Approximate ESPs on surfaces of polytopes using a rubberband algorithm

Let p and q be two points on the surface of a polytope Pi. This paper provides a rubberband algorithm for computing a Euclidean shortest path between p and q (a so-called surface ESP) that is contained on the surface of Pi. The algorithm has k(1)(epsilon).k(2)(epsilon).O(n(2)) time complexity, where n is the number of vertices of Pi, k(i)(epsilon) = (L-0i - L-i)/epsilon, for the true length L-i of some shortest path with initial (polygonal path) length L-0i (used when approximating this shortest path), for i = 1, 2. Rubberband algorithms follow a straightforward design strategy, and the proposed algorithm is easy to implement and thus of importance for applications, for example, when analyzing 3D objects in 3D image analysis, such as in biomedical or industrial image analysis, using 3D image scanners

Approximate ESPs on Surfaces of Polytopes Using a Rubberband Algorithm

Abstract. Let p and q be two points on the surface of a polytope Π. This paper provides a rubberband algorithm for computing a Euclidean shortest path between p and q (a so-called surface ESP) that is contained on the surface of Π. The algorithm has κ1(ε) · κ2(ε) · O(n 2) time complexity, where n is the number of vertices of Π, κi(ε) = (L0i − Li)/ε, for the true length Li of some shortest path with initial (polygonal path) length L0i (used when approximating this shortest path), for i = 1, 2. Rubberband algorithms follow a straightforward design strategy, and the proposed algorithm is easy to implement and thus of importance for applications, for example, when analyzing 3D objects in 3D image analysis, such as in biomedical or industrial image analysis, using 3D image scanners. Keywords: rubberband algorithm, Euclidean shortest path, surface ESP.