TY - GEN

T1 - Nonoverlap of the star unfolding

AU - Aronov, Boris

AU - O'Rourke, Joseph

N1 - Funding Information:
“Part of the research for this paper was carried out while the first author was at the DIMACS Center, Rut-gers University. The second author’s research was supported by NSF grant tXX-882194. t c~~puter Science Department, Polytechnic University, Brooklyn, NY 11201 USA tDePartment of Computer Science, Smith colfege, Northampton, MA 01063 USA

PY - 1991/6/1

Y1 - 1991/6/1

N2 - The star unfolding of a convex polytope with respect to a point x is obtained by cutting the surface along the shortest paths from x to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding: (1) It does not self-overlap: its boundary is a simple polygon. (2) The ridge tree in the unfolding, which is the locus of points with more than one shortest path from x, is precisely the Voronoi diagram of the images of x, restricted to the unfolding. These two properties permit the conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: for constructing the ridge tree, for finding the exact set of all shortest-path "edge sequences," and for computing the geodesic diameter of a polytope. Our results suggest conjectures on "unfoldings" of general convex surfaces.

AB - The star unfolding of a convex polytope with respect to a point x is obtained by cutting the surface along the shortest paths from x to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding: (1) It does not self-overlap: its boundary is a simple polygon. (2) The ridge tree in the unfolding, which is the locus of points with more than one shortest path from x, is precisely the Voronoi diagram of the images of x, restricted to the unfolding. These two properties permit the conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: for constructing the ridge tree, for finding the exact set of all shortest-path "edge sequences," and for computing the geodesic diameter of a polytope. Our results suggest conjectures on "unfoldings" of general convex surfaces.

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U2 - 10.1145/109648.109660

DO - 10.1145/109648.109660

M3 - Conference contribution

AN - SCOPUS:0042535794

SN - 0897914260

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 105

EP - 114

BT - Proceedings of the Annual Symposium on Computational Geometry

PB - Association for Computing Machinery

T2 - 7th Annual Symposium on Computational Geometry, SCG 1991

Y2 - 10 June 1991 through 12 June 1991

ER -