TITLE: Interactive model of the hyperbolic plane for a general public lecture
QUESTION [48 upvotes]: The following is not quite a research level question, but I still find this site appropriate for asking it. I hope I get it right here.
I am preparing a talk for a general public and I want to discuss some hyperbolic geometry. I wish I had a good illustration device. I imagine a dynamical version of one of Escher's tessellations of the Poincare model (e.g the one in How might M.C. Escher have designed his patterns?) which changes isometrically when I slide the computer mouse.
Question: Could you please make any recommendation regarding any device of a nature similar to the one I describe above?
In fact, I will be happy to have whatever interactive model of whatever geometry, not necessarily hyperbolic.
Subquestion: if you're kind enough to make a recommendation, could you also advice regarding copyright issues (if applicable)? 
Sidequestion: Any other recommendation regarding presentation of geometry will be appreciated. Please note that my concern is more about the quality of the presentation than the actual mathematical content...

UPDATE: Thank you! I am thrilled to get in less than 24 hours so many excellent answers and comments. Fortunately, Arnaud Chéritat provided EXACTLY what I asked for, and I happily accept his answer. Indeed, I am going to use his tool for my presentation. However, there are other excellent tools here which could be useful elsewhere. It seems to me a good idea to keep collecting those and MOF is an excellent platform for that.
I suppose one should post one of these big-list questions which has a broader scope than this one (but not too broad), something like "Visualizing tools for lectures on geometry". I am not sure how this is done, so you can pick up the glove!

REPLY [2 votes]: I've used my program MagicTile during a few presentations, and it has many capabilities that make it a nice choice.

Supports all regular 2-dimensional tilings across the 3 geometries (spherical, Euclidean, and hyperbolic).
Many models: Poincare, Klein, Upper Half Plane, Orthographic, and their analogues in spherical geometry (Stereographic, Gnomonic, etc.), and some less common models...
All these models are dynamically pannable. I've found this a nice way to demonstrate classes of conformal Möbius transformations. 
Spherical/Euclidean geometries also include quotient surface views, including the non-orientable Klein bottle and Boy's surface.  
It is a very colorful program (as it is centered around analogues of Rubik's cube), which could heighten public interest.
It is completely free, in active development since 2009, and the opensource code has the most permissive GitHub license.

One downside is that it is a Windows program, though some have it running on Macs as well.

This would also be a good place to recommend Henry Segerman and Saul Schleimer's physical models, which they have proved work well for public lectures.  For an example presentation, see their YouTube video Illuminating hyperbolic geometry, which has links to purchase the models.