Elliptical Billiard Configuration Space, 3 bounces

The x axis is t, the location on the ellipse P[t] = {1.5 Cos[t], Sin[t]}, and the y axis is s, the angle with the CCW tangent to the curve. Each (dashed black, red, green, blue) curve represents the map for the point shown on the ellipse after 0, 1, 2, and 3 bounces. The (lower) intersection of the blue curve with the black curve is shown, and it represents a particular injection angle which forms a 3-link closed path.