unfolding polyhedral surfaces joseph orourke smith college

Unfolding Unfolding Polyhedral Polyhedral SurfacesSurfacesJoseph O’RourkeJoseph O’Rourke

Smith CollegeSmith College

OutlineOutline

What is an unfolding?What is an unfolding? Why study?Why study? Main questions: statusMain questions: status Convex: edge unfoldingConvex: edge unfolding Convex: general unfoldingConvex: general unfolding Nonconvex: edge unfoldingNonconvex: edge unfolding Nonconvex: general unfoldingNonconvex: general unfolding

Orthogonal polyhedraOrthogonal polyhedra

ten1.mov

What is an unfolding?What is an unfolding?

Cut surface and unfold to a Cut surface and unfold to a single nonoverlapping single nonoverlapping piece in the plane.piece in the plane.


Cut surfaceCut surface and unfold to a and unfold to a single nonoverlapping single nonoverlapping piece in the plane.piece in the plane.

Unfolding PolyhedraUnfolding Polyhedra Two types of unfoldings:Two types of unfoldings:

EdgeEdge unfoldings unfoldings: Cut only along edges: Cut only along edges GeneralGeneral unfoldings unfoldings: Cut through faces too: Cut through faces too


Cut surface and unfold to Cut surface and unfold to a a single nonoverlapping single nonoverlapping piecepiece in the plane. in the plane.

Cube with one corner Cube with one corner truncatedtruncated

““Sliver” TetrahedronSliver” Tetrahedron

Cut Edges form Cut Edges form Spanning TreeSpanning Tree

Lemma: The cut edges of an edge Lemma: The cut edges of an edge unfolding of a convex polyhedron unfolding of a convex polyhedron to to a simple polygona simple polygon form a spanning form a spanning tree of the 1-skeleton of the tree of the 1-skeleton of the polyhedron.polyhedron.

Polygons: Simple vs. Polygons: Simple vs. Weakly SimpleWeakly Simple

Nonsimple PolygonsNonsimple Polygons

Andrea Mantler exampleAndrea Mantler example

Cut edges: strengtheningCut edges: strengthening

Lemma: The cut edges of an edge Lemma: The cut edges of an edge unfolding of a unfolding of a convexconvex polyhedron polyhedron to to a single, connected piecea single, connected piece form a form a spanning tree of the 1-skeleton of spanning tree of the 1-skeleton of the polyhedron.the polyhedron.

[Bern, Demaine, Eppstein, Kuo, Mantler, O’Rourke, [Bern, Demaine, Eppstein, Kuo, Mantler, O’Rourke, Snoeyink 01]Snoeyink 01]


Cut surface and Cut surface and unfoldunfold to a to a single nonoverlapping single nonoverlapping piece piece in the planein the plane..

Cuboctahedron unfoldingCuboctahedron unfolding[Matthew Chadwick][Matthew Chadwick]

[Biedl, Lubiw, Sun, 2005]

OutlineOutline



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

OutlineOutline



Status of main questionsStatus of main questions

ShapesShapes Edge Edge Unfolding?Unfolding?

General General Unfolding?Unfolding?

Convex Convex polyhedrapolyhedra

?????? Yes: always Yes: always possiblepossible

Nonconvex Nonconvex polyhedrapolyhedra

No: not No: not always always

possiblepossible

??????


ShapesShapes Edge Edge UnfoldingUnfolding

??






possiblepossible

??????

Open: Open: Edge-Unfolding Convex Edge-Unfolding Convex

PolyhedraPolyhedra

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

[Shephard, 1975]

Albrecht DAlbrecht Düürer, 1425rer, 1425

Melancholia I

Albrecht DAlbrecht Düürer, 1425rer, 1425

Snub Cube

Unfolding the Platonic Unfolding the Platonic SolidsSolids

Some nets:http://www.cs.washington.edu/homes/dougz/polyhedra/

Archimedian SolidsArchimedian Solids [Eric Weisstein]

SclickenriederSclickenrieder11::steepest-edge-unfoldsteepest-edge-unfold

“Nets of Polyhedra”TU Berlin, 1997

SclickenriederSclickenrieder22::flat-spanning-tree-unfoldflat-spanning-tree-unfold

SclickenriederSclickenrieder33::rightmost-ascending-edge-rightmost-ascending-edge-

unfoldunfold

SclickenriederSclickenrieder44::normal-order-unfoldnormal-order-unfold

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

Classes of Convex Classes of Convex PolyhedraPolyhedra

with Edge-Unfolding with Edge-Unfolding AlgorithmsAlgorithms PrismsPrisms

PrismoidsPrismoids ““Domes”Domes” ““Bands”Bands” Radially monotone lattice Radially monotone lattice

quadrilateralsquadrilaterals Prismatoids???Prismatoids???

PrismoidsPrismoids

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

Overlapping UnfoldingOverlapping Unfolding

Volcano UnfoldingVolcano Unfolding

Unfolding “Domes”Unfolding “Domes”

Proof via degree-3 leaf Proof via degree-3 leaf truncationtruncation

[Benton, JOR, 2007]dodec.wmv

Lattice Quadrilateral Lattice Quadrilateral Convex CapsConvex Caps

Classes of Convex Classes of Convex PolyhedraPolyhedra

with Edge-Unfolding with Edge-Unfolding AlgorithmsAlgorithms PrismsPrisms

PrismoidsPrismoids ““Domes”Domes” ““Bands”Bands” Radially monotone lattice Radially monotone lattice

quadrilateralsquadrilaterals Prismatoids???Prismatoids???

Unfolding Smooth Unfolding Smooth PrismatoidsPrismatoids

[Benbernou, Cahn, JOR 2004]

Open: Fewest NetsOpen: Fewest Nets

For a convex For a convex polyhedron of polyhedron of nn vertices and vertices and FF faces, what is the faces, what is the fewest number of fewest number of nets (simple, nets (simple, nonoverlapping nonoverlapping polygons) into polygons) into which it may be cut which it may be cut along edges?along edges?

≤ F

≤ (2/3)F [Spriggs][Spriggs]

≤ (1/2)F [Dujmenovi[Dujmenovic, Moran, c, Moran, Wood]Wood]

≤ (3/8)F [Pincu, [Pincu, 2007]2007]

OutlineOutline





General General UnfoldingUnfolding

??





possiblepossible

??????

TheoremTheorem: Every : Every convexconvex polyhedron polyhedron has a has a generalgeneral nonoverlapping nonoverlapping unfolding (a net).unfolding (a net).

General Unfoldings of General Unfoldings of Convex PolyhedraConvex Polyhedra

1)1) Source unfolding [Sharir & Schorr ’86, Source unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87]Mitchell, Mount, Papadimitrou ’87]

2)2) Star unfolding [Aronov & JOR ’92]Star unfolding [Aronov & JOR ’92]

3)3) Quasigeodesic unfolding [Itoh, JOR, Quasigeodesic unfolding [Itoh, JOR, Vilcu, 2007]Vilcu, 2007]

[Poincare?]

Shortest paths from x to all Shortest paths from x to all verticesvertices

Source UnfoldingSource Unfolding

Star UnfoldingStar Unfolding

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

[Alexandrov, 1950]

Geodesics & Closed Geodesics & Closed GeodesicsGeodesics

GeodesicGeodesic: locally shortest path; : locally shortest path; straightest lines on surfacestraightest lines on surface

Simple geodesicSimple geodesic: non-self-intersecting: non-self-intersecting Simple, Simple, closed geodesicclosed geodesic::

Closed geodesic: returns to start w/o cornerClosed geodesic: returns to start w/o corner Geodesic loop: returns to start at cornerGeodesic loop: returns to start at corner

(closed geodesic = simple, closed (closed geodesic = simple, closed geodesic)geodesic)

Lyusternick-Schnirelmann Lyusternick-Schnirelmann TheoremTheorem

Theorem:Theorem: Every closed surface Every closed surface homeomorphic to a sphere has at least homeomorphic to a sphere has at least three, distinct closed geodesics.three, distinct closed geodesics.

Birkoff 1927: at least one closed geodesicBirkoff 1927: at least one closed geodesic LS 1929: at least threeLS 1929: at least three ““gaps” filled in 1978 [BTZ83]gaps” filled in 1978 [BTZ83] Pogorelov 1949: extended to polyhedral Pogorelov 1949: extended to polyhedral

surfacessurfaces

QuasigeodesicQuasigeodesic

Aleksandrov 1948Aleksandrov 1948 left(p) = total incident face angle left(p) = total incident face angle

from leftfrom left quasigeodesic: curve s.t. quasigeodesic: curve s.t.

left(p) ≤ left(p) ≤ right(p) ≤ right(p) ≤

at each point p of curve.at each point p of curve.

Closed QuasigeodesicClosed Quasigeodesic

[Lysyanskaya, O’Rourke 1996]

Open: Find a Closed Open: Find a Closed QuasigeodesicQuasigeodesic

Is there an algorithmIs there an algorithmpolynomial timepolynomial time

or efficient numerical algorithmor efficient numerical algorithm

for finding a closed quasigeodesic on a for finding a closed quasigeodesic on a (convex) polyhedron?(convex) polyhedron?

Exponential Number of Exponential Number of Closed GeodesicsClosed Geodesics

Theorem: 2(n) distinct closed quasigeodesics.

[Aronov & JOR 2002]


ShapesShapes Edge Edge UnfoldingUnfolding

??




NonconveNonconvex x polyhedrapolyhedra


possiblepossible

??????

Edge-Ununfoldable Orthogonal Edge-Ununfoldable Orthogonal PolyhedraPolyhedra

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

Spiked TetrahedronSpiked Tetrahedron

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

Unfoldability of Spiked Unfoldability of Spiked TetrahedronTetrahedron

TheoremTheorem: Spiked tetrahedron is: Spiked tetrahedron isedge-ununfoldableedge-ununfoldable

(BDEKMS ’99)(BDEKMS ’99)

Overlapping Star-Overlapping Star-UnfoldingUnfolding

OutlineOutline










possiblepossible

??????

Overlapping Source Overlapping Source UnfoldingUnfolding

[Kineva, JOR 2000]

Status of main questionsStatus of main questionsShapesShapes Edge Edge

Unfolding?Unfolding?General General

UnfoldingUnfolding??





possiblepossible

??????

OrthogonOrthogonal al

polyhedrapolyhedra


possiblepossible

Yes: always Yes: always possiblepossible

Orthogonal Polygon / Orthogonal Polygon / PolyhedronPolyhedron

Grid refinement: Orthogonal Grid refinement: Orthogonal PolyhedraPolyhedra

[DIL04] [DM04]

Types of UnfoldingsTypes of Unfoldings

Gridding HierarchyGridding Hierarchy

Edge Unfolding

1 x 1

k1 x k2

polycubes/lattice

O(1) x O(1)

2O(n) x 2O(n)

Original polyhedral edges

1 x 1

k1 x k2

polycubes/lattice

O(1) x O(1)

2O(n) x 2O(n)

All genus-0 o-polyhedrao-terrains;o-convex o-stacks

All genus-0 o-polyhedra

o-stacks: 1 x 2Manhattan towers: 4 x 5

o-stacks

Not always possibleNot always possible

Edge-Edge-UnfUnf

Vertex-Vertex-UnfUnf

Orthogonal TerrainOrthogonal Terrain

Four algorithmic Four algorithmic techniquestechniques

1)1) Strip/Staircase unfoldingsStrip/Staircase unfoldings

2)2) Recursive unfoldingsRecursive unfoldings

3)3) Spiraling unfoldingsSpiraling unfoldings

4)4) Nested spiralsNested spirals





4)4) Nested spiralsNested spirals[Damian, Flatland, JOR ’05]

4x5 grid unfolding Manhattan towers


1 x 1

k1 x k2

polycubes/lattice

O(1) x O(1)

2O(n) x 2O(n)

All genus-0 o-polyhedrao-terrains


o-stacks


Edge-Edge-UnfUnf



Manhattan TowerManhattan Tower

Single Box UnfoldingSingle Box Unfolding

Suturing two spiralsSuturing two spirals

Suture AnimationSuture Animation

Animation by Robin Flatland & Ray Navarette, Siena College

rdsuture_4x.wmv





4)4) Nested spiralsNested spirals

[Damian, Flatland, JOR ’06b]

-unfolding orthogonal polyhedra


1 x 1

k1 x k2

polycubes/lattice

O(1) x O(1)

2O(n) x 2O(n)

All genus-0 o-polyhedrao-terrains


o-stacks


Edge-Edge-UnfUnf



Band unfolding tree (for Band unfolding tree (for extrusion)extrusion)

Visiting front childrenVisiting front children

Visiting back childrenVisiting back children

Retrace entire path back to Retrace entire path back to entrance pointentrance point

Deeper recursionDeeper recursion

4-block 4-block exampleexample

b0b1b2b3b4

ResultResult

Arbitrary genus-0 Orthogonal Arbitrary genus-0 Orthogonal Polyhedra have a general unfolding Polyhedra have a general unfolding into one piece,into one piece, which may be viewed as a 2which may be viewed as a 2nn x 2 x 2nn grid grid

unfoldingunfolding (and so is, in places, 1/2(and so is, in places, 1/2nn thin). thin).

Arbitrary (non-orthogonal) polyhedra: Arbitrary (non-orthogonal) polyhedra: still open.still open.








possiblepossible

??????

unfolding polyhedral surfaces joseph orourke smith college

Documents

general unfolding convex

volcano unfolding slide

overlapping unfolding

general unfolding nonconvex

unfolding domes

status convex

weakly simple slide

prismoids convex