spatial embedding of pseudo-triangulations
DESCRIPTION
Spatial Embedding of Pseudo-Triangulations. Oswin Aichholzer Institute for Software Technology Graz University of Technology Graz, Austria. Franz Aurenhammer Hannes Krasser Institute for Theoretical Computer Science Graz University of Technology Graz, Austria. Peter Braß - PowerPoint PPT PresentationTRANSCRIPT
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Spatial Embedding of Pseudo-Triangulations
Peter BraßInstitut für Informatik
Freie Universität BerlinBerlin, Germany
Franz AurenhammerHannes Krasser
Institute for Theoretical Computer Science
Graz University of TechnologyGraz, Austria
Oswin Aichholzer
Institute for Software TechnologyGraz University of Technology
Graz, Austria
supported by Apart, FWF, DFG
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Pseudo-Triangle
3 corners
non-corners
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Pseudo-Triangulation
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Applications
ray shooting B.Chazelle, H.Edelsbrunner, M.Grigni, L.J.Guibas, J.Hershberger, M.Sharir, J.Snoeyink. Ray shooting in polygons using geodesic triangulations. 1994M.T.Goodrich, R.Tamassia. Dynamic ray shooting and shortest paths in planar subdivisions via balanced geodesic triangulations. 1997
visibility M.Pocchiola, G.Vegter. Minimal tangent visibility graphs. 1996M.Pocchiola, G.Vegter. Topologically sweeping visibility complexes via pseudo-triangulations. 1996
kinetic collision detectionP.K.Agarwal, J.Basch, L.J.Guibas, J.Hershberger, L.Zhang. Deformable free space tilings for kinetic collision detection. 2001D.Kirkpatrick, J.Snoeyink, B.Speckmann. Kinetic collision detection for simple polygons. 2002D.Kirkpatrick, B.Speckmann. Kinetic maintenance of context-sensitive hierarchical representations for disjoint simple polygons. 2002
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Applications
rigidity I.Streinu. A combinatorial approach to planar non-colliding robot arm motion planning. 2000G.Rote, F.Santos, I.Streinu. Expansive motions and the polytope of pointed pseudo-triangulations. 2001R.Haas, F.Santos, B.Servatius, D.Souvaine, I.Streinu, W.Whiteley. Planar minimally rigid graphs have pseudo-triangular embeddings. 2002
guardingM.Pocchiola, G.Vegter. On polygon covers. 1999B.Speckmann, C.D.Toth. Allocating vertex Pi-guards in simple polygons via pseudo-triangulations. 2002
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Overview
- pseudo-triangulation surfaces- new flip type- locally convex functions
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Triangulations
set of points in the plane
assume general position
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Triangulations
triangulation in the plane
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Triangulations
assign heights to each point
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Triangulations
lift points to assigned heights
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Triangulations
spatial surface
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Triangulations
spatial surface
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Projectivity
projectiveedges of surface project vertically to edges of graph
regularsurface is in convex position
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more general: polygon with interior points
Pseudo-Triangulations
set of points in the plane
pending points non-corner in one incident pseudo-triangle
partition points
rigid points corner in all incident pseudo-triangles
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Surface Theorem
Theorem. Let (P,S) be a polygon with interior points, and let PT be any pseudo-triangulation thereof. Let h be a vector assigning a height to each rigid vertex of PT. For each choice of h, there exists a unique polyhedral surface F above P, that respects h and whose edges project vertically to (a subset of) the edges of PT.
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Surface Theorem
pseudo-triangulation in the plane
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Surface Theorem
surface
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Surface Theorem
surfacesurface
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Surface Theorem
sketch of proof:
pending points: co-planar with 3 corners
rigid points: fixed height
linear system: bzA
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Surface Theorem
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rigid points
pending points
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Surface Theorem
Theorem. Let (P,S) be a polygon with interior points, and let PT be any pseudo-triangulation thereof. Let h be a vector assigning a height to each rigid vertex of PT. For each choice of h, there exists a unique polyhedral surface F above P, that respects h and whose edges project vertically to (a subset of) the edges of PT.
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Surface Theorem
Theorem. Let (P,S) be a polygon with interior points, and let PT be any pseudo-triangulation thereof. Let h be a vector assigning a height to each rigid vertex of PT. For each choice of h, there exists a unique polyhedral surface F above P, that respects h and whose edges project vertically to (a subset of) the edges of PT.
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Projectivity
not projective edges
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Projectivity
A pseudo-triangulation is stable if no subset of pending points can be eliminated with their incident edges s.t.
(1) a valid pseudo-triangulation remains
(2) status of each point is unchanged
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Projectivity
Theorem. A pseudo-triangulation PT of (P,S) is projective only if PT is stable. If PT is stable then the point set S can be perturbed (by some arbitrarily small ε) such that PT becomes projective.
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Surface Flips
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Surface Flips
triangulations: tetrahedral flips, Lawson flips
edge-exchangingpoint removing/inserting
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Surface Flips
flips in pseudo-triangulations
edge-exchanging, geodesics
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Surface Flips
flip reflex edge
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Surface Flips
convexifying flip
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Surface Flips
new flip type in pseudo-triangulations
edge-removing/inserting
independently introduced by D.Orden, F.Santos. The polyhedron of non-crossing graphs on a planar point set. 2002also in O. Aichholzer, F. Aurenhammer, and H. Krasser. Adapting (pseudo-) triangulations with a near-linear number of edge flips. WADS 2003
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Surface Flips
flip reflex edge
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Surface Flips
planarizing flip
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Locally Convex Functions
P … polygon in the plane
f … real-valued function with domain P
locally convex function: convex on each line segment interior to P
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Locally Convex Functions
optimization problem:(P,S) … polygon with interior pointsh … heights for points in S
f * … maximal locally convex function with f*(vi) ≤ hi for each viS
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Locally Convex Functions
properties of f *:- unique and piecewise linear- corresponding surface F * projects to a pseudo-triangulation of (P,S‘), S‘S
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Optimality Theorem
Theorem: Let F*(T,h) be a surface obtained from F(T,h) by applying convexifying and planarizing surface flips (in any order) as long as reflex edges do exist. Then F*(T,h)=F*, for any choice of the initial triangulation T. The optimum F* is reached after a finite number of surface flips.
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Optimality Theorem
initial surface
flip
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Optimality Theorem
flip 1: convexifying
flip
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Optimality Theorem
flip 2: planarizing
flip
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Optimality Theorem
flip 3: planarizing
flip
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Optimality Theorem
flip 4: convexifyingoptimum
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reflex
convex
Optimality Theorem
tetrahedral flips are not sufficient toreach optimality
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Optimality Theorem
initial triangulation
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Optimality Theorem
lifted surface
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Optimality Theorem
lifted surface
flip
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Optimality Theorem
flip 1: planarizing
flip
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Optimality Theorem
flip 2: planarizing
flip
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Optimality Theorem
flip 3: planarizing
remove edges
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Optimality Theorem
optimum
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Optimality Theorem
every triangulation surface can beflipped to regularity with surface flips
generalization of situation for Delaunaytriangulation (convex heights)
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admissible planar straight-line graph: each component is connected to the boundary
Constrained Regularity
collection of polygons with interior points
Optimality Theorem:f* piecewise linear, but notcontinuous in general
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Polytope Representation
convex polytope: all regular pseudo-triangulations constrained by an admissible planar straight-line graph
generalization of associahedron (secondary polytope) for regulartriangulations
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Spatial Embedding of Pseudo-Triangulations
Thank you!