Euclidean length of a slope on a horospherical cusp section

Let $M$ be the interior of a compact connected orientable three-manifold with a complete finite-volume hyperbolic metric and torus cusps $C_1,\dots ,C_k$. For a chosen cusp $C_i$, choose a horospherical section $T_i$ and identify its Euclidean universal cover with ${\mathbb{R}}^2$ so its deck group is a rank-two translation lattice. A slope on $T_i$ is represented by a primitive lattice vector $v$ modulo sign. Its Euclidean slope length on the chosen section is ${\mathrm{\ell }}_{T_i}(v)=\parallel v\parallel$ in the induced Euclidean metric. In upper half-space coordinates, if the section is the quotient of the horosphere of height $h>0$ by lattice $\mathrm{\Lambda }$, then ${\mathrm{\ell }}_{T_i}(p,q)=\mathrm{\mid }pu+qw\mathrm{\mid }\mathrm{/}h$, where $u,w$ are the translation vectors representing the chosen basis. Replacing height $h$ by $h^{\mathrm{\prime }}>0$ multiplies every slope length by $h\mathrm{/}{h}^{\mathrm{\prime }}$; therefore the length depends on the chosen section and is not an intrinsic number until that section is fixed.

Primitive torus slopes and generalized cusp parameters

Complete finite-volume hyperbolic structures and torus cusps

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 4, §§4.3–4.5

Danny Calegari, Kleinian Groups, §§2.3.2–2.4.1