Transcript Gridding Hierarchy - Geometric Folding Algorithms

Unfolding Polyhedral
Surfaces
Joseph O’Rourke
Smith College
Outline
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What is an unfolding?
Why study?
Main questions: status
Convex: edge unfolding
Convex: general unfolding
Nonconvex: edge unfolding
Nonconvex: general unfolding
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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
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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.
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
[Matthew Chadwick]
[Biedl, Lubiw, Sun, 2005]
Outline
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What is an unfolding?
Why study?
Main questions: status
Convex: edge unfolding
Convex: general unfolding
Nonconvex: edge unfolding
Nonconvex: general unfolding
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Orthogonal polyhedra
Lundström Design,
http://www.algonet.se/~ludesign/index.html
Outline
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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
Convex
polyhedra
Nonconvex
polyhedra
Edge
Unfolding?
???
General
Unfolding?
Yes: always
possible
No: not always
possible
???
Status of main questions
Shapes
Convex
polyhedra
Nonconvex
polyhedra
Edge
Unfolding?
???
General
Unfolding?
Yes: always
possible
No: not always
possible
???
Open:
Edge-Unfolding Convex Polyhedra
Does every convex polyhedron have an edgeunfolding 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
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
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Prisms
Prismoids
“Domes”
“Bands”
Radially monotone lattice quadrilaterals
Prismatoids???
Prismoids
Convex top A and bottom B, equiangular.
Edges parallel; lateral faces quadrilaterals.
Overlapping Unfolding
Volcano Unfolding
Unfolding “Domes”
Proof via degree-3 leaf
truncation
dodec.wmv
[Benton, JOR, 2007]
Lattice Quadrilateral Convex Caps
Classes of Convex Polyhedra
with Edge-Unfolding Algorithms
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Prisms
Prismoids
“Domes”
“Bands”
Radially monotone lattice quadrilaterals
Prismatoids???
Unfolding Smooth Prismatoids
[Benbernou, Cahn, JOR 2004]
Open: Fewest Nets
≤F
For a convex polyhedron
of n vertices and F faces,
what is the fewest
≤ (2/3)F
number of nets (simple,
nonoverlapping
≤ (1/2)F
polygons) into which it
may be cut along edges? ≤ (3/8)F
[Spriggs]
[Dujmenovic,
Moran, Wood]
[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
Convex
polyhedra
Nonconvex
polyhedra
Edge
Unfolding?
???
General
Unfolding?
Yes: always
possible
No: not always
possible
???
General Unfoldings of Convex
Polyhedra
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Theorem: Every convex polyhedron has a
general nonoverlapping unfolding (a net).
1) Source unfolding [Sharir & Schorr ’86, Mitchell,
Mount, Papadimitrou ’87]
[Poincare?]
2) Star unfolding [Aronov & JOR ’92]
3) Quasigeodesic unfolding [Itoh, JOR, Vilcu, 2007]
Shortest paths from x to all vertices
Source Unfolding
Star Unfolding
Star-unfolding of 30-vertex convex
polyhedron
[Alexandrov, 1950]
Geodesics & Closed Geodesics
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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
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(closed geodesic = simple, closed geodesic)
Lyusternick-Schnirelmann Theorem
Theorem: Every closed surface homeomorphic to
a sphere has at least three, distinct closed
geodesics.
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Birkoff 1927: at least one closed geodesic
LS 1929: at least three
“gaps” filled in 1978 [BTZ83]
Pogorelov 1949: extended to polyhedral surfaces
Quasigeodesic
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Aleksandrov 1948
left(p) = total incident face angle from left
quasigeodesic: curve s.t.
left(p) ≤ 
 right(p) ≤ 
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at each point p of curve.
Closed Quasigeodesic
[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
Convex
polyhedra
Nonconvex
polyhedra
Edge
Unfolding?
???
General
Unfolding?
Yes: always
possible
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
(BDEKMS ’99)
Tetrahedron
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Theorem: Spiked tetrahedron is
edge-ununfoldable
Overlapping Star-Unfolding
Outline
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




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
Convex
polyhedra
Nonconvex
polyhedra
Edge
Unfolding?
???
General
Unfolding?
Yes: always
possible
No: not always
possible
???
Overlapping Source Unfolding
[Kineva, JOR 2000]
Status of main questions
Shapes
Edge
Unfolding?
General
Unfolding?
???
Yes: always
possible
Nonconvex
polyhedra
No: not always
possible
???
Orthogonal
polyhedra
No: not always
possible
Yes: always
possible
Convex
polyhedra
Orthogonal Polygon / Polyhedron
Grid refinement: Orthogonal Polyhedra
[DIL04] [DM04]
Types of Unfoldings
Gridding Hierarchy
Edge Unfolding
1x1
polycubes/
lattice
k1 x k2
O(1) x O(1)
2O(n) x 2O(n)
Edge-Unf
Not always possible
o-terrains;
o-convex o-stacks
Vertex-Unf
Original polyhedral edges
1x1
o-stacks: 1 x 2
Manhattan towers: 4 x 5
o-stacks
All genus-0 o-polyhedra
polycubes/
lattice
k1 x k2
O(1) x O(1)
All genus-0 o-polyhedra
Not always possible
2O(n) x 2O(n)
Orthogonal Terrain
Four algorithmic techniques
1)
2)
3)
4)
Strip/Staircase unfoldings
Recursive unfoldings
Spiraling unfoldings
Nested spirals
Four algorithmic techniques
1)
2)
3)
Strip/Staircase unfoldings
Recursive unfoldings
Spiraling unfoldings
4x5 grid unfolding Manhattan towers
4)
Nested spirals
[Damian, Flatland, JOR ’05]
Edge-Unf
Not always possible
o-terrains
Vertex-Unf
Original polyhedral edges
1x1
o-stacks: 1 x 2
Manhattan towers: 4 x 5
o-stacks
All genus-0 o-polyhedra
polycubes/
lattice
k1 x k2
O(1) x O(1)
All genus-0 o-polyhedra
Not always possible
2O(n) x 2O(n)
Manhattan Tower
Single Box Unfolding
Suturing two spirals
Suture Animation
rdsuture_4x.wmv
Animation by Robin Flatland & Ray Navarette, Siena College
Four algorithmic techniques
1)
2)
3)
4)
Strip/Staircase unfoldings
Recursive unfoldings
Spiraling unfoldings
Nested spirals
-unfolding orthogonal polyhedra
[Damian, Flatland, JOR ’06b]
Edge-Unf
Not always possible
o-terrains
Vertex-Unf
Original polyhedral edges
1x1
o-stacks: 1 x 2
Manhattan towers: 4 x 5
o-stacks
All genus-0 o-polyhedra
polycubes/
lattice
k1 x k2
O(1) x O(1)
All genus-0 o-polyhedra
Not always possible
2O(n) x 2O(n)
Band unfolding tree (for extrusion)
Visiting front children
Visiting back children
Retrace entire path back to entrance point
Deeper recursion
4-block example
b0b1b2b3b4
Result
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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).
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Arbitrary (non-orthogonal) polyhedra: still open.
Status of main questions
Shapes
Convex
polyhedra
Nonconvex
polyhedra
Edge
Unfolding?
???
General
Unfolding?
Yes: always
possible
No: not always
possible
???