There is the following result on page 160 of Thurston's book "The Geometry and Topology of Three-Manifolds", as follows.



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> The volume of an ideal simplex in $\mathbb{H}^3$ with dihedral angles $\alpha$, $\beta$, $\gamma$ is equal to$$\Lambda(\alpha) + \Lambda(\beta) + \Lambda(\gamma),$$where $\Lambda: \mathbb{R} \to \mathbb{R}$ is the Lobachevsky function$$\Lambda(\theta) = -\int\_0^\theta \log|2\sin t|\,dt.$$
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What is the idea/intuition behind the proof of this result, i.e. why does the volume of an orthoscheme have a formula in terms of the Lobachevsky function? I've read the proof in Thurston's book, but I'd still be interested in hearing what others have to say on this issue.



Volume of an ideal simplex in $\mathbb{H}^3$, idea/intuition behind result
A longer but more intuitive proof of this formula is by "integrating" the Schlaefli identity for ideal tetrahedra.


One starts from a euclidean triangle with angles $\alpha$, $\beta$, $\gamma$ and side lengths $a$, $b$, $c$. Put $x=\log a$, $y=\log b$, $z=\log c$ and consider the function $$f(x,y,z) = \alpha x + \beta y + \gamma z + \Lambda(\alpha) + \Lambda(\beta) + \Lambda(\gamma)$$ One computes (using that $x - \log(2\sin\alpha) = \log R$) the partial derivatives: $$\frac{\partial f}{\partial x} = \alpha, \quad \frac{\partial f}{\partial y} = \beta, \quad \frac{\partial f}{\partial z} = \gamma$$
Now take the Legendre transform of $f$: $$F(\alpha,\beta,\gamma) = \Lambda(\alpha)+\Lambda(\beta)+\Lambda(\gamma)$$
It follows that $$\frac{\partial F}{\partial \alpha} - \frac{\partial F}{\partial\beta} = y-x = \log\frac{b}{a} \quad \text{ etc.}$$
This $\log\frac{b}{a}$ turns out to be related to the truncated edge lengths of the tetrahedron, so that the function $F$ satisfies the Schlaefli differential identity. Therefore it is the volume up to an additive constant.


For more details see Sections 4 and 5 in


[Bobenko, Alexander I.; Pinkall, Ulrich; Springborn, Boris A. Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19 (2015), no. 4, 2155–2215.](http://msp.org/gt/2015/19-4/p07.xhtml)


[The arxiv version.](http://arxiv.org/abs/1005.2698)


