Reference request: a differential equation in elementary geometry

Question

15 hours and four up-votes but not a word from anybody. That's the result of this question 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 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 $$ \cos\beta=\frac{\cos\alpha+n}{n\cos\alpha+1}. $$

Answer 1

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 (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 (On the Non-Euclidean Interpretation of the Theory of Relativity, by Vladimir Varicak, 1912).