Visualization of hyperbolic surfaces via generic fundamental polygons, and its applications on the theory of the canonical decomposition

为了在计算机上实现双曲曲面的可视化，需要对双曲曲面进行合适的符号化处理。我得到了这样一个原型，不仅是双曲的，而且是一般的，表面的象征性实现。这种实现的一个优点是可以检测曲面的双曲度。本研究的另一个结果是得到了Fuchsian群的一般基本多边形的组合型在双曲平面无穷远圆上的稳定性证明。这个结果与中心在双曲平面上的一般基本多边形的相同性质的已知结果相对应。

To visualize hyperbolic surfaces by using of computer, it is required to decide suitable symbolical treatment of such surfaces in this research. I obtained a prototype of such symbolical realization of, not only hyperbolic but also general, surfaces. An advantage of this realization is to detect hyperbolicity of surfaces. Another result is this research is that I obtained a proof of stability of the combinatorial type of generic fundamental polygons of Fuchsian groups when their centers are on the circle at infinity of the hyperbolic plane. This result corresponds to a known result of the same property of generic fundamental polygons with their centers in the hyperbolic plane.