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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!