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Page 1: Spatial Embedding of  Pseudo-Triangulations

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!