Hi Tom,
Thank you for your question. Pilou is correct.
Hyperbolic helper only creates the uniform central hyperbolic polygon. It is up to the user to divide it up, if desired, using MoI commands
such as polyline to create a bunch of triangles. Or create outlines of Escher fishes...
The 3pt inversion script and the central inversion script do essentially the same thing, with different picks.
Using the 3pt inversion script, which starts out with much of the code of the 3 point circle tool of MoI, the two end points of one side of the polygon are picked,
and then a point on the side of the polygon in between is picked. A new circle pops up, the circle of inversion. The user then picks the center
of this circle. Then the line segments making up the central polygon are selected. They should have previously been separated.
The inversion of the selected central polygons causes them to be "replicated," (sort of mirrored), across from the side arc.

The polygons in the Poincare disk, projected onto half of a two sheeted hyperboloid which hovers above the disk. (rounded cone-like surface, sort of),
have straight lines, and are all the same size and have the same angles. They just look like they have arced sides, and look "distorted," lying flat on the
Poincare disk.

Computer programs to do such tessellations were done 30 years ago, but not with NURBS, and students are still doing projects.

I'm working on a version to do planar faces.
In the literature, there are also non-uniform tessellations, and Weierstrass projection and math, which can be relatively easy, or be written with "strange"
math symbols.

It was hard for me, but I am very happy to have the "fast" versions working, using previous help from Michael, and others. There are lots of comments
in the .js and .htm code, visible with notepad++

- Brian