Given a circular billiards table with center O,
radius r, and no pockets. A cue ball (with a
radius negligible compared to r) is placed at
point A such that 0<|OA|≤r. The ball is struck
and bounces off the cushion twice before
returning to point A.

How was the direction of the shot determined?

From wherever the cue ball lies it will clearly return to its rest twice if its path is directly over the the central point and struck firmly enough.  The points of contact will be at tangents which are parallel to the diameter along the path if its travel.


            t1
            |\
            | \ 
      |     |   \b     .
   C |     |   /       o
            |  /
            |/
            t2
In the above diagram the red lines represent the circumference.  t1 and t2 are points on tangents and form a chord perpendicular to the diameter on which the ball lies and is between the ball and the circumference. The path so formed does not encompass the centre of the table.  The location of t1 and t2 will vary in accord with the position of b but:
the perpendicular to the tangents will pass through the incentre of the triangle formed by t1, t2 and b.
The ball will only pass over its initial spot once.

Edited on April 23, 2013, 10:48 pm

Posted by brianjn on 2013-04-23 21:17:58