CSCE 620 : Edge-Unfolding Convex Polyhedra

1. CSCE 620 : Edge-Unfolding Convex Polyhedra Open Problem 9 http://maven.smith.edu/~orourke/TOPP/P9.html#Problem.9 Yoosun Song 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 14 34 13 15 8 4 12 26 5 6 16 17 7 27 3 9 10 17 16 33 18 11 25 9 18 2 32 37 8 19 20 3 19 24 15 7 1 21 10 37 31 0 2 0 6 22 14 4 1 36 28 11 20 23 23 1 24 5 35 12 35 34 33 13 25 21 32 22 28 26 29 27 29 30 30 31 36 (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,Ununfoldablepolyhedra 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