The unit disk graph (UDG) is a widely employed model for the study of wireless networks. In this model, wireless nodes are represented by points in the plane and there is an edge between two points if and only if their Euclidean distance is at most one. A hop spanner for the UDG is a spanning subgraph H such that for every edge $(p,q)$ in the UDG the topological shortest path between p and q in H has a constant number of edges. The hop stretch factor of H is the maximum number of edges of these paths. A hop spanner is plane (i.e. embedded planar) if its edges do not cross each other.

The problem of constructing hop spanners for the UDG has received considerable attention in both computational geometry and wireless ad hoc networks. Despite this attention, there has not been significant progress on getting hop spanners that (i) are plane, and (ii) have low hop stretch factor. Previous constructions either do not ensure the planarity or have high hop stretch factor. The only construction that satisfies both conditions is due to Catusse, Chepoi, and Vaxès (2010); their plane hop spanner has hop stretch factor at most 449.

Our main result is a simple algorithm that constructs a plane hop spanner for the UDG. In addition to the simplicity, the hop stretch factor of the constructed spanner is at most 341. Even though the algorithm itself is simple, its analysis is rather involved. Several results on the plane geometry are established in the course of the proof. These results are of independent interest.

单位磁盘图（UDG）是研究无线网络的一种广泛采用的模型。在此模型中，无线节点由平面中的点表示，并且当且仅当其欧几里得距离最大为1时，两个点之间才存在一条边。甲一跳扳手用于UDG为生成子图ħ使得对于每一个边缘$（p，q）$在UDG中，H中p和q之间的拓扑最短路径具有恒定数量的边。该跳拉伸因子的ħ是这些路径的边的最大数目。如果跃点扳手的边缘彼此不交叉，则为平面（即嵌入式平面）。

在计算几何结构和无线自组织网络中，为UDG构造跃点扳手的问题都受到了相当大的关注。尽管有这种注意，但在获得（i）是平面的和（ii）具有低跳变拉伸因子的跳变扳手方面，并没有取得重大进展。先前的构造要么不能确保平面性，要么具有高的跳变拉伸因子。满足这两个条件的唯一结构是Catusse，Chepoi和Vaxès（2010）提出的；他们的飞机跃点扳手的跃点拉伸因子最多为449。

我们的主要结果是一个简单的算法，该算法为UDG构造了一个平面跃点扳手。除了简单之外，构造的扳手的跳变拉伸因子最大为341。尽管算法本身很简单，但其分析相当复杂。在证明过程中建立了关于平面几何的几个结果。这些结果具有独立意义。