The surface of the tetrahedron is a 4-punctured sphere, made of 4 ideal triangles. In fact, this is a complete hyperbolic metric, as may be seen by sending $z_1$ to $\infty$ (in fact, we may assume $z_1=\infty, z_2=0, z_3=1, z_4=z$, for $z$ the cross-ratio of the 4 points).

The hyperbolic metrics on the 4-punctured sphere may be parameterized by shearing coordinates for the triangulation. For the edge connecting $0$ and $\infty$, the shearing coordinate will be $\ln(|z|)$, since one triangle is scaled by a factor of $|z|$. Opposite edges will have the same shear coordinate because of the symmetries, and the other cross-ratios are $\frac{1}{1-z}$ and $1-1/z$, from which one obtains the other shearing coordinates. The fact that the product of the cross-ratios $|z \cdot \frac{1}{1-z} \cdot (1-1/z)|=1$ corresponds to the fact that the cusps will be complete.