Billiards nearby the 45-45-90 triangle

This popup enumerates all periodic billiard paths in isosceles triangles which vanish at the 45-45-90 triangle. Here "vanish" means billiard paths that exist in an open interval of isosceles triangles with the 45-45-90 as an end point, but do not exist in the 45-45-90 triangle itself.

First, choose whether you wish to list billiard paths in the acute or obtuse triangles.

Then, you choose a slope. This should be a reduced rational number which must be a ratio of two odd numbers. Here, you are choosing a direction to flow on the square torus. It determines the Hausdorff limit of the billiard paths in the 45-45-90 triangle.

Then, choose an "infinitesimal slope." This should be a reduced rational number between zero and one, whose denominator is odd. In the obtuse case, the billiard path will be stable if and only if the numerator is even. This corresponds to a direction on the "geometric derivative." In this case, the geometric derivative is covered by a infinite genus surface with the lattice property.

Finally, there is a cylinder decomposition of this infinite genus surface in your chosen direction. The cylinders are naturally numbered by the positive integers. You can choose which cylinders to list.

All billiard paths in acute and obtuse isosceles triangles which vanish at the 45-45-90 triangle arise in this way. The resulting billiard paths are uniquely determined by a slope p/q so that 0

, an infinitesimal slope r/s with 0, and a cylinder choice of n>=1.