# Unfolding Polyhedral Surfaces

## Presentation on theme: "Unfolding Polyhedral Surfaces"— Presentation transcript:

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? 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.

Cube with one corner truncated

“Sliver” Tetrahedron

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.

Polygons: Simple vs. Weakly Simple

Nonsimple Polygons

Andrea Mantler example

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

[Biedl, Lubiw, Sun, 2005]

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

Status of main questions
Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

Status of main questions
Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

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:

Archimedian Solids [Eric Weisstein]

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

Sclickenrieder2: flat-spanning-tree-unfold

Sclickenrieder3: rightmost-ascending-edge-unfold

Sclickenrieder4: normal-order-unfold

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

Classes of Convex Polyhedra with Edge-Unfolding Algorithms

Prismoids Convex top A and bottom B, equiangular.

Overlapping Unfolding

Volcano Unfolding

Unfolding “Domes”

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

Classes of Convex Polyhedra with Edge-Unfolding Algorithms

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

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

Status of main questions
Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

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?]

Shortest paths from x to all vertices

Source Unfolding

Star Unfolding

Star-unfolding of 30-vertex convex polyhedron

[Alexandrov, 1950]

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)

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

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.

Closed Quasigeodesic [Lysyanskaya, O’Rourke 1996]

Grayson Curve Shortening
[Lysyanskaya, O’Rourke 1996]

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

Exponential Number of Closed Geodesics
Theorem: 2(n) distinct closed quasigeodesics. [Aronov & JOR 2002]

Status of main questions
Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

Edge-Ununfoldable Orthogonal Polyhedra
[Biedl, Demaine, Demaine, Lubiw, JOR, Overmars, Robbins, Whitesides, ‘98]

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

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

Overlapping Star-Unfolding

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

Status of main questions
Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

Overlapping Source Unfolding
[Kineva, JOR 2000]

Status of main questions
Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible Orthogonal polyhedra

Orthogonal Polygon / Polyhedron

Grid refinement: Orthogonal Polyhedra
[DIL04] [DM04]

Types of Unfoldings

Gridding Hierarchy

Edge-Unf Vertex-Unf Not always possible

Orthogonal Terrain

Four algorithmic techniques
Strip/Staircase unfoldings Recursive unfoldings Spiraling unfoldings Nested spirals

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

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

Manhattan Tower

Base & Recursion Tree

Single Box Unfolding

Suturing two spirals

Suture Animation rdsuture_4x.wmv
Animation by Robin Flatland & Ray Navarette, Siena College

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

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

Dents vs. Protrusions

Band unfolding tree (for extrusion)

Visiting front children

Visiting back children

Retrace entire path back to entrance point

Deeper recursion

All strips thickened to fill surface

4-block example b0b1b2b3b4

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.

Status of main questions
Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible

Similar presentations