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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 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
What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.
What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.
Unfolding Polyhedra • Two types of unfoldings: • Edge unfoldings: Cut only along edges • General unfoldings: Cut through faces too
What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.
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.
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]
What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.
Cuboctahedron unfolding [Matthew Chadwick]
Outline • What is an unfolding? • Why study? • Main questions: status • Convex: edge unfolding • Convex: general unfolding • Nonconvex: edge unfolding • Nonconvex: general unfolding • Orthogonal polyhedra
Lundström Design, http://www.algonet.se/~ludesign/index.html
Outline • What is an unfolding? • Why study? • Main questions: status • Convex: edge unfolding • Convex: general unfolding • Nonconvex: edge unfolding • Nonconvex: general unfolding • Orthogonal polyhedra
Open: Edge-Unfolding Convex Polyhedra Does every convex polyhedron have an edge-unfolding to a simple, nonoverlapping polygon? [Shephard, 1975]
Albrecht Dürer, 1425 Melancholia I
Albrecht Dürer, 1425 Snub Cube
Unfolding the Platonic Solids Some nets: http://www.cs.washington.edu/homes/dougz/polyhedra/
Archimedian Solids [Eric Weisstein]
“Nets of Polyhedra” TU Berlin, 1997 Sclickenrieder1:steepest-edge-unfold
Percent Random Unfoldings that Overlap [O’Rourke, Schevon 1987]
Classes of Convex Polyhedrawith Edge-Unfolding Algorithms • Prisms • Prismoids • “Domes” • “Bands” • Radially monotone lattice quadrilaterals • Prismatoids???
Prismoids Convex top A and bottom B, equiangular. Edges parallel; lateral faces quadrilaterals.
Proof via degree-3 leaf truncation dodec.wmv [Benton, JOR, 2007]
Classes of Convex Polyhedrawith Edge-Unfolding Algorithms • Prisms • Prismoids • “Domes” • “Bands” • Radially monotone lattice quadrilaterals • Prismatoids???
Unfolding Smooth Prismatoids [Benbernou, Cahn, JOR 2004]
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?
Outline • What is an unfolding? • Why study? • Main questions: status • Convex: edge unfolding • Convex: general unfolding • Nonconvex: edge unfolding • Nonconvex: general unfolding • Orthogonal polyhedra
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?]