The Riemann surface S can be divided up into 2g − 2 pairs of pants by cutting along 3g − 3 disjoint simple closed curves.
For each of these 3g − 3 curves γ, choose an arc crossing it that ends in other boundary components of the pairs of pants with boundary containing γ.
The Fenchel–Nielsen coordinates for a point of the Teichmüller space of S consist of 3g − 3 positive real numbers called the lengths and 3g − 3 real numbers called the twists.
A point of Teichmüller space is represented by a hyperbolic metric on S. The lengths of the Fenchel–Nielsen coordinates are the lengths of geodesics homotopic to the 3g − 3 disjoint simple closed curves.
The twist is the (positive or negative) distance the middle segment travels along the geodesic of γ.