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

Pseudo-Triangle

3 corners

non-corners

Pseudo-Triangulation

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

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

Overview

- pseudo-triangulation surfaces- new flip type- locally convex functions

Triangulations

set of points in the plane

assume general position

Triangulations

triangulation in the plane

Triangulations

assign heights to each point

Triangulations

lift points to assigned heights

Triangulations

spatial surface

Triangulations

spatial surface

Projectivity

projectiveedges of surface project vertically to edges of graph

regularsurface is in convex position

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

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.

Surface Theorem

pseudo-triangulation in the plane

Surface Theorem

surface

Surface Theorem

surfacesurface

Surface Theorem

sketch of proof:

pending points: co-planar with 3 corners

rigid points: fixed height

linear system: bzA

Surface Theorem

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rigid points

pending points

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.

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.

Projectivity

not projective edges

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

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.

Surface Flips

Surface Flips

triangulations: tetrahedral flips, Lawson flips

edge-exchangingpoint removing/inserting

Surface Flips

flips in pseudo-triangulations

edge-exchanging, geodesics

Surface Flips

flip reflex edge

Surface Flips

convexifying flip

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

Surface Flips

flip reflex edge

Surface Flips

planarizing flip

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

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

Locally Convex Functions

properties of f *:- unique and piecewise linear- corresponding surface F * projects to a pseudo-triangulation of (P,S‘), S‘S

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.

Optimality Theorem

initial surface

flip

Optimality Theorem

flip 1: convexifying

flip

Optimality Theorem

flip 2: planarizing

flip

Optimality Theorem

flip 3: planarizing

flip

Optimality Theorem

flip 4: convexifyingoptimum

reflex

convex

Optimality Theorem

tetrahedral flips are not sufficient toreach optimality

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Optimality Theorem

initial triangulation

Optimality Theorem

lifted surface

Optimality Theorem

lifted surface

flip

Optimality Theorem

flip 1: planarizing

flip

Optimality Theorem

flip 2: planarizing

flip

Optimality Theorem

flip 3: planarizing

remove edges

Optimality Theorem

optimum

Optimality Theorem

every triangulation surface can beflipped to regularity with surface flips

generalization of situation for Delaunaytriangulation (convex heights)

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

Polytope Representation

convex polytope: all regular pseudo-triangulations constrained by an admissible planar straight-line graph

generalization of associahedron (secondary polytope) for regulartriangulations

Spatial Embedding of Pseudo-Triangulations

Thank you!