Presentation is loading. Please wait.

Presentation is loading. Please wait.

Unfolding Polyhedral Surfaces Joseph ORourke Smith College.

Similar presentations


Presentation on theme: "Unfolding Polyhedral Surfaces Joseph ORourke Smith College."— Presentation transcript:

1 Unfolding Polyhedral Surfaces Joseph ORourke Smith College

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

3 ten1.mov

4

5 What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

6 What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

7 Unfolding Polyhedra Two types of unfoldings: Two types of unfoldings: Edge unfoldings: Cut only along edges Edge unfoldings: Cut only along edges General unfoldings: Cut through faces too General unfoldings: Cut through faces too

8 What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

9 Cube with one corner truncated

10 Sliver Tetrahedron

11 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.

12 Polygons: Simple vs. Weakly Simple

13 Nonsimple Polygons

14 Andrea Mantler example

15 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, ORourke, Snoeyink 01]

16 What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

17 Cuboctahedron unfolding [Matthew Chadwick]

18 [Biedl, Lubiw, Sun, 2005]

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

20 Lundström Design,

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

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

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

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

25 Albrecht Dürer, 1425 Melancholia I

26 Albrecht Dürer, 1425 Snub Cube

27 Unfolding the Platonic Solids Some nets:

28 Archimedian Solids [Eric Weisstein]

29 Sclickenrieder 1 : steepest-edge-unfold Nets of Polyhedra TU Berlin, 1997

30 Sclickenrieder 2 : flat-spanning-tree-unfold

31 Sclickenrieder 3 : rightmost-ascending-edge-unfold

32 Sclickenrieder 4 : normal-order-unfold

33 Percent Random Unfoldings that Overlap [ORourke, Schevon 1987]

34 Classes of Convex Polyhedra with Edge-Unfolding Algorithms Prisms Prisms Prismoids Prismoids Domes Domes Bands Bands Radially monotone lattice quadrilaterals Radially monotone lattice quadrilaterals Prismatoids??? Prismatoids???

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

36 Overlapping Unfolding

37 Volcano Unfolding

38 Unfolding Domes

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

40 Lattice Quadrilateral Convex Caps

41

42 Classes of Convex Polyhedra with Edge-Unfolding Algorithms Prisms Prisms Prismoids Prismoids Domes Domes Bands Bands Radially monotone lattice quadrilaterals Radially monotone lattice quadrilaterals Prismatoids??? Prismatoids???

43 Unfolding Smooth Prismatoids [Benbernou, Cahn, JOR 2004]

44 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]

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

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

47 Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). General Unfoldings of Convex Polyhedra 1)Source unfolding [Sharir & Schorr 86, Mitchell, Mount, Papadimitrou 87] 2)Star unfolding [Aronov & JOR 92] 3)Quasigeodesic unfolding [Itoh, JOR, Vilcu, 2007] [Poincare?]

48 Shortest paths from x to all vertices

49

50 Source Unfolding

51 Star Unfolding

52 Star-unfolding of 30-vertex convex polyhedron

53

54 [Alexandrov, 1950]

55 Geodesics & Closed Geodesics Geodesic: locally shortest path; straightest lines on surface Geodesic: locally shortest path; straightest lines on surface Simple geodesic: non-self-intersecting Simple geodesic: non-self-intersecting Simple, closed geodesic: Simple, closed geodesic: Closed geodesic: returns to start w/o corner Closed geodesic: returns to start w/o corner Geodesic loop: returns to start at corner Geodesic loop: returns to start at corner (closed geodesic = simple, closed geodesic)

56 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 Birkoff 1927: at least one closed geodesic LS 1929: at least three LS 1929: at least three gaps filled in 1978 [BTZ83] gaps filled in 1978 [BTZ83] Pogorelov 1949: extended to polyhedral surfaces Pogorelov 1949: extended to polyhedral surfaces

57 Quasigeodesic Aleksandrov 1948 Aleksandrov 1948 left(p) = total incident face angle from left left(p) = total incident face angle from 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.

58 Closed Quasigeodesic [Lysyanskaya, ORourke 1996]

59 Grayson Curve Shortening [Lysyanskaya, ORourke 1996]

60

61

62

63

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

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

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

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

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

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

70 Overlapping Star-Unfolding

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

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

73 Overlapping Source Unfolding [Kineva, JOR 2000]

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

75 Orthogonal Polygon / Polyhedron

76 Grid refinement: Orthogonal Polyhedra [DIL04] [DM04]

77 Types of Unfoldings

78 Gridding Hierarchy Edge Unfolding 1 x 1 k1 x k2 O(1) x O(1) 2 O(n) x 2 O(n) polycubes/ lattice

79 All genus-0 o-polyhedra o-terrains; o-convex o-stacks All genus-0 o- polyhedra o-stacks: 1 x 2 Manhattan towers: 4 x 5 o-stacks Not always possible Edge-UnfVertex-Unf

80 Orthogonal Terrain

81 Four algorithmic techniques 1) Strip/Staircase unfoldings 2) Recursive unfoldings 3) Spiraling unfoldings 4) Nested spirals

82 Four algorithmic techniques 1) Strip/Staircase unfoldings 2) Recursive unfoldings 3) Spiraling unfoldings 4) Nested spirals [Damian, Flatland, JOR 05] 4x5 grid unfolding Manhattan towers

83 All genus-0 o-polyhedra o-terrains o-stacks: 1 x 2 Manhattan towers: 4 x 5 o-stacks Not always possible Edge-UnfVertex-Unf All genus-0 o-polyhedra

84 Manhattan Tower

85 Base & Recursion Tree

86 Single Box Unfolding

87 Suturing two spirals

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

89

90 Four algorithmic techniques 1) Strip/Staircase unfoldings 2) Recursive unfoldings 3) Spiraling unfoldings 4) Nested spirals [Damian, Flatland, JOR 06b] -unfolding orthogonal polyhedra

91 All genus-0 o-polyhedra o-terrains o-stacks: 1 x 2 Manhattan towers: 4 x 5 o-stacks Not always possible Edge-UnfVertex-Unf All genus-0 o-polyhedra

92 Dents vs. Protrusions

93 Band unfolding tree (for extrusion)

94 Visiting front children

95 Visiting back children

96 Retrace entire path back to entrance point

97

98 Deeper recursion

99 All strips thickened to fill surface

100 4-block example b 0 b 1 b 2 b 3 b 4

101 Result Arbitrary genus-0 Orthogonal Polyhedra have a general unfolding into one piece, Arbitrary genus-0 Orthogonal Polyhedra have a general unfolding into one piece, which may be viewed as a 2 n x 2 n grid unfolding which may be viewed as a 2 n x 2 n grid unfolding (and so is, in places, 1/2 n thin). (and so is, in places, 1/2 n thin). Arbitrary (non-orthogonal) polyhedra: still open. Arbitrary (non-orthogonal) polyhedra: still open.

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


Download ppt "Unfolding Polyhedral Surfaces Joseph ORourke Smith College."

Similar presentations


Ads by Google