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Consider the strip 0 ≤ y ≤ . Its images have equal areas, and their interiors are disjoint. Hence there is an image of this narrow strip that intersects the upper boundary y = 1; let k be the least number of iterations needed. Let Pk be the point of the upper boundary of the k-th image of the -strip with the greatest y-coordinate, and let P0 = T˜−k (Pk ) be the corresponding point of the line y = 0. Set T˜i (P0 ) = Pi . Join P0 and P1 by a segment and consider its consecutive images under T˜ . They constitute a simple arc from P0 to Pk .

This chord is a 2-periodic trajectory, and their totality is the invariant curve α = π/2. What are other possible values of the constant angle α? fig. 42 Theorem 1. There exists a non-circular billiard table with an invariant curve α = const, 0 < 49 α < π/2, if and only if this constant angle α satisfies the equation tan nα = n tan α for some integer n > 1. We outline the proof, leaving aside some computations. Proof. Parametrize the billiard curve by the angle φ it makes with a fixed direction.

The lines that have the eigendirections of the second quadratic form of the surface of the ellipsoid. 1, one concludes that these lines are analogous to confocal ellipses and hyperbolas in the the following sense: the 30 sum (or difference) of distances from all points of a line of curvature to two umbilic points of the ellipsoid is constant (umbilic points are the points where the principal curvatures are equal ; they are the singularities of the foliation by the lines of curvature) – see [H-CV].

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Billiards by Tabachnikov S.


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