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Billiards are semi-focusing if their boundary components consist of pieces of cylinders and possibly some flat components.
In words, cylindric billiards are toric billiards where the scatterers are cylinders.
Uniform estimates on the number of collisions in semidispersing billiards.
We need to recall the first inputs on rational polygonal billiards.
Here we start the study of a wider class of billiards called projective billiards, free from this shortcoming.
Projective billiards include the usual billiards along with the dual, or outer, billiards.
In this section we summarize some basic properties of semi-dispersing billiards.
Estimates like that play an important role in various questions about billiards.
The motivation for these definitions, in the realm of billiards, is the following.
Our proof can be readily extended to a broader class of cylindrical billiards, and very likely to other highdimensional billiards (see 5).
All three papers address various problems in the theory of semi-dispersing billiards using the geometric approach outlined above.
To obtain our results, we will re-prove in a more general setting many technical results established earlier for semi-dispersing hyperbolic billiards.
Note that for cylindrical semi-focusing billiards, the same happens for consecutive collisions with the flat boundary components.
These curves form a large family of focusing curves that can be used to construct hyperbolic billiards.
Next we turn our attention to projective billiards in a circle.
