Some algorithms studied by computational origami does not have an implementation. JOrigami solves the famous and funny fold-and-cut using Demaine's (et al) disk packing implementation:

http://theory.lcs.mit.edu/~edemaine/foldcut/

There is also another solution for this problem, using straigth skeletons and other computational geometry algorithms and data structures.

JOrigami plans to implement Lang's Three method for origami design.

http://www.langorigami.com/