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Let $(\zeta_i) \subset \hat{\mathbb{C}}$, for $i = 1, \ldots, 4$, be $4$ distinct points on the Riemann sphere $\hat{\mathbb{C}}$.

We will use the following convention for the cross-ratio $CR$ of these four points: $$ CR(\zeta_1, \dots, \zeta_4) = \frac{(\zeta_1 - \zeta_4) (\zeta_2 - \zeta_3)}{(\zeta_1 - \zeta_3) (\zeta_2 - \zeta_4)}.$$

In the previous formula, I assumed that all the $\zeta_i$ are finite, but it is well known how to extend the formula for the case where say one of the $\zeta_i$ is infinite.

We identify the Riemann sphere with the sphere at infinity $S^2_\infty$ for the $3$-dimensional hyperbolic space $H^3$. This amounts to using a complex coordinate on $S^2_\infty$ using stereographic projection from the North pole $(0, 0, 1)^T$ (I am using the Poincaré open 3-ball model for $H^3$).

Let $l_1$ denote the oriented hyperbolic line going from $\zeta_1$ to $\zeta_3$ and $l_2$ be the oriented hyperbolic line going from $\zeta_2$ to $\zeta_4$.

We denote by $\delta$ the hyperbolic distance between the lines $l_1$ and $l_2$. There are points, say $p_1 \in l_1$ and $p_2 \in l_2$, which realize this hyperbolic distance. Let $l_{12}$ denote the hyperbolic line segment joining $p_1$ and $p_2$.

Let $v_1$, resp. $v_2$, be a unit vector at $p_1$, resp. $p_2$, tangent to $l_1$, resp. $l_2$, and in the direction corresponding to the orientation of $l_1$, resp. $l_2$.

Using the geodesic line segment $l_{12}$, one may parallel transport $v_2$ to a vector $v'_2$ based at $p_1$.

Let $n$ denote a unit vector at $p_1$ that is tangent to $l_{12}$ and pointing towards $p_2$. We remark that $n$ defines an orientation on the tangent plane which is orthogonal to $n$. We also remark that the orthogonal complement of $n$ contains both $v_1$ and $v'_2$. Using this orientation, let $\alpha$ be the angle between $v_1$ and $v'_2$.

I obtained the following beautiful formula $$ CR(\zeta_1, \dots, \zeta_4) = \operatorname{cosh}^2 \left(\delta + i \frac{\alpha}{2} \right). $$

We thus see that it is natural to define the complex distance between $l_1$ and $l_2$ to be $$ \delta + i \frac{\alpha}{2} \in \mathbb{C} / (i \pi \mathbb{Z}).$$

This reminds me of the notion of a complex volume for compact hyperbolic 3-manifolds, although this is a much simpler situation, of course!

Is this formula new? It is a classical topic, so I kind of have doubts that it is new. If it is known, could someone please point me to the relevant literature where it previously appeared? Is it interesting (I certainly think so, but this is just my opinion)?

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Your formula can be found on page 355 (near the end of Section 7.4) of Marden's book Outer circles: an introduction to hyperbolic 3-manifolds. In the second edition of the book, with the title, Hyperbolic manifolds: an introduction in 2 and 3 dimensions, the formula appears on page 432. [To be precise, Marden is talking about the translation distance of the product of the half-rotations about the given lines. So his complex distance is twice yours and lives in $\mathbb{C} / 2 \pi i \mathbb{Z}$.]


The reference took me a very long time to find. This frustrated me a little bit! I am quite sure that I have seen this formula before in papers and/or in lectures... there must be more uses/mentions of this formula in the literature.

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    $\begingroup$ Thank you so much! I thought it had to be known, especially since many more complicated things are known about 3-dimensional geometry! I really appreciate your time and energy. If I am able to use this formula in my work, I will make sure to acknowledge your help! $\endgroup$
    – Malkoun
    Feb 11 at 13:51

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