 Fenchel–Nielsen coordinates

For a fixed pants decomposition, the positive cuff lengths and real twist parameters give coordinates on Teichmüller space. With twists measured in length units, a full Dehn twist changes ${\tau }_i$ by ${\ell }_i$; with angle units it changes the twist by $2\pi$.

Fenchel–Nielsen coordinates are a method used in the study of hyperbolic surfaces and Riemann surfaces, particularly in the context of the deformation spaces of these surfaces. They provide a parametrization of the moduli space of hyperbolic surfaces with a fixed topological type, such as a surface with a given number of punctures or boundaries.