Finding points along a catenary curve As I am no mathematician, I have been struggling to find an equation to accurately predict points spaced along a curve separated by distance d. Given two points, assume a string with a length equal to 125% of the distance between the two points.  If the string is "hooked" at its ends to the two points, what is the equation to find a third point p at some distance along the downward arc created by the hanging string?
https://en.wikipedia.org/wiki/Catenary
 A: Call your known points $(0, y_{1})$ and $(x_{2}, y_{2})$. There exist constants $a$, $b$, and $c$ such that the desired catenary has equation
$$
y = c + a\cosh \frac{x - b}{a}.
\tag{1}
$$
(This is the standard catenary of Wikipedia translated horizontally by $b$ and vertically by $c$.)

The arc length element of the graph is easily checked to be
$$
ds = \cosh \frac{x - b}{a}\, dx.
\tag{2}
$$
The arc length from $(0, y_{1})$ to a general point $(x, y)$ with $0 \leq x \leq x_{2}$ is therefore
$$
\ell(x)
  = \int_{0}^{x} \cosh \frac{t - b}{a}\, dt
  = a\left[\sinh \frac{x - b}{a} + \sinh \frac{b}{a}\right].
\tag{3}
$$
Since the points $(0, y_{1})$ and $(x_{2}, y_{2})$ lie on the curve (1), you have
$$
\left.
\begin{aligned}
y_{1} &= c + a \cosh \frac{b}{a}, \\
y_{2} &= c + a \cosh \frac{x_{2} - b}{a}.
\end{aligned}
\right\}
\tag{4}
$$
Since the arc length of the catenary is $125$% of the straight-line distance between the points $(0, y_{1})$ and $(x_{2}, y_{2})$, (3) gives
$$
1.25\sqrt{x_{2}^{2} + (y_{2} - y_{1})^{2}}
  = a\left[\sinh \frac{x_{2} - b}{a} + \sinh \frac{b}{a}\right].
\tag{5}
$$
Equations (4) and (5) allow you to express $a$ and $b$ in terms of $x_{2}$, $y_{1}$, and $y_{2}$ (in principle; haven't tried to solve analytically in practice). Equation (3), which is easily solved for $x$ in terms of the distance to the point $p = (x, y)$ then gives the desired information.
A: The difficult part
of fitting a catenary
of a specified length
between two specified points
is solving
$y = \dfrac{\sinh(x)}{x}
$
for $x$
in terms of $y$.
This can only be done numerically;
there is no analytic solution.
Here is question
with answers
that tell you how:
Solving $\sinh(ax) = bx$
