Geodesic complexity of the octahedron, and an algorithm for cut loci on convex polyhedra
Florian Frick and Pranav Rajbhandari
长度空间 X 的大地测量复杂性量化了所需的案例区分数量,以持续选择连接任何给定起点和终点的最短路径。 我们证明了通过将简单化嵌入到 X× X 中获得的 X 的地缘复杂度的局部下限。 我们还创建并证明了算法的正确性,以便在凸多面体表面上找到切割位点,因为空间切割位点的结构与其大地测量复杂性有关。 我们用这些技术来证明八面体的大地测量复杂性是4。 我们的方法受到Recio-Mitter和Davis早期工作的启发,因此分别恢复了对n-torus和四面体的大地测量复杂性的结果。
The geodesic complexity of a length space X quantifies the required number of case distinctions to continuously choose a shortest path connecting any given start and end point. We prove a local lower bound for the geodesic complexity of X obtained by embedding simplices into X× X. We additionally create and prove correctness of an algorithm to find cut loci on surfaces of convex polyhedra, as the structure of a space's cut loci is related to its geodesic complexity. We use these techniques to pr...
