Catenary equations solving for sag given distance and cable length I know the distance between points, and the length of the cable between them, how can I calculate the sag?
How to solve this eqn for x = f(s,d)? since I know both s and d.
s = cable length = 2 * x * sinh ( d / ( 2 * x ))
 I tried the free versions of wolfram alpha and various other equation simplifiers, but they wont do this one.  I wouldn't mind paying, if I knew the paid version would give me the answer.
From https://www.easycalculation.com/analytical/cable-sag-error.php
I find these equations below:  I have been clumsily using excel solver to find the x that gives my known value of cable length based on distance, which then determines sag in one step.
q = w * g,
x = n / q = angle between x axis and cable sector,  ??? what is this ???
h = cable sag = x ( cosh ( d / ( 2* x )) - 1),
s = cable length = 2 * x * sinh ( d / ( 2* x )) is a function of d (distance)
Where,
q = cable weight per unit length,
w = cable mass per unit length,
g = force perpendicular to cable length,
n = cable tension,
d = straight line distance,
 A: As Wikipedia explains, the equation of a catenary in Cartesian $x$-$y$ coordinates is
$$y(x)=a\cosh{\frac{x}{a}}$$
where $a$ is some constant.
The arc length along a curve in the $x$-$y$ plane can be calculated by integrating $ds$, where
$$ds^2=dx^2+dy^2.$$
Since
$$dy=\sinh{\frac{x}{a}}\,dx,$$
we have
$$ds^2=\left(1+\sinh^2{\frac{x}{a}}\right)dx^2=\cosh^2{\frac{x}{a}}\,dx^2$$
or
$$ds=\cosh{\frac{x}{a}}\,dx.$$
Integrating $ds$ from $x_1$ to $x_2$, we find that the length is
$$s=\int_{x_1}^{x_2}ds=\int_{x_1}^{x_2}\cosh{\frac{x}{a}}\,dx=a\sinh{\frac{x_2}{a}}-a\sinh{\frac{x_1}{a}}.$$
Take the catenary to be suspended from level supports at $x_1=-d/2$ and $x_2=d/2$, so that the distance between the supports is $d$, and take the length of the catenary between the supports to be $L$. Then
$$L=2a\sinh{\frac{d}{2a}}.$$
If you know $d$ and $L$, this equation determines $a$. But this equation can only be solved for $a$ numerically, not analytically.
Once you have solved numerically for the value of $a$, the sag can then be calculated as
$$\Delta y=y(d/2)-y(0)=a\left(\cosh\frac{d}{2a}-1\right).$$
As a numerical example, let $d=1.0$ and $L=2.0$. Using Mathematica's NSolve function, we find that $a=0.22964$. Then $\Delta y=0.79639$.
Excel has an add-in called Solver which you can use to solve for $a$.
A: Have a look at Atlantic Electronic Journal of Mathematics. THE HANGING CABLE PROBLEM FOR PRACTICAL APPLICATIONS. Chatterjee & Nita
http://euclid.trentu.ca/aejm/V4N1/Chatterjee.V4N1.pdf
Abstract:
"We investigate the `hanging cable' problem for practical applica-
tions. We focus on determining the minimum distance between two vertical
poles which will prevent a cable, hanging from the top of these poles, to touch
the ground. We consider two set-ups, starting with the case of equal poles then
generalizing to unequal poles. In both cases we assume that the only known
quantities are the heights of the poles and the length of the cable."
