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 !
- 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.