Outer billiard

Outer billiards is a dynamical system based on a convex shape in the plane. Classically, this system is defined for the Euclidean plane but one can also consider the system in the hyperbolic plane or in other spaces that suitably generalize the plane. Outer billiards differs from a usual dynamical billiard in that it deals with a discrete sequence of moves outside the shape rather than inside of it.

Let P be a convex shape in the plane. Given a point x0 outside P, there is typically a unique point x1 (also outside P) so that the line segment connecting x0 to x1 is tangent to P at its midpoint and a person walking from x0 to x1 would see P on the right. (See Figure.) The map F: x0 -> x1 is called the outer billiards map.

The inverse (or backwards) outer billiards map is also defined, as the map x1 -> x0. One gets the inverse map simply by replacing the word right by the word left in the definition given above. The figure shows the situation in the Euclidean plane, but the definition in the hyperbolic plane is essentially the same.

An outer billiards orbit is the set of all iterations of the point, namely ... x0 <--> x1 <--> x2 <--> x3 ... That is, start at x0 and iteratively apply both the outer billiards map and the backwards outer billiards map. When P is a strictly convex shape, such as an ellipse, every point in the exterior of P has a well defined orbit. When P is a polygon, some points might not have well-defined orbits, on account of the potential ambiguity of choosing the midpoint of the relevant tangent line. Nevertheless, in the polygonal case, almost every point has a well-defined orbit.

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