Determining the Caustic Generated by a Cylindrical Cup Gage Bonner and James Alban MacDonald Queen’s University Department of Mathematics and Statistics Abstract We show analytically that the caustic generated by a cylindrical cup of radius R is a nephroid generated by a circle of radius R/4. Furthermore, it is shown that said nephroid illuminates exactly 1/8 of the cup. Introduction In any ordinary cylindrical cup with a sufficiently reflective interior, it can be seen that a single overhead light source generates a bright heart-shaped curve at the bottom of the cup when it is tilted at various angles. It is known that parallel lines incident on a semi-circular reflective surface generate a nephroid; it thus remains to be determined how the consideration of the three-dimensional case will affect this result. Model We shall assume that the light source is essentially an infinite distance away so that the incident rays are parallel and that the light rays obey all the standard classical optics conventions. Figures 1 and 2 show the overhead respectively the side (in the plane of the reflected ray) view of the light rays, where R is the radius of the cup, φ is the incident angle in the horizontal plane, θ is the angle between the incident ray and the vertical, L is the planar length of the reflected ray and h is the height of an arbitrary ray. Figure 1: The overhead view of the horizontal plane in which the light’s collision with the cup occurs. 1