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Teichmuller explorer

The applet below lets you explore the Teichmuller space of a compact surface of genus two, describing hyperbolic metrics by exhibiting a hyperbolic octagon in the Poincaré disk model. The octagon is chosen to be symmetric about the origin, with two vertices along te real axis. There are then three natural complex parameters, the position of the three of the other vertices (they will always be chosen on the lower half of the disk).

You can adjust the position of the three vertices that appear as small blue disks, and the applet will try to adjust the remaining vertices to keep the area of the polygon constant (the area should be 4π according to Gauss-Bonnet). If it manages to perform this adjustment, it will draw part of the tiling of the hyperbolic plane.