Some sliders could be put on the selection boxes.
Also, try negative for curvature and/or alpha. (s is forced to be positive. wonder what a negative distance would do?)
A better script would select the start point, and a direction,
and the end point and a direction, and use LM to create the curve.
I guess javascript would be very slow (?) for interactive use, but C++ would be fast enough?
(I'm just an amateur programmer, but I gather that a Jacobian is just a matrix of derivatives.) so some matrix
routines would be needed...

Maybe the algorithm would work for a better 2 point catenary script also?
Brian.