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Focusing on Nephroids

By Mark Levi

Figure 1. Imperfect focusing creates a caustic.
The bright strip in Figure 1 is illuminated by the incoming parallel beam reflected from the inner surface of the cup. The rays do not focus at a single point, as they would if the wall were parabolic. Instead, they “focus” at a curve, with the cusp at the focus of the osculating parabola.

Figure 2 sheds some light on the situation: the density of reflected rays spikes at the envelope, referred to as the caustic. This explains the caustic’s brightness.

Remarkably, this caustic is an epicycloid — more precisely, the path of a particle on the rim of the wheel rolling without sliding on another wheel, with a \(1:2\) ratio of radii (see Figure 3). Because of its vaguely kidney-like shape, this epicycloid is referred to as a nephroid (kidney\(=\nu \varepsilon \varphi \rho o\)).

Figure 2. The density of reflected rays spikes at the caustic. This caustic is half of the nephroid in Figure 3.
The precise statement is the following. For every ray from a pencil of parallel rays striking the inside of a circular mirror of radius \(R\) the reflected ray is tangent to the nephroid generated by rolling a circle of radius \(R/4\) on the stationary circle of radius \(R/2\) concentric with the mirror. The cusps of this nephroid lie on the ray passing through the center of the mirror.

To restate this result, imagine walking around the circle with constant angular velocity \(1\) and twirling a baton in the horizontal plane with angular velocity \(2\). The envelope of the resulting family of lines is a nephroid.

 

Figure 3. The nephroid: an epicycloid with a 1:2 ratio of radii.

Proof of the Claim

Consider the nephroid generated by rolling the circle, as seen in Figures 3 and 4; \(T\) is the point tracing out the nephroid. We must prove that \(PT\) is the reflected ray, i.e., that

\[\angle APO = \angle OPT, \tag1 \]

and that \(PT\) is tangent to the nephroid.

Note that \(\textrm{arclength}(CS)=\textrm{arclength}(CT)\) due to the non-slip condition, so that

Figure 4. Proving the tangency of reflected rays to the nephroid, which is traced by point T on the rolling wheel. The starting position of T is S.
\[\angle COS=\frac{1}{2} \angle CQT,\]

because the radii are in the \(1:2\) ratio. In turn, \(\angle CQT=2\angle CPT\), since these angles subtend the same arc. This proves \((1)\).

Why is \(TP\) tangent to the nephroid? As the smaller circle rolls on the larger one, the velocity \({\bf v}_T \perp CT,\) because \(C\) is the rolling wheel’s instantaneous center of rotation. But \(TP\perp CT,\) since \(CP\) is a diameter. Thus, \({\bf v}_t \parallel TP\); \(TP\) is indeed tangent. \(\square\)

Figure 5. The cardioid—an epicycloid with 1:1 ratio—is the caustic created by the source of light lying on the circular mirror.
On a related note, if the source of light lies on the circle, then the resulting caustic is a cardioid, i.e., the epicycloid generated by the circle rolling on another circle of equal radius (see Figure 5). This proof is the same as the one above.

To conclude, here is a small challenge: show that the length of the thick line in Figure 5 is independent of the choice of \(B\), namely

\[AB+BT+TC = \frac{8}{3}R, \tag2 \]

\(R\) being the radius of the mirror. Assuming that \((2)\) holds, setting \(B=A\) causes the first two terms to vanish, and we conclude that the cardioid’s length is \(16R/3\).

The figures in this article were provided by the author.

Mark Levi (levi@math.psu.edu) is a professor of mathematics at the Pennsylvania State University.

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