Approximating shortest paths on a convex polytope in three dimensions

Given a convex polytope P with n faces in ℝ³, points s, t ∈ ∂P, and a parameter 0 < ∈ ≤ 1, we present an algorithm that constructs a path on ∂P from s to t whose length is at most (1 + ∈)d_P(s, t), where d_P(s, t) is the length of the shortest path between s and t on ∂P. The algorithm runs in O(n log 1/∈ + 1/∈³) time, and is relatively simple. The running time is O(n + 1/∈³) if we only want the approximate shortest path distance and not the path itself. We also present an extension of the algorithm that computes approximate shortest path distances from a given source point on ∂P to all vertices of P.