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Unfolding Polyhedral Surfaces

Unfolding Polyhedral Surfaces. Joseph O’Rourke Smith College. Outline. What is an unfolding? Why study? Main questions: status Convex: edge unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Orthogonal polyhedra. ten1.mov. What is an unfolding?.

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Unfolding Polyhedral Surfaces

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  1. Unfolding Polyhedral Surfaces Joseph O’Rourke Smith College

  2. Outline • What is an unfolding? • Why study? • Main questions: status • Convex: edge unfolding • Convex: general unfolding • Nonconvex: edge unfolding • Nonconvex: general unfolding • Orthogonal polyhedra

  3. ten1.mov

  4. What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

  5. What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

  6. Unfolding Polyhedra • Two types of unfoldings: • Edge unfoldings: Cut only along edges • General unfoldings: Cut through faces too

  7. What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

  8. Cube with one corner truncated

  9. “Sliver” Tetrahedron

  10. Cut Edges form Spanning Tree Lemma: The cut edges of an edge unfolding of a convex polyhedron to a simple polygon form a spanning tree of the 1-skeleton of the polyhedron.

  11. Polygons: Simple vs. Weakly Simple

  12. Nonsimple Polygons

  13. Andrea Mantler example

  14. Cut edges: strengthening Lemma: The cut edges of an edge unfolding of a convex polyhedron to a single, connected piece form a spanning tree of the 1-skeleton of the polyhedron. [Bern, Demaine, Eppstein, Kuo, Mantler, O’Rourke, Snoeyink 01]

  15. What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

  16. Cuboctahedron unfolding [Matthew Chadwick]

  17. [Biedl, Lubiw, Sun, 2005]

  18. Outline • What is an unfolding? • Why study? • Main questions: status • Convex: edge unfolding • Convex: general unfolding • Nonconvex: edge unfolding • Nonconvex: general unfolding • Orthogonal polyhedra

  19. Lundström Design, http://www.algonet.se/~ludesign/index.html

  20. Outline • What is an unfolding? • Why study? • Main questions: status • Convex: edge unfolding • Convex: general unfolding • Nonconvex: edge unfolding • Nonconvex: general unfolding • Orthogonal polyhedra

  21. Status of main questions

  22. Status of main questions

  23. Open: Edge-Unfolding Convex Polyhedra Does every convex polyhedron have an edge-unfolding to a simple, nonoverlapping polygon? [Shephard, 1975]

  24. Albrecht Dürer, 1425 Melancholia I

  25. Albrecht Dürer, 1425 Snub Cube

  26. Unfolding the Platonic Solids Some nets: http://www.cs.washington.edu/homes/dougz/polyhedra/

  27. Archimedian Solids [Eric Weisstein]

  28. “Nets of Polyhedra” TU Berlin, 1997 Sclickenrieder1:steepest-edge-unfold

  29. Sclickenrieder2:flat-spanning-tree-unfold

  30. Sclickenrieder3:rightmost-ascending-edge-unfold

  31. Sclickenrieder4:normal-order-unfold

  32. Percent Random Unfoldings that Overlap [O’Rourke, Schevon 1987]

  33. Classes of Convex Polyhedrawith Edge-Unfolding Algorithms • Prisms • Prismoids • “Domes” • “Bands” • Radially monotone lattice quadrilaterals • Prismatoids???

  34. Prismoids Convex top A and bottom B, equiangular. Edges parallel; lateral faces quadrilaterals.

  35. Overlapping Unfolding

  36. Volcano Unfolding

  37. Unfolding “Domes”

  38. Proof via degree-3 leaf truncation dodec.wmv [Benton, JOR, 2007]

  39. Lattice Quadrilateral Convex Caps

  40. Classes of Convex Polyhedrawith Edge-Unfolding Algorithms • Prisms • Prismoids • “Domes” • “Bands” • Radially monotone lattice quadrilaterals • Prismatoids???

  41. Unfolding Smooth Prismatoids [Benbernou, Cahn, JOR 2004]

  42. Open: Fewest Nets For a convex polyhedron of n vertices and F faces, what is the fewest number of nets (simple, nonoverlapping polygons) into which it may be cut along edges?

  43. Outline • What is an unfolding? • Why study? • Main questions: status • Convex: edge unfolding • Convex: general unfolding • Nonconvex: edge unfolding • Nonconvex: general unfolding • Orthogonal polyhedra

  44. Status of main questions

  45. General Unfoldings of Convex Polyhedra • Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). • Source unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87] • Star unfolding [Aronov & JOR ’92] • Quasigeodesic unfolding [Itoh, JOR, Vilcu, 2007] [Poincare?]

  46. Shortest paths from x to all vertices

  47. Source Unfolding

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