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 by ; with angle units it changes the twist by .
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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.