This script for the CAD program Moment of Inspiration, (Moi), creates Elastica curves on the surface of a unit sphere. The curves look very similar to the planar elastica curves, but wrap around the sphere. It is possible for the curves to be closed, under certain circumstances. A regular sphere in 3 dimension R3 space, is referred to as S2. The sphere is 3 dimensional, but in some respects, the surface of the sphere seems to have 2 dimensions. The Gaussian curvature of the unit sphere is +1.
The primary reference paper for this script is: "Lectures on Elastic Curves and Rods," by Dr. David A Singer: http:www.case.edu/artsci/math/singer/publish/elaslecs.pdf
A Sabban frame was used.

See the pdf for more information.

I've written about 5 versions, but finally got this faster version to work.

I would recommend 1000 to 3000 points, which takes a few seconds to draw. And a rebuild with tolerance .001.






Reference paper #3, by J. Langer & D. Singer, is: "The total squared curvature of closed curves": http:projecteuclid.org/euclid.jdg/1214438990
Only the first 5 of 22 pages could be downloaded. The rest of the paper could not be read because Euclid has access restricted to institutional subscribers of that journal.
If someone has access, I would like to read the paper.

- Brian

Edit 12/19/14 Version3 posted. It includes "figure8" curve, looping waves, and incorporates Join and Rail and Orbit option boxes, and is relatively fast)
Uploaded new draft .pdf
Edit 12/20/14, Version3b posted. Corrected case of wavelike and pSquared = .5, which caused crashes.