# Further Upon Reflection: Caustics in a Ball

**Figure 1.**A circular mirror “focuses” the incoming beam on a caustic.

*caustics*.

Interestingly, the exact same caustic is the locus of the sun’s reflections from the outside surface of the same circular mirror (see Figure 2). The caustic in Figure 2 is the envelope of (the lines defined by) reflected rays. Since the same words describe the caustic in Figure 1, the two caustics are indeed identical; Figure 3 depicts the outer and inner reflections side-by-side, illustrating the identity of the two caustics in one picture.

**Figure 2.**As the observer walks around the circular mirror, the sun’s reflections slide along the caustic.

**Figure 3.**The locus of the sun’s images in a circular mirror (A) coincides with the caustic on the bottom of a coffee cup (B).

**Figure 4.**The nephroid is a hypercycloid traced by a point of a wheel rolling on a circle, with the 1:2 ratio of the radii.

*every reflection in any shiny mirror lies on a caustic, i.e., on the envelope of reflected rays*. This holds true for a mirror of any shape, not only a circle.

For the circular mirrors in Figures 1 and 2, the caustic happens to be a nephroid, i.e., a hypercycloid traced by a point on the rolling wheel, as shown in Figure 4. I provided a short proof of this fact in my April column.

The figures in this article were provided by the author.