I'm no matematician, I hope my description of the problem with common words will be clear:
I wish to represent the relationship between volume ($x$) and pressure ($y$) inside a container with elastic walls with a function, rules are as follows:
- The container has a resting volume at wich pressure inside it should be 0 (gravity is ignored)
- The container also has max and min possible volumes, wich should be limits for wich the pressure tends to infinity.
- The container has a compliance gradient, which dictates how "fast" the function transitions from -inf to + inf.
Having no training in mathematics beyond high school I resorted to good old wikipedia plus try and error and came up this formula, which almost satisfies my needs:
$y\ =\log\left(\frac{-l_{1}\ +\ x}{l_{2}\ -\ x}\right)\ r\ +s$
I can use $l_{1}$ and $l_{2}$ to nicely dictate the min and max volume of the container, and $r$ for the rate of change. My problem is that in this form $s$ represents "the value of $y$ at $x = (l_{1} + l_{2})/2$", in other terms "the pressure at the midpoint between min and max volume" ; I whish instead it represented "the value of $x$ for which $y = 0$" which whould be "the resting volume of the container".
My intuition is that there should be a way to "rephrase" the function to give $s$ that meaning, because when I play with it in desmos I can see that $s$ does shift the point where the function crosses 0, but I have no idea how to translate that in numbers.