论文标题
装饰双曲线表面的典型镶嵌
Canonical tessellations of decorated hyperbolic surfaces
论文作者
论文摘要
有限类型的双曲表面的装饰是每个圆锥点，表面或表面的圆形圆形或圆周的圆形，烟节或超循环的选择。在本文中，我们表明，装饰引起了基础表面的独特规范镶嵌和双重分解。它们是欧几里得平面中加权的Delaunay Tessellation和Voronoi分解的类似物。我们根据Delaunay的空盘和Laguerre的切线距离（也称为功率距离）的双曲线几何等效物进行了表征。此外，提出了Minkowski空间中的镶嵌和凸壳之间的关系，从而推广了爱泼斯坦 - 彭纳式凸壳的结构。这种关系使我们能够将数周的翻转算法扩展到装饰有限型双曲线表面的情况。最后，我们简单地描述了装饰的配置空间，并表明任何固定双曲线表面都只能接受有限数量的组合不同的规范镶嵌。
A decoration of a hyperbolic surface of finite type is a choice of circle, horocycle or hypercycle about each cone-point, cusp or flare of the surface, respectively. In this article we show that a decoration induces a unique canonical tessellation and dual decomposition of the underlying surface. They are analogues of the weighted Delaunay tessellation and Voronoi decomposition in the Euclidean plane. We develop a characterisation in terms of the hyperbolic geometric equivalents of Delaunay's empty-discs and Laguerre's tangent-distance, also known as power-distance. Furthermore, the relation between the tessellations and convex hulls in Minkowski space is presented, generalising the Epstein-Penner convex hull construction. This relation allows us to extend Weeks' flip algorithm to the case of decorated finite type hyperbolic surfaces. Finally, we give a simple description of the configuration space of decorations and show that any fixed hyperbolic surface only admits a finite number of combinatorially different canonical tessellations.
