I currently study discrete conformal maps and read the paper "Discrete conformal maps an ideal hyperbolic polyhedra" by Bobenko, Pinkall and Springborn. Consider the following definitions:
Two Riemannian metrics $g$ and $\tilde{g}$ on a Riemann surface $M$ are called conformally equivalent if $$\tilde{g} = e^{2u} g$$ for a smooth function $u \colon M \rightarrow \mathbb{R}$.
Two combinatorially equivalent Euclidean triangulations $(T,l)$ and $(T,\tilde{l})$ (where $l$ and $\tilde{l}$ are discrete metrics on a surface triangulation $T$) are called discretely conformally equivalent if $$\tilde{l}_{ij} = e^{\frac{1}{2}(u_i+u_j)} l_{ij}$$ holds for all edges $ij \in E(T)$ for some function $u \colon V(T) \rightarrow \mathbb{R}$ on the vertices.
The discrete definition is inspired by the continuous one and yields a nice theory. I wonder for which reason the coefficients $ 2 $ and $ \frac12 $ differ. If I did not overlook something, they do not mention this in the paper. I see that both definitions are independent of the concrete coefficient because one can simply include the factor in $u$. But why do they differ and why do they not forego both the $2$ and the $\frac12$ in the first place?
The Riemannian metrics measure squared length. Consider scaling all lengths by a constant factor $a$. This corresponds to constant $u=\log a$ in both the discrete and the smooth theory. So $u$ is the logarithmic length scale factor.
More precisely, in the smooth theory, $e^u$ is the local length scale factor, in the discrete theory, the scale factor for an edge is the geometric mean of $e^u$ at its vertices.