Folding%20 - PowerPoint PPT Presentation

About This Presentation
Title:

Folding%20

Description:

Cut along the surface of a polyhedron. Unfold into a simple planar polygon without overlap ... Cut Edges form Spanning Tree. Lemma: The cut edges of an edge ... – PowerPoint PPT presentation

Number of Views:249
Avg rating:3.0/5.0
Slides: 47
Provided by: ClarkScie9
Learn more at: http://gfalop.org
Category:
Tags: cut | folding

less

Transcript and Presenter's Notes

Title: Folding%20


1
Folding Unfolding in Computational
GeometryIntroduction
Joseph ORourke Smith College (Many slides made
by Erik Demaine)
2
Folding and Unfolding in Computational Geometry
  • 1D Linkages
  • Preserve edge lengths
  • Edges cannot cross
  • 2D Paper
  • Preserve distances
  • Cannot cross itself
  • 3D Polyhedra
  • Cut the surface while keeping it connected

3
Characteristics
  • Tangible
  • Applicable
  • Elementary
  • Deep
  • Frontier Accessible

4
Outline
  • Topics
  • 1D Linkages
  • 2D Paper
  • 3D Polyhedra

5
Lectures Schedule
Sunday 730-830 0 Introduction and Overview
Monday 900-950 1 Part Ia Linkages and Universality
Monday 1000-1050 2 Part Ib Pantographs and Pop-ups
Monday 130-230 Discussion
Monday 240-330 3 Part Ic Locked Chains
Monday 340-430 4 Part IIa Flat Origami
Tuesday 900-950 5 Part IIb One-Cut Theorem
Tuesday 1000-1050 6 Part IIIa Folding Polygons to Polyhedra
Tuesday 130-230 Discussion
Tuesday 240-330 7 Part IIIb Unfolding Polyhedra to Nets
Tuesday 340-430 Guest Lecture Jane Sangwine-Yeager
Wednesday 900-950 8 Part Id Protein Folding Fixed-angle Chains
Wednesday 1000-1050 9 Part Ie Unit-Length Chains Locked?
Thursday 900-950 10 Part IIc Skeletons, Roofs, Medial Axis
Thursday 1000-1050 11 Part IId Medial Axis Models
Friday 900-950 12 Part IIIc Cauchys Rigidity Theorem
Friday 1000-1050 13 Part IIId Bellows, Volume, Reconstruction
6
Outline Tonight
  • Topics
  • 1D Linkages
  • 2D Paper
  • 3D Polyhedra
  • Within each
  • Definitions
  • One application
  • One open problem

7
Outline1 ? 1D Linkages
  • Definitions
  • Configurations
  • Locked chain in 3D
  • Fixed-angle chains
  • Application Protein folding
  • Open Problem unit-length locked chains?

8
Linkages / Frameworks
  • Link / bar / edge line segment
  • Joint / vertex connection between
    endpoints of bars

Closed chain / cycle / polygon
Open chain / arc
Tree
General
9
Configurations
  • Configuration positions of the vertices that
    preserves the bar lengths
  • Non-self-intersecting No bars cross

Non-self-intersecting configurations
Self-intersecting
10
Locked Question
  • Can a linkage be moved between any
    twonon-self-intersecting configurations?
  • Can any non-self-intersecting configuration be
    unfolded, i.e., moved to canonical
    configuration?
  • Equivalent by reversing and concatenating motions

?
11
Canonical Configurations
  • Chains Straight configuration
  • Polygons Convex configurations
  • Trees Flat configurations

12
Locked 3D Chains Cantarella Johnston 1998
Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke,
Overmars, Robbins, Streinu, Toussaint, Whitesides
1999
  • Cannot straighten some chains,
  • even with universal joints.

13
Locked 2D TreesBiedl, Demaine, Demaine, Lazard,
Lubiw, ORourke, Robbins, Streinu, Toussaint,
Whitesides 1998
  • Theorem Not all trees can be flattened
  • No petal can be opened unless all others are
    closed significantly
  • No petal can be closed more than a little unless
    it has already opened

14
Can Chains Lock?
  • Can every chain, with universal joints, be
    straightened?

Chains Straightened?
2D Yes
3D No some locked
4D beyond Yes
Polygonal Chains Cannot Lock in 4D. Roxana
Cocan and J. O'RourkeComput. Geom. Theory Appl.,
20 (2001) 105-129.
15
Open1 Can Equilateral Chains Lock?
  • Does there exist an open polygonal chain embedded
    in 3D, with all links of equal length, that is
    locked?

16
ProteinFolding
17
Protein Folding
18
Fixed-angle chain
19
Flattenable
  • A configuration of a chain if flattenable if it
    can be reconfigured, without self-intersection,
    so that it lies flat in a plane.
  • Otherwise the configuration is unflattenable, or
    locked.

20
Unflattenable fixed-angle chain
21
Open Problems1 Locked Equilateral Chains?
  • Is there a configuration of a chain with
    universal joints, all of whose links have the
    same length, that is locked?
  • Is there a configuration of a 90o fixed-angle
    chain, all of whose links have the same length,
    that is locked?

Perhaps No?
Perhaps Yes for 1e?
22
Outline2 ? 2D Paper
  • Definitions
  • Foldings
  • Crease patterns
  • Application Map Folding
  • Open Problem Complexity of Map Folding

23
Foldings
  • Piece of paper 2D surface
  • Square, or polygon, or polyhedral surface
  • Folded state isometric embedding
  • Isometric preserve intrinsic distances
    (measured alongpaper surface)
  • Embedding no self-intersections exceptthat
    multiple surfacescan touch withinfinitesimal
    separation

Nonflat folding
Flat origami crane
24
Structure of Foldings
  • Creases in folded state discontinuities in the
    derivative
  • Crease pattern planar graph drawn with straight
    edges (creases) on the paper, corresponding
    tounfolded creases
  • Mountain-valleyassignment specifycrease
    directions as? or ?

Nonflat folding
Flat origami crane
25
Map Folding
  • Motivating problem
  • Given a map (grid of unit squares),each crease
    marked mountain or valley
  • Can it be folded into a packet(whose silhouette
    is a unit square)via a sequence of simple folds?
  • Simple fold fold along a line

26
Map Folding
  • Motivating problem
  • Given a map (grid of unit squares),each crease
    marked mountain or valley
  • Can it be folded into a packet(whose silhouette
    is a unit square)via a sequence of simple folds?
  • Simple fold fold along a line

27
Easy?
28
Hard?
29
Map Folding
  • Motivating problem
  • Given a map (grid of unit squares),each crease
    marked mountain or valley
  • Can it be folded into a packet(whose silhouette
    is a unit square)via a sequence of simple folds?
  • Simple fold fold along a line

6
7
1
30
Map Folding
  • Motivating problem
  • Given a map (grid of unit squares),each crease
    marked mountain or valley
  • Can it be folded into a packet(whose silhouette
    is a unit square)via a sequence of simple folds?
  • Simple fold fold along a line

7
6
1
31
Map Folding
  • Motivating problem
  • Given a map (grid of unit squares),each crease
    marked mountain or valley
  • Can it be folded into a packet(whose silhouette
    is a unit square)via a sequence of simple folds?
  • Simple fold fold along a line

1
7
6
32
Map Folding
  • Motivating problem
  • Given a map (grid of unit squares),each crease
    marked mountain or valley
  • Can it be folded into a packet(whose silhouette
    is a unit square)via a sequence of simple folds?
  • Simple fold fold along a line

7
6
33
Map Folding
  • Motivating problem
  • Given a map (grid of unit squares),each crease
    marked mountain or valley
  • Can it be folded into a packet(whose silhouette
    is a unit square)via a sequence of simple folds?
  • Simple fold fold along a line

9
6
  • More generally Given an arbitrary crease
    pattern, is it flat-foldable by simple folds?

34
Open2 Map Folding Complexity?
  • Given a rectangular map, with designated
    mountain/valley folds in a regular grid pattern,
    how difficult is it to decide if there is a
    folded state of the map realizing those crease
    patterns?

35
Outline3 ? 3D Polyhedra
  • Edge-Unfolding
  • Definitions
  • Cut tree spanning tree
  • Net
  • Applications Manufacturing
  • Open Problem Does every polyhedron have a net?

36
Unfolding Polyhedra
  • Cut along the surface of a polyhedron
  • Unfold into a simple planar polygon without
    overlap

37
Edge Unfoldings
  • Two types of unfoldings
  • Edge unfoldings Cut only along edges
  • General unfoldings Cut through faces too

38
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.
  • spanning to flatten every vertex
  • forest cycle would isolate a surface piece
  • tree connected by boundary of polygon

39
Commercial Software
Lundström Design, http//www.algonet.se/ludesign/
index.html
40
Open3 Edge-Unfolding Convex Polyhedra
  • Does every convex polyhedron have an
    edge-unfolding to a net (a simple, nonoverlapping
    polygon)?

Shephard, 1975
41
Archimedian Solids
42
Nets for Archimedian Solids
43
Cube with one corner truncated
44
Sclickenrieder1steepest-edge-unfold
Nets of Polyhedra TU Berlin, 1997
45
Sclickenrieder3rightmost-ascending-edge-unfold
46
Open3 Edge-Unfolding Convex Polyhedra
  • Does every convex polyhedron have an
    edge-unfolding to a net (a simple, nonoverlapping
    polygon)?

Shephard, 1975
Write a Comment
User Comments (0)
About PowerShow.com