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Revisited David Eppstein Elena Mumford Curves as Surface - - PowerPoint PPT Presentation
Revisited David Eppstein Elena Mumford Curves as Surface - - PowerPoint PPT Presentation
Self-overlapping Curves Revisited David Eppstein Elena Mumford Curves as Surface Boundaries Immersion An immersion of a disk D in the plane is a continuous mapping i: D R 2 disk immersed disk in the plane in the plane i i n(p) : n(p)
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Immersion
An immersion of a disk D in the plane is a continuous mapping i: D → R2 in(p): n(p) → i(n(p)) is a homeomorphism. i disk in the plane immersed disk in the plane
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Immersion
An immersion of a disk D in the plane is a continuous mapping i disk in the plane n(p) i(n(p)) n(p) i(n(p)) in(p) immersed disk in the plane i: D → R2 in(p): n(p) → i(n(p)) is a homeomorphism.
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Immersion
i disk boundary in the plane self-intersecting curve in the plane i disk in the plane immersed disk in the plane The image of the boundary of the disk is a (self-intersecting) closed curve.
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Embedding
An embedding of a disk We consider a special type of embeddings:
- ne side of e(D) consistently points up.
e: D → R3 e: D → e(D) is a homeomorphism. e disk in the plane disk embedded in space as a generalized terrain
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Embedding
An embedding of a disk We consider a special type of embeddings:
- ne side of e(D) consistently points up.
e: D → R3 e: D → e(D) is a homeomorphism. e disk in the plane disk embedded in space that is NOT a generalized terrain
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Embedding
An embedding of a disk We consider a special type of embeddings:
- ne side of e(D) consistently points up.
e: D → R3 e: D → e(D) is a homeomorphism. e disk in the plane disk embedded in space that is NOT a generalized terrain
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Examples
By Flickr user Mark Wheeler from http://www.flickr.com/photos/markwheeler/246569058/
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Examples
from http://www.flickr.com/photos/clocky/257469851/ By Flickr user Mark McLaughlin
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Embedding
prz disk embedded in space as a generalized terrain immersed disk in the plane the boundary of a disk embedded in space self-intersecting curve in the plane prz
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Problem Statement
disk immersed in the plane (self-intersecting) curve in the plane disk embedded in space as a generalized terrain
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Problem Statement
disk immersed in the plane (self-intersecting) curve in the plane ? ? ? disk embedded in space as a generalized terrain
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Problem Statement
disk immersed in the plane (self-intersecting) curve in the plane ? ? ? ? disk embedded in space as a generalized terrain
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Bennequin Disk i
An immersed disk that is not a projection of a disk embedded in space
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Bennequin Disk i
An immersed disk that is not a projection of a disk embedded in space
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Bennequin Disk i
An immersed disk that is not a projection of a disk embedded in space
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Bennequin Disk i
An immersed disk that is not a projection of a disk embedded in space
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Problem Statement
disk immersed in the plane (self-intersecting) curve in the plane ? ? ? Whitney(1937) Shor and van Wyk(1992) disk embedded in space as a generalized terrain
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Problem Statement
disk immersed in the plane (self-intersecting) curve in the plane ? ? Whitney(1937) Shor and van Wyk(1992) NP-complete disk embedded in space as a generalized terrain
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Bennequin Disk II
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Bennequin Disk II
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Problem Statement
? disk embedded in space as a generalized terrain disk immersed in the plane (self-intersecting) curve in the plane ? Whitney(1937) Shor and van Wyk(1992) NP-complete
- pen
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Generalize the Problem
disk with a boundary surface (two-dimensional manifold) with a boundary closed curve multiple closed curves
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Disk → Manifold
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Multiple Curves
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Generalize the Problem
surface immersed in the plane multiple (self-intersecting) curve in the plane ? ? ? surface embedded in space as a generalized terrain
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Results
[Shor and van Dyk]
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Lift a Disk
u v w u v w Theorem Lifting an immersed disk is NP-complete. Proof: By reduction from ACYCLIC PARTITION.
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Acyclic Partition
Given: a digraph G = (V, E) Find: partition of V into sets V1 and V2 : G(V1) and G(V2) in G are acyclic. u v w v w u Theorem ACYCLIC PARTITION is NP-complete. Proof: By reduction from PLANAR 3-SAT.
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Lift a Disk
u v w u v w Theorem Lifting an immersed disk is NP-complete. Proof: By reduction from ACYCLIC PARTITION.
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Lift a Disk
u v w u v w u v w vertices: edges: (w, u) w v (v, u) (v, w) u (u, v)
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Lift a Disk
u w
- n the same side of the purple disk
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Lift a Disk
u w
- n different sides of the purple disk
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Lift a Disk
u v
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Lift a Disk
u v w v w u
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Summary
[Shor and van Dyk]
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Cased Curves
A cased curve can be decided in O(min(nk,n+k3)).
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Summary
[Shor and van Dyk]
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