# Approximating shortest paths on a convex polytope in three dimensions

Journal Article

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 + ∈)dP(s, t), where dP(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.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Har-Peled, S; Sharir, M; Varadarajan, KR

### Published Date

- January 1, 1997

### Published In

### Volume / Issue

- 44 / 4

### Start / End Page

- 567 - 584

### International Standard Serial Number (ISSN)

- 0004-5411

### Digital Object Identifier (DOI)

- 10.1145/263867.263869

### Citation Source

- Scopus