Projective billiards

Projective billiards are bounded domains whose boundary is endowed with a field of transverse lines called projective lines (see here, here or here for more details).

The following interactive animation shows polygonal examples of projective billiards, in which

• boundary = black lines;
• projective lines = dotted lines;
• trajectory = black and green lines.

Please click and move the points you want !

You can:

• increase or decrease the number of reflections: arrows UP/DOWN;
• change the polygon's number of vertices: arrows LEFT/RIGHT;
• change the type of projective lines: SPACE BAR;
• constrain the vertices' moves to the great diagonal: C.

You can observe the following properties (which are proved here):

PROP. 1: The trajectories of the projective billiard called right-spherical are periodic of period 3 in the case of a triangle.

PROP. 2: The trajectories of the projective billiards called centrally polygonal are:

• periodic of period k, if the polygon is regular with an even number k of vertices;
• periodic of period 2k, if the polygon has an odd number k of vertices.