Finding the catenary curve through three given points I have been searching for this, but all I found where solutions, in which the arch length is given, or the sack at the lowest point. I tried also Wolfram Alpha, but it doesn't come up with anything useful.
This is the problem: Find the equation for the catenary that goes through three points. It can be assumed that $x_1 < x_3 < x_2$ and $y_1 <= y_2$.

Here is what I got so far. We can shift the curve so that $P_1$ has coordinates $(0, 0)$. This gives the equations:
$$
0 = a \cosh\left(\frac{-x_0}{a}\right) + y_0 \\
y_2 = a \cosh\left(\frac{x_2-x_0}{a}\right) + y_0 \\
y_3 = a \cosh\left(\frac{x_3-x_0}{a}\right) + y_0
$$
Its easy to substitute $y_0$:
$$
\frac{y_2}{a} - \cosh\left(\frac{x_2-x_0}{a}\right) + \cosh\left(\frac{x_0}{a}\right) = 0\\
\frac{y_3}{a} - \cosh\left(\frac{x_3-x_0}{a}\right) + \cosh\left(\frac{x_0}{a}\right) = 0
$$
But then I am stuck. I know at some point I need to use some numerical method. Can this be simplified further?
From here I thought I could use some multi-dimensional root solver or formulate it as a minimization problem.
$$\min_{a, \,x_0}{ \left(\frac{y_2}{a} - \cosh\frac{x_2-x_0}{a} + \cosh\frac{x_0}{a}\right)^2 + 
\left(\frac{y_3}{a} - \cosh\frac{x_3-x_0}{a} + \cosh\frac{x_0}{a}\right)^2 }$$
This leaves the question of good start values and if this space is convex, which I honestly doubt.
 A: Since you can have some errors for the values of the $(x_i,y_i)$ and a single variable $a$, I should let the problem as it is and  consider the norm (to be minimized)
$$\Phi(a)=\Bigg[a \cosh\left(\frac{x_1-x_0}{a}\right) + (y_0-y_1) \Bigg]^2+
\Bigg[ a \cosh\left(\frac{x_2-x_0}{a}\right) + (y_0-y_2) \Bigg]^2+$$ $$\Bigg[a \cosh\left(\frac{x_3-x_0}{a}\right) + (y_0-y_3)\Bigg]^2$$ which corresponds to something like data validation or data reconciliation.  This is very similar to what you propose.
We can compute the derivative which, simplified, leads to the problem of finding the zero of the expression
$$\left(a \cosh \left(\frac{x_1-x_0}{a}\right)+(y_0-y_1)\right)
   \left((x_0-x_1) \sinh \left(\frac{x_1-x_0}{a}\right)+a \cosh
   \left(\frac{x_1-x_0}{a}\right)\right)+$$
$$\left(a \cosh \left(\frac{x_2-x_0}{a}\right)+(y_0-y_2)\right)
   \left((x_0-x_2) \sinh \left(\frac{x_2-x_0}{a}\right)+a \cosh
   \left(\frac{x_2-x_0}{a}\right)\right)+$$
$$\left(a \cosh \left(\frac{x_3-x_0}{a}\right)+(y_0-y_3)\right)
   \left((x_0-x_3) \sinh \left(\frac{x_3-x_0}{a}\right)+a \cosh
   \left(\frac{x_3-x_0}{a}\right)\right)$$
Any root finding algorithm will be fine.
