A Voronoi diagram is a basic geometric structure that partitions the space into regions associated with a given set of sites, such that all points in a region are closer to the corresponding site than to all other sites. While being thoroughly studied in Euclidean space, they are also of interest in hyperbolic space. In fact, there are several algorithms for computing hyperbolic Voronoi diagrams that work with the various models used to describe hyperbolic geometry. However, the polar-coordinate model has not been considered before, despite its popularity in the network science community. While Voronoi diagrams have the potential to advance this field, the model is geometrically not as approachable as other models, which impedes the development of geometric algorithms. In this paper, we present an algorithm for computing Voronoi diagrams natively in the polar-coordinate model of the hyperbolic plane. The approach is based on Fortune's sweep line algorithm for Euclidean Voronoi diagrams. We characterize the hyperbolic counterparts of the concepts it utilizes and introduce adaptations necessary to account for the differences. We implemented our algorithm and compared it with the corresponding CGAL implementation. While not being as numerically stable, our method has proven to be useful as a reference, which helped resolving fundamental issues in the implementation of the state-of-the-art method.

翻译：沃罗诺伊图是一种基本的几何结构，它将空间划分为与给定站点集合相关联的区域，使区域内所有点与对应站点的距离均小于到其他任何站点的距离。尽管该结构在欧几里得空间中已得到深入研究，其在双曲空间中也具有重要价值。事实上，现有多种算法可用于计算双曲沃罗诺伊图，这些算法基于描述双曲几何的不同模型。然而，极坐标模型虽在网络科学领域广泛应用，却尚未被纳入相关研究。尽管沃罗诺伊图有望推动该领域发展，但该模型的几何特性较其他模型更难处理，这阻碍了几何算法的开发。本文提出了一种在双曲平面极坐标模型中原生计算沃罗诺伊图的算法。该方法以欧几里得沃罗诺伊图的福琼扫描线算法为基础，刻画了该算法所利用概念在双曲空间中的对应形式，并引入适应双曲差异的修正机制。我们实现了该算法，并与相应的CGAL实现进行对比。尽管数值稳定性稍逊，但本方法可作为有效参考，帮助解决现有最优方法实现中的根本性问题。