Direct floating-point billiard: the examples of the two source papers
(300 random starts on the large arc; period = A1 + A2 + segments; 'open' = not closed
 after 20000 reflections)
  (1/3, 1/4):  7 = 3+4+0: 158 starts;  12 = 6+4+2: 142 starts
  (1/4, 1/6):  5 = 2+3+0: 193 starts;  6 = 4+0+2: 107 starts
  (1/3, 1/5):  13 = 6+5+2: 127 starts;  21 = 9+10+2: 173 starts
  (1/4, 1/sqrt(30)):  6 = 4+0+2: 77 starts;  open: 223 starts
  ((5-sqrt5)/10, sqrt5/10):  4 = 2+2+0: 199 starts;  14 = 6+6+2: 101 starts
  (5/11+1/(22pi), 5/11-1/(220pi)):  open: 299 starts
