Open Problem 9

http://maven.smith.edu/~orourke/TOPP/P9.html#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

0

11

2

3

4

5

6

7

89

10

1112

13

14

15

161718

19

20

21 22

23

25

24

26

27

28

2930

31

32

33

34

35

36

37

01

23

456

7

8

9 1011

1213 14

15

16 1718

19 2021

2223 24

25

2627 28 2930

31

32333435

36

37

(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. 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