DescriptionThe Teichmüller space of a surface with boundary is the space of all isotopy classes of hyperbolic metrics with totally geodesic boundary. Using the cosine law of a hyperbolic
right-angled hexagon, Feng Luo introduced a continuous family of new coordinates of the Teichmüller space: the $psi_{lambda}$
coordinate. He proved that for $lambda geq 0$, the image of the Teichmüller space under the $psi_{lambda}$ coordinate is an open convex polytope independent of $lambda$. In this
dissertation, we verify Luo's conjecture that for $lambda [less than]0$, the image of the Teichmüller space under the $psi_{lambda}$
coordinate is a bounded open convex polytope.