AC$ is of unit length, $DF$ is of length $d$, and $AB = r$. Then along $DF$, $DG = r$, $DE = r d$, and $GE = r(d-1)$. The next point $X_1$ is at a ratio $r$ of the distance between $G$ and $E$, so $DX_1 = DG + r\cdot GE = r + r^2(d-1)$. Similarly, $DX_2 = DG + r\cdot GX_1 = r + r^3(d-1)$. At this point you've probably spotted that this is a geometric series: $DX_n = r + r^{n+1}(d-1) = r[1+r^n(d-1)]$.