csce 620 : edge-unfolding convex polyhedra
DESCRIPTION
CSCE 620 : Edge-Unfolding Convex Polyhedra. Open Problem 9 http ://maven.smith.edu/~ orourke/TOPP/P9.html#Problem.9 Yoosun Song. Yoosun Song. Problem Description. What’s Unfolding? Cut surface and unfold to a single non-overlapping piece in the plane . - PowerPoint PPT PresentationTRANSCRIPT
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