15 hours and four up-votes but not a word from anybody. That's the result of [this question](https://math.stackexchange.com/questions/778010/reference-request-a-differential-equation-arising-in-geometry) to stackexchange.


My question is where the following differential equation arises naturally and where it appears in the literature:
$$
\frac{d\alpha}{\sin\alpha} = \frac{d\beta}{\sin\beta}. \tag 1
$$


[Villarceau's theorem](http://en.wikipedia.org/wiki/Villarceau_circles) says that a plane bitangent to a torus (embedded in the usual way in $\mathbb R^3$) intersects the torus in two circles. I have found that as a point with **longitude** $\alpha$ and **latitude** $\beta$ moves along one of the Villarceau circles, equation $(1)$ holds. One could say Villarceau's circles are solutions to this differential equation. The equation is invariant under permutation of the two variables. I have found the geometric meaning of that symmetry.


**So I'm wondering whether** this equation has appeared in print or folklore in the context either of that bit of geometry or of anything else?


**PS:** The solution to $(1)$ can be defined implicitly by $$\tan\frac\beta2=m\tan\frac\alpha2$$ or by
$$
\cos\beta=\frac{\cos\alpha+n}{n\cos\alpha+1}.
$$



Reference request: a differential equation in elementary geometry
Let us consider two light rays of the same frequency in the $x$-$y$ plane forming the angles $\alpha$ and $\alpha+d\alpha$ with the axis $x$. The four-momenta of the corresponding photons are
$$p\_1=\left (\frac{E}{c}, \frac{E}{c}\cos{\alpha}, \frac{E}{c}\sin{\alpha}
\right ),\;\;\;\mathrm{and}\;\;\; 
p\_2=\left (\frac{E}{c}, \frac{E}{c}\cos{(\alpha+d\alpha)}, 
\frac{E}{c}\sin{(\alpha+d\alpha)}\right ).$$
From the invariance of the scalar product
$$p\_1\cdot p\_2=\frac{E^2}{c^2}(1-\cos{(d\alpha)})\approx \frac{1}{2}
\frac{E^2}{c^2}(d\alpha)^2,$$
we get $$E\,d\alpha=E^\prime\,d\alpha^\prime, \tag 1$$ where the primed quantities are measured in the inertial frame $S^\prime$ moving with some velocity $V$ along the $x$-axis. On the other hand, transverse components of the 3-momentum is not changed under the Lorentz transformations and, therefore, $$\frac{E}{c}\sin{\alpha}=\frac{E^\prime}{c}\sin{\alpha^\prime}.\tag 2$$ Combining (1) and (2), we get
$$\frac{d\alpha}{\sin{\alpha}}=\frac{d\alpha^\prime}{\sin{\alpha^\prime}}. \tag 3$$ Hence this differential equation describes the (relativistic) aberration of light. An integration constant $n$ in its solution $$\cos{\alpha}=\frac{\cos{\alpha^\prime}+n}{1+n\,\cos{\alpha^\prime}}\tag 4$$ can be fixed in the following way. Let $\alpha^\prime=\pi/2$, then it is evident that the $x$-component of the photon's velocity $c\_x=c\,\cos{\alpha}$ in the original "stationary" frame equals to the $S^\prime$ frames velocity $V$. on the other hand, from (4) in this case we have $\cos{\alpha}=n$. Therefore, $c\,n=V$ and $n=V/c$. 


P.S. The relation with the spherical trigonometry (see William Chauvenet's book mentioned in the comments) can be established, I think, in light of Arnold Sommerfeld's this old idea: [http://en.wikisource.org/wiki/Translation:On\_the\_Composition\_of\_Velocities\_in\_the\_Theory\_of\_Relativity](http://en.wikisource.org/wiki/Translation%3aOn_the_Composition_of_Velocities_in_the_Theory_of_Relativity) (On the Composition of Velocities in the Theory of Relativity). Of course, in fact, the real geometry of Einstein's velocity addition is the hyperbolic (Lobachevsky-Boljai) geometry - "the imaginary counter-image of the spherical geometry": [http://en.wikisource.org/wiki/Translation:On\_the\_Non-Euclidean\_Interpretation\_of\_the\_Theory\_of\_Relativity](http://en.wikisource.org/wiki/Translation%3aOn_the_Non-Euclidean_Interpretation_of_the_Theory_of_Relativity) (On the Non-Euclidean Interpretation of the Theory of Relativity, by Vladimir Varicak, 1912).


