穿孔双曲曲面可以由理想双曲三角形拼接而成。本课题将从三个方面进行研究。 首先，研究理想三角形的shearing坐标与双曲曲面的几何性质、以及黎曼模空间的Thurston度量之间的关系。其次，针对同一个双曲曲面，我们将比较不同理想三角剖分的拓扑类型所对应的shearing坐标，探索最小shearing坐标所对应的最优化理想三角剖分对双曲曲面的依赖关系，从而研究Teichmuller空间与三角剖分复形之间的拟等距关系。 最后，给定一个理想三角剖分以及shearing坐标，我们可以通过伸缩shearing坐标的方式得到Teichmuller空间的一条射线。我们将研究该条射线在黎曼模空间中的分布情况。本课题的开展有助于研究 stretch 映射，继而研究Thurston度量的测地线等。

It is known that every punctured hyperbolic surface can be constructed via ideal triangulations with shearing coordinates. This program will focus on the following three perspectives. First, we will investigate how the shearing coordinates of a fixed triangulation change the geometric properties of hyperbolic surfaces and the Thurston distance between hyperbolic surfaces. Secondly, we will compare the shearing coordinates corresponding to different triangulations in order to find the optimal triangulation for which the shearing is minimal. In this way, we are able to compare the Teichmuller space and triangulation complex. Thirdly, for a fixed triangulation and a shearing coordinate, by scaling the shearing coordinate, we obtain a ray in the Teichmuller space. We will investigate the distribution of this ray in the moduli space. This program is motivated by the research on the Thurston geodesic.