Shortest Path Problems on a Polyhedral Surface

Abstract

We develop algorithms to compute edge sequences, Voronoi diagrams, shortest path maps, the Fréchet distance, and the diameter for a polyhedral surface. Distances on the surface are measured either by the length of a Euclidean shortest path or by link distance.

Geodesic diameter of a polygonal domain in O(n^4 log n) time

“…An interesting open problem is whether our ideas can be applied to diameters of polytopes in R 3 [1,6,10]. Shortest paths on polyhedral surfaces do not bend at vertices [11]; the combinatorial type of a shortest path is the sequence of edges that it visits (the path is uniquely defined by the sequence due to the unfolding property -the shortest path becomes a line segment if the polytope is unfolded along the edges in the sequence).…”

“…The problem is non-trivial already in simple polygons, where it was examined decades ago [4,12] culminating in a linear-time algorithm [7]. Similarly, for convex polytopes polynomial-time algorithms have been known since 1990's [1,10]; the current best running time is O(n 7 log n) (for a polytope with n vertices) [6]. However, for polygonal domains with holes, no algorithms existed until very recently.…”

An intrinsic algorithm for computing geodesic distance fields on triangle meshes with holes

“…They also presented practical algorithms to construct geodesic Voronoi diagrams on discrete meshes, which has potential to a wide range of shape analysis applications. There are also a large body of literatures of computing discrete geodesics in other formats, such as geodesic loops [15,16], offsets [17], and all-pairs geodesic distance [18,19].…”

Querying two boundary points for shortest paths in a polygonal domain

“…They also considered the problem where the query points are restricted to lie on the edges of the polytope, reducing the bounds by a factor of n from the general case. Recently, Cook IV and Wenk [8] presented an improved method using kinetic Voronoi diagrams.…”