Reference request: a differential equation arising in geometry

Question

$$ \frac{d\beta}{d\alpha} = \frac {\sin\beta}{\sin\alpha} $$ In what contexts (if any) is this equation known to occur?

Answer 1

This isn't a perfect match, as you can see, but since no one else has made a contribution here I thought you might be interested.

I'm a physics professor, and the first thing that came to my mind when I looked at your problem was Snell's Law of Refraction. It describes the refraction of light at the interface of two substances with different indices of refraction, $n_1$ and $n_2$. The angle $\theta_1$ is the angle of incidence (deviation from perpendicular to the surface of the interface) within substance 1 and $\theta_2$ is the angle of refraction within substance 2.
$$ \frac{n_1}{n_2}=\frac{\sin\theta_1}{\sin\theta_2} $$ Modern optical cables use a medium whose index of refraction is low in the middle of the cable and higher as the radius increases, redirecting laser light toward the center of the fiber as the fiber bends and twists. A continually changing index of refraction COULD (and I emphasize "could" because I haven't actually seen this application) be described by your equation. From an engineering standpoint it may be more commercially viable to use radially layered media with definite boundaries instead of a smooth media with a continually changing index, but if one were to use the latter then an equation similar to the one you describe could play an important role.

Answer 2

Here's my own partial answer. Each point on the Villarceau circle on a torus embedded in $\mathbb R^3$ in the usual way has a longitude $\alpha$ and a latituted $\beta$, and as one moves along the curve, the equation $\dfrac{d\alpha} \alpha = \dfrac{d\beta} \beta$ holds.