In this paper, we consider enumeration of geodesics on a polyhedron, where a geodesic means locally-shortest path between two points. Particularly, we consider the following preprocessing problem: given a point $s$ on a polyhedral surface and a positive real number $r$, to build a data structure that enables, for any point $t$ on the surface, to enumerate all geodesics from $s$ to $t$ whose length is less than $r$. First, we present a naive algorithm by removing the trimming process from the MMP algorithm (1987). Next, we present an improved algorithm which is practically more efficient on a non-convex polyhedron, in terms of preprocessing time and memory consumption. Moreover, we introduce a single-pair geodesic graph to succinctly encode a result of geodesic query. Lastly, we compare these naive and improved algorithms by some computer experiments.

翻译：本文研究多面体上测地线的枚举问题，其中测地线指两点间局部最短路径。特别地，我们考虑如下预处理问题：给定多面体表面上的点$s$和正实数$r$，构建一种数据结构，使得对于表面上任意点$t$，能枚举从$s$到$t$且长度小于$r$的所有测地线。首先，我们提出一种朴素算法，该算法通过移除MMP算法（1987）中的剪枝过程实现。其次，我们提出一种改进算法，该算法在非凸多面体上预处理时间和内存消耗方面实际效率更高。此外，我们引入单对测地线图来简洁编码测地线查询结果。最后，通过计算机实验对比了这两种朴素算法与改进算法。