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**Unfolding Polyhedral Surfaces**

Joseph O’Rourke Smith College

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**Outline What is an unfolding? Why study? Main questions: status**

Convex: edge unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Orthogonal polyhedra

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ten1.mov

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

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

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**Unfolding Polyhedra Two types of unfoldings:**

Edge unfoldings: Cut only along edges General unfoldings: Cut through faces too

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

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**Cube with one corner truncated**

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“Sliver” Tetrahedron

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**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.

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**Polygons: Simple vs. Weakly Simple**

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Nonsimple Polygons

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**Andrea Mantler example**

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**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]

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

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**Cuboctahedron unfolding**

[Matthew Chadwick]

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[Biedl, Lubiw, Sun, 2005]

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**Outline What is an unfolding? Why study? Main questions: status**

Convex: edge unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Orthogonal polyhedra

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

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**Outline What is an unfolding? Why study? Main questions: status**

Convex: edge unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Orthogonal polyhedra

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**Status of main questions**

Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

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**Status of main questions**

Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

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**Open: Edge-Unfolding Convex Polyhedra**

Does every convex polyhedron have an edge-unfolding to a simple, nonoverlapping polygon? [Shephard, 1975]

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Albrecht Dürer, 1425 Melancholia I

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Albrecht Dürer, 1425 Snub Cube

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**Unfolding the Platonic Solids**

Some nets:

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Archimedian Solids [Eric Weisstein]

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**Sclickenrieder1: steepest-edge-unfold**

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

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**Sclickenrieder2: flat-spanning-tree-unfold**

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**Sclickenrieder3: rightmost-ascending-edge-unfold**

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**Sclickenrieder4: normal-order-unfold**

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**Percent Random Unfoldings that Overlap**

[O’Rourke, Schevon 1987]

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**Classes of Convex Polyhedra with Edge-Unfolding Algorithms**

Prisms Prismoids “Domes” “Bands” Radially monotone lattice quadrilaterals Prismatoids???

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**Prismoids Convex top A and bottom B, equiangular.**

Edges parallel; lateral faces quadrilaterals.

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**Overlapping Unfolding**

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Volcano Unfolding

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Unfolding “Domes”

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**Proof via degree-3 leaf truncation**

dodec.wmv [Benton, JOR, 2007]

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**Lattice Quadrilateral Convex Caps**

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**Classes of Convex Polyhedra with Edge-Unfolding Algorithms**

Prisms Prismoids “Domes” “Bands” Radially monotone lattice quadrilaterals Prismatoids???

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**Unfolding Smooth Prismatoids**

[Benbernou, Cahn, JOR 2004]

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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? ≤ F ≤ (2/3)F [Spriggs] ≤ (1/2)F [Dujmenovic, Moran, Wood] ≤ (3/8)F [Pincu, 2007]

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**Outline What is an unfolding? Why study? Main questions: status**

Convex: edge unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Orthogonal polyhedra

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**Status of main questions**

Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

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**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?]

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**Shortest paths from x to all vertices**

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Source Unfolding

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Star Unfolding

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**Star-unfolding of 30-vertex convex polyhedron**

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[Alexandrov, 1950]

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**Geodesics & Closed Geodesics**

Geodesic: locally shortest path; straightest lines on surface Simple geodesic: non-self-intersecting Simple, closed geodesic: Closed geodesic: returns to start w/o corner Geodesic loop: returns to start at corner (closed geodesic = simple, closed geodesic)

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**Lyusternick-Schnirelmann Theorem**

Theorem: Every closed surface homeomorphic to a sphere has at least three, distinct closed geodesics. Birkoff 1927: at least one closed geodesic LS 1929: at least three “gaps” filled in 1978 [BTZ83] Pogorelov 1949: extended to polyhedral surfaces

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**Quasigeodesic Aleksandrov 1948**

left(p) = total incident face angle from left quasigeodesic: curve s.t. left(p) ≤ right(p) ≤ at each point p of curve.

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Closed Quasigeodesic [Lysyanskaya, O’Rourke 1996]

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**Grayson Curve Shortening**

[Lysyanskaya, O’Rourke 1996]

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**Open: Find a Closed Quasigeodesic**

Is there an algorithm polynomial time or efficient numerical algorithm for finding a closed quasigeodesic on a (convex) polyhedron?

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**Exponential Number of Closed Geodesics**

Theorem: 2(n) distinct closed quasigeodesics. [Aronov & JOR 2002]

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**Status of main questions**

Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

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**Edge-Ununfoldable Orthogonal Polyhedra**

[Biedl, Demaine, Demaine, Lubiw, JOR, Overmars, Robbins, Whitesides, ‘98]

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Spiked Tetrahedron [Tarasov ‘99] [Grünbaum ‘01] [Bern, Demaine, Eppstein, Kuo ’99]

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**Unfoldability of Spiked Tetrahedron**

(BDEKMS ’99) Theorem: Spiked tetrahedron is edge-ununfoldable

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**Overlapping Star-Unfolding**

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**Outline What is an unfolding? Why study? Main questions: status**

Convex: edge unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Orthogonal polyhedra

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**Status of main questions**

Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

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**Overlapping Source Unfolding**

[Kineva, JOR 2000]

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**Status of main questions**

Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible Orthogonal polyhedra

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**Orthogonal Polygon / Polyhedron**

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**Grid refinement: Orthogonal Polyhedra**

[DIL04] [DM04]

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Types of Unfoldings

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Gridding Hierarchy

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Edge-Unf Vertex-Unf Not always possible

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Orthogonal Terrain

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**Four algorithmic techniques**

Strip/Staircase unfoldings Recursive unfoldings Spiraling unfoldings Nested spirals

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**Four algorithmic techniques**

Strip/Staircase unfoldings Recursive unfoldings Spiraling unfoldings Nested spirals 4x5 grid unfolding Manhattan towers [Damian, Flatland, JOR ’05]

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Edge-Unf Vertex-Unf Not always possible All genus-0 o-polyhedra

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Manhattan Tower

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Base & Recursion Tree

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Single Box Unfolding

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Suturing two spirals

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**Suture Animation rdsuture_4x.wmv**

Animation by Robin Flatland & Ray Navarette, Siena College

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**Four algorithmic techniques**

Strip/Staircase unfoldings Recursive unfoldings Spiraling unfoldings Nested spirals -unfolding orthogonal polyhedra [Damian, Flatland, JOR ’06b]

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Edge-Unf Vertex-Unf Not always possible All genus-0 o-polyhedra

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Dents vs. Protrusions

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**Band unfolding tree (for extrusion)**

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**Visiting front children**

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**Visiting back children**

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**Retrace entire path back to entrance point**

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Deeper recursion

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**All strips thickened to fill surface**

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4-block example b0b1b2b3b4

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Result Arbitrary genus-0 Orthogonal Polyhedra have a general unfolding into one piece, which may be viewed as a 2n x 2n grid unfolding (and so is, in places, 1/2n thin). Arbitrary (non-orthogonal) polyhedra: still open.

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**Status of main questions**

Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

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