Note: this is an Unofficial script, by an amateur programmer...
The script works pretty good. It needs an adjustment to "aGuess" if the two points are close together, "relative" to the arclength S.
Values of .1 to .5 seem good in that situation. Some additional code to assist NewtonRaphson may be in order.

Here is an updated script which will Also accept points with different z values.
Gravity is still in the -y direction. The two points cannot be vertically one above the other.

Also attached the rambling PDF about catenaries, which has been edited numerous times, but may still have errors...

Just added in a coordinate frame, using previous explainations of Michael, to temporarily rotate the x-axis about the new parallel y-axis, which effectively converts
the z coordinates to zero, so that the prior Catenary2PtXY code is re-used.

Changing the direction of gravity, or finding S given "a," have not been done.