# 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.