This webpage provides interactive graphics illustrating the "elliptic billiards" phenomena discussed in the exercises on page 354 of What is Mathematics? (Second Edition) by Courant, Robbins, and Stewart.
This exercise concerns reflections in an ellipse when the initial ray passes through one of the two foci. The ray is repeatedly reflected back and forth through alternate foci and approaches the major axis. In the graphic below, the ray always begins at the left focus. The eccentricity of the ellipse, the initial heading of the ray, and the number of reflections are all adjustable. The initial ray before reflection is shown in red; the reflected rays are shown in blue.
This exercise concerns reflections in an ellipse when the initial ray does not pass exactly through one of the two foci. If the ray passes between the two foci, all reflections do the same, and are all tangent to a confocal hyperbola. If the ray does not pass between the two foci, all reflections do the same, and are all tangent to a confocal ellipse. This is illustrated in the graphic below. The eccentricity of the ellipse, the initial point and heading of the ray, and the number of reflections are all adjustable. The initial ray before reflection is shown in red; the reflected rays are shown in blue.