The geodesic edge center of a polygon is a point c inside the polygon that minimizes the maximum geodesic distance from c to any edge of the polygon, where geodesic distance is the shortest path distance inside the polygon. We give a linear-time algorithm to find a geodesic edge center of a simple polygon. This improves on the previous O(n log n) time algorithm by Lubiw and Naredla [European Symposium on Algorithms, 2021]. The algorithm builds on an algorithm to find the geodesic vertex center of a simple polygon due to Pollack, Sharir, and Rote [Discrete & Computational Geometry, 1989] and an improvement to linear time by Ahn, Barba, Bose, De Carufel, Korman, and Oh [Discrete & Computational Geometry, 2016]. The geodesic edge center can easily be found from the geodesic farthest-edge Voronoi diagram of the polygon. Finding that Voronoi diagram in linear time is an open question, although the geodesic nearest edge Voronoi diagram (the medial axis) can be found in linear time. As a first step of our geodesic edge center algorithm, we give a linear-time algorithm to find the geodesic farthest-edge Voronoi diagram restricted to the polygon boundary.

翻译：多边形的地测边缘中心是多边形内部的一点c，它使得从c到多边形任意边的最大测地距离最小化，其中测地距离是多边形内部的最短路径距离。我们提出了一种线性时间算法来寻找简单多边形的测地边缘中心。这改进了Lubiw和Naredla在[欧洲算法研讨会，2021]中提出的O(n log n)时间算法。该算法基于Pollack、Sharir和Rote在[离散与计算几何，1989]中提出的寻找简单多边形测地顶点中心的算法，以及Ahn、Barba、Bose、De Carufel、Korman和Oh在[离散与计算几何，2016]中改进的线性时间算法。测地边缘中心可以很容易地从多边形的测地最远边缘Voronoi图中找到。在线性时间内找到该Voronoi图是一个开放问题，尽管测地最近边缘Voronoi图（中轴）可以在线性时间内找到。作为我们测地边缘中心算法的第一步，我们提出了一种线性时间算法，用于寻找限制在多边形边界上的测地最远边缘Voronoi图。