Computing Fenchel-Nielsen Coordinates in Teichmüller Shape Space

Abstract

Teichmüller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichmüller space, only if they can be conformally mapped to each other. Teichmüller shape space can be used for surface classification purpose in shape modeling.

This work focuses on the computation of the coordinates of high genus surfaces in the Teichmüller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to deform the surface conformally using surface Ricci flow, such that the Gaussian curvature is −1 everywhere. The surface is decomposed to several pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary.

The algorithms are based on Teichmüller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichmüller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real world showed the practical value of the method for geometric database indexing, shape comparison and classification.