  We perform fractal analysis on a closed classical hard-wall billiard, the
circular billiard with a straight cut, assuming there are two openings on the
boundary. We use a two-dimensional set of initial conditions that produce all
possible trajectories of a particle injected from one opening, and numerically
compute the fractal dimension of singular points of a function that maps an
initial condition to the number of collisions with the wall before the exit. We
introduce a simplified box-counting algorithm, which uses points from a
rectangular grid inside the two-dimensional set of the initial conditions, to
simplify the calculation, and observe the classical chaotic properties while
varying the parameters of the billiard.
