Download presentation

Presentation is loading. Please wait.

Published byAmia Rice Modified over 3 years ago

1
Folding & Unfolding in Computational Geometry: Introduction Joseph ORourke Smith College (Many slides made by Erik Demaine)

2
Folding and Unfolding in Computational Geometry z1D: Linkages z2D: Paper z3D: Polyhedra yPreserve edge lengths yEdges cannot cross yPreserve distances yCannot cross itself yCut the surface while keeping it connected

3
Characteristics zTangible zApplicable zElementary zDeep zFrontier Accessible

4
Outline Topics: z1D: Linkages z2D: Paper z3D: Polyhedra

5
Lectures Schedule Sunday 7:30-8:300Introduction and Overview Monday 9:00-9:501Part Ia: Linkages and Universality Monday 10:00-10:502Part Ib: Pantographs and Pop-ups Monday 1:30-2:30Discussion Monday 2:40-3:303Part Ic: Locked Chains Monday 3:40-4:304Part IIa: Flat Origami Tuesday 9:00-9:505Part IIb: One-Cut Theorem Tuesday 10:00-10:506Part IIIa: Folding Polygons to Polyhedra Tuesday 1:30-2:30Discussion Tuesday 2:40-3:307Part IIIb: Unfolding Polyhedra to Nets Tuesday 3:40-4:30Guest Lecture: Jane Sangwine-Yeager Wednesday 9:00-9:508Part Id: Protein Folding: Fixed-angle Chains Wednesday 10:00-10:509Part Ie: Unit-Length Chains: Locked? Thursday 9:00-9:5010Part IIc: Skeletons, Roofs, Medial Axis Thursday 10:00-10:5011Part IId: Medial Axis Models Friday 9:00-9:5012Part IIIc: Cauchys Rigidity Theorem Friday 10:00-10:5013Part IIId: Bellows, Volume, Reconstruction

6
Outline: Tonight Topics: z1D: Linkages z2D: Paper z3D: Polyhedra Within each: zDefinitions zOne application zOne open problem

7
Outline 1 1D: Linkages zDefinitions yConfigurations yLocked chain in 3D yFixed-angle chains zApplication: Protein folding zOpen Problem: unit-length locked chains?

8
Linkages / Frameworks zLink / bar / edge = line segment zJoint / vertex = connection between endpoints of bars Closed chain / cycle / polygon Open chain / arc TreeGeneral

9
Configurations zConfiguration = positions of the vertices that preserves the bar lengths Non-self-intersecting configurations Self-intersecting zNon-self-intersecting = No bars cross

10
Locked Question zCan a linkage be moved between any two non-self-intersecting configurations? ? zCan any non-self-intersecting configuration be unfolded, i.e., moved to canonical configuration? yEquivalent by reversing and concatenating motions

11
Canonical Configurations zChains: Straight configuration zPolygons: Convex configurations zTrees: 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 Trees [Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke, Robbins, Streinu, Toussaint, Whitesides 1998] zTheorem: Not all trees can be flattened yNo petal can be opened unless all others are closed significantly yNo petal can be closed more than a little unless it has already opened

14
Can Chains Lock? zCan every chain, with universal joints, be straightened? Chains Straightened? 2DYes 3D No: some locked 4D & beyondYes Polygonal Chains Cannot Lock in 4D. Roxana Cocan and J. O'Rourke Comput. Geom. Theory Appl., 20 (2001) 105-129.

15
Open 1 : Can Equilateral Chains Lock? Does there exist an open polygonal chain embedded in 3D, with all links of equal length, that is locked?

16
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 Problems 1 : Locked Equilateral Chains? (1)Is there a configuration of a chain with universal joints, all of whose links have the same length, that is locked? (2)Is there a configuration of a 90 o fixed- angle chain, all of whose links have the same length, that is locked? Perhaps: No? Perhaps: Yes for 1+ ?

22
Outline 2 2D: Paper zDefinitions yFoldings yCrease patterns zApplication: Map Folding zOpen Problem: Complexity of Map Folding

23
Foldings zPiece of paper = 2D surface ySquare, or polygon, or polyhedral surface zFolded state = isometric embedding yIsometric = preserve intrinsic distances (measured along paper surface) yEmbedding = no self- intersections except that multiple surfaces can touch with infinitesimal separation Flat origami crane Nonflat folding

24
Structure of Foldings zCreases in folded state = discontinuities in the derivative zCrease pattern = planar graph drawn with straight edges (creases) on the paper, corresponding to unfolded creases zMountain-valley assignment = specify crease directions as or Nonflat folding Flat origami crane

25
Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 1 67 2 58 3 49

26
Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 1 67 2 58 3 49

27
Easy?

28
Hard?

29
Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 2 58 3 49 1 67

30
Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 2 58 1 76

31
Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 1 7 6

32
Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 7 6

33
Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 6 9 zMore generally: Given an arbitrary crease pattern, is it flat-foldable by simple folds?

34
Open 2 : 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
Outline 3 3D: Polyhedra zEdge-Unfolding yDefinitions xCut tree: spanning tree xNet yApplications: Manufacturing yOpen Problem: Does every polyhedron have a net?

36
Unfolding Polyhedra zCut along the surface of a polyhedron zUnfold into a simple planar polygon without overlap

37
Edge Unfoldings zTwo types of unfoldings: yEdge unfoldings: Cut only along edges yGeneral 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. o spanning: to flatten every vertex o forest: cycle would isolate a surface piece o tree: connected by boundary of polygon

39
Commercial Software Lundström Design, http://www.algonet.se/~ludesign/index.html http://www.algonet.se/~ludesign/index.html

40
Open 3 : 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
Sclickenrieder 1 : steepest-edge-unfold Nets of Polyhedra TU Berlin, 1997

45
Sclickenrieder 3 : rightmost-ascending-edge-unfold

46
Open 3 : Edge-Unfolding Convex Polyhedra Does every convex polyhedron have an edge- unfolding to a net (a simple, nonoverlapping polygon)? [Shephard, 1975]

Similar presentations

Presentation is loading. Please wait....

OK

12.1 Exploring Solids.

12.1 Exploring Solids.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on limits and continuity examples Ppt on object-oriented programming tutorial Ppt on eddy current test Ppt on sources of energy for class 8th physics Ppt on natural numbers define Ppt on data collection methods quantitative research Ppt on united nations organization Ppt on voice recognition system using matlab Ppt on classical economics books Ppt on conservation of nonrenewable resources