There is a deep connection between the formula for aberration,

$$\cos\omega=\frac{\cos\omega'-\beta}{1-\beta\cos\omega'},$$

and the Mobius automorphism that swaps the relative state of motion, $\beta$, with one at rest, $0$,

$$\mathcal{M}(\cos\omega')=\frac{\cos\omega'-\beta}{\beta\cos\omega'-1},$$

except for the negative sign. If $\cos\omega'=\beta$, then $\mathcal{M}(\beta)=0$, while if $\omega'$ is a right-angle, then $\mathcal{M}(0)=\beta$. But, there is more to this than what meets the eye.

A Mobius transform takes any triplet, $v,v',v"$ int any other triplet. The proof rests on the facts that values of the Mobius transform are $0$, $1$, and $\infty$. These values are the range of the velocities $\beta$ and $1/\beta$; the former reduces to the numerator of the Mobius transform to zero, while the latter does the same for the denominator.

The property of inversion guarantees that if $\beta$ exists so does its inverse, $1/\beta$. It is here that the Mobius transformation and aberration part company. Aberration does not allow the denominator to vanish, i.e., the inverse $1/\beta$. However, we have seen that the equivalent expressions for the speed,

$$\frac{1-\beta^2}{\beta\cos\omega+K(\omega)},$$

and

$$\frac{1-\beta\cos\omega'}{K(\omega)}$$

break the symmetry of the aberration formula, which is now given by

$$\frac{\cos\omega}{K(\omega)}=\frac{\cos\omega'-\beta}{1-\beta\cos\omega'}.$$

This applies to one way speeds. For a two-way speed, we would find $\beta=\cos\omega'$ and $\beta=0$ on equating the two expressions since the dipole variant $\pm\beta\cos\omega$ cancel one another leaving the speed as

$$\frac{1-\beta^2}{K(\omega)}$$

which when equated to the second expression gives $\beta^2=\beta\cos\omega'$.

It is the two-way speed of light which prevents the third point which sends the Mobius transform to infinity from appearing. The symmetry in past and future, $\omega\rightarrow\omega'$ and $\beta\rightarrow-\beta$, which reduces the speed to a two-way speed of light that eliminates the third conjugate point. But we *know* that the inverse speed exists!

According to the usual aberration formula, $\cos\omega'=\beta$ implies that the triangle is a right-triangle since $\omega=\pi/2$. As we have seen, this is Einstein's choice, while the choice of Bolyai and Lobaschevsky was

$$\omega'(\overline{\beta})=2\tan^{-1}e^{-\overline{\beta}},$$

where

$$\overline{\beta}=\frac{1}{2}\ln\left(\frac{1+\beta}{1-\beta}\right),$$

is the hyperbolic velocity. The angle $\omega'$ is equal to its hyperbolic counterpart $\overline{\omega'}$ since it is in the direction of the motion. No so with the opposite angle whose distance to the right angle undergoes contraction by an amount $\surd(1-\beta^2)$.

The angle $\omega$ is sole distance of the hyperbolic distance $\overline{\beta}$ according to the formula

$$\cos\omega'=\beta=\tanh\overline{\beta}=\frac{2u/c}{1+u^2/c^2},$$

which is the relative speed of two systems with equal and opposite speeds, $\vec{v}+\vec{v'}=0$, with $u=|\vec{v}|=|\vec{v'}|$, The hyperbolic tangent is the Lobachevskian segment of a straight line in hyperbolic geometry.

Imagine two circles that intersect orthogonally. Draw a secant through the centers of both circles. Where one circle that intersects the secant, its position is given by the smaller root of

$$v_{\pm}=\frac{1\pm\surd(1-\beta^2)}{\beta}.$$

$v_{-}$ divides the hyperbolic distance between $\beta$ and the origin in two. The inverse of $\beta$ is found at the origin of the second circle, and $v_{+}$ lies at the intersection where the secant cuts the second circle. The latter can rightly be considered the inverse of the former since $v_{-}\cdot v_{+}=1$. Hence, $v_{+}=1/v_{-}$. Both the speed and its inverse is guaranteed in this manner, where the inverse speeds lie outside the principal circle which contains both $\beta$ and $v_{-}$.

Lobachevsky gave $\omega'$ a special symbol $\Pi(\overline{\beta})$, which is a sole function of the hyperbolic distance from the vertex of the angle to the right-triangle. In our optical analogy, $\Pi$ is the critical angle, beyond which no refraction takes place. For values greater than the critical angle, there is total internal reflection.

So for all angles of incidence leading up to the critical angle, or the angle of parallelism, refraction takes place, and the lack of symmetry in past and future shows up as a one-way speed of light. There is a conspiracy insofar as the two-way speed of light, the symmetry in past and future, or Einstein's convention of simultaneity destroy the anisotropic properties of light propagation.