Open Problem 9 Yoosun Song CSCE 620 : EDGE-UNFOLDING CONVEX POLYHEDRA Yoosun Song

PROBLEM DESCRIPTION What’s Unfolding? Cut surface and unfold to a single non-overlapping piece in the plane. Edge unfolding : Cut only along edges General unfolding: Cut through face too

ORIGINS Does every convex polyhedron have an edge-unfolding to a simple, non-overlapping polygon? [Shephard, 1975] [Albrecht Dürer, 1425]

UNFOLDING ARCHEMEDEAN POLYHEDRON

UNFOLDING ALGORITHMS Simple trees Breadth-first unfolding Depth first unfolding Left-first unfolding Shortest Path unfolding Steepest edge cut unfolding Greatest increase cut unfolding Normal order unfolding Backtrack unfolding

UNFOLDING RULES(DFS, BFS)

STEPS TO UNFOLDING (a) BFS (b) DFS

STEEPEST EDGE UNFOLDING Choose a cut tree which is the steepest edge in vertex v in polyhedron. Heuristically, we cut “the most upward edge”

STEEPEST EDGES We have direction unit vector c, and if c faces top of the pages. As follow the Steepest edge cutting rules, we have steepest edges drawn in bold like next figure.

UNFOLDING RULES

2 LAYER OVERLAP Suppose P′ is an unfolding of a convex polyhedron. Let e1, e2, and e3 be incident edges on the boundary of P′, where e1 and e2 have common vertex v and e2 and e3 have common vertex w. Further suppose that |e3| = |e2|. Let φ be the exterior angle at v, and let θ be the exterior angle at w. If 1. θ + 2φ < π, and 2. |e1| ≥ |e2|*sin θ/sin(π−θ−φ) then P′ will contain a 2-local overlap

COUNTER EXAMPLES TO UNFOLDING ALGORITHMS Counter example to Steepest Edge cutting algorithm

REFERENCES W. Schlickenrieder, Nets of Polyhedra. Diplomarbeit at TU-Berlin (1997) M. Bern, E. D. Demaine, D. Eppstein, E. Kuo, A. Mantler, and J. Snoeyink, Ununfoldable polyhedra with convex faces. Comput. Geom. Theory Appl., 24 (2):51-62 (2003) Joseph O'Rourke. Folding and unfolding in computational geometry. In Proc Japan Conf. Discrete Comput. Geom., volume 1763 of Lecture Notes Comput. Sci., pages Springer-Verlag, 2000 B. Lucier. Unfolding and Reconstructing Polyhedra. M.Math Thesis, University of Waterloo,