1. Open Problem 9 http://maven.smith.edu/~orourke/TOPP/P9.html#Problem.9 Yoosun Song CSCE 620 : EDGE-UNFOLDING CONVEX POLYHEDRA Yoosun Song

2. 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

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

4. UNFOLDING ARCHEMEDEAN POLYHEDRON

5. 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

6. UNFOLDING RULES(DFS, BFS)

7. STEPS TO UNFOLDING 0 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 24 26 27 28 29 30 31 32 33 34 35 36 37 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 (a) BFS (b) DFS

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

9. 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.

10. UNFOLDING RULES

11. 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

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

13. 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. 1998 Japan Conf. Discrete Comput. Geom., volume 1763 of Lecture Notes Comput. Sci., pages 258-266. Springer-Verlag, 2000 B. Lucier. Unfolding and Reconstructing Polyhedra. M.Math Thesis, University of Waterloo, 2006 http://isotropic.org//polyhedra/ http://erikdemaine.org/papers/Ununfoldable/paper.pdf http://www.cs.toronto.edu/~blucier/misc/thesis.pdf