The latest version 3 is posted in post #1. The .js and .htm files are named _ElasticaSpherical, without the 3.

This version is fast, but 3000 points takes a few seconds to form the curve.
The "missing" elastica cases are included, which include "figure8" and some "wavelike" curves which loop back on themselves. "Pleats" can be more curvy.
Corrected an error in the orbitlike formula.
The borderline case can also be formed, but cycles should be made fractional. It is sort of a circle with a "90 degree" loop on one side.
No crashes have been encountered, so the script should cover all of the cases.
Negative lambda can occur in some cases, which tends to make the curves smaller in size.

Curves with lambda less than 8/7 may be made to close.
Curves with lambda greater than 8/7 are not supposed to close with "perfection", but they can be made to close, and be very nearly uniformly symmetrical.
The last loop, with the join does not quite equal the first curve, as may be seen by increasing the cycle number, or doing rotation, and seeing slight displacement.

I've rewritten, and am still rewriting the .pdf documentation, including the curve Cases, and more sample closed curves.
Edit: uploaded Dec. 19 draft of .pdf, as the old one was so out of date. Has parameters for a "figure8" and a "looping wave."

- Brian