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Spatial Embedding of Pseudo-Triangulations Peter Braß Institut für Informatik Freie Universität Berlin Berlin, Germany Franz Aurenhammer Hannes Krasser Institute for Theoretical Computer Science Graz University of Technology Graz, Austria Oswin Aichholzer Institute for Software Technology Graz 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. 1994 M.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. 1996 M.Pocchiola, G.Vegter. Topologically sweeping visibility complexes via pseudo-triangulations. 1996 kinetic collision detection P.K.Agarwal, J.Basch, L.J.Guibas, J.Hershberger, L.Zhang. Deformable free space tilings for kinetic collision detection. 2001 D.Kirkpatrick, J.Snoeyink, B.Speckmann. Kinetic collision detection for simple polygons. 2002 D.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. 2000 G.Rote, F.Santos, I.Streinu. Expansive motions and the polytope of pointed pseudo-triangulations. 2001 R.Haas, F.Santos, B.Servatius, D.Souvaine, I.Streinu, W.Whiteley. Planar minimally rigid graphs have pseudo-triangular embeddings. 2002 guarding M.Pocchiola, G.Vegter. On polygon covers. 1999 B.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 projective edges of surface project vertically to edges of graph regular surface 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 surface

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Surface Theorem sketch of proof: pending points: co-planar with 3 corners rigid points: fixed height linear system:

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Surface Theorem 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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Stability

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remove both pending points: no valid pseudo-triangulation

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Stability remove right pending point: no valid pseudo-triangulation

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Stability remove left pending point: status changes

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Stability stable

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Stability not stable

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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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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. 2002 also 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 points h … heights for points in S f * … maximal locally convex function with f * (v i ) ≤ h i for each v i S

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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 special cases: O(n 2 ) surface flips - (P,S) convex: lower convex hull - (P,S) polygon without interior points

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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 to reach optimality 0 0 0 1 1 1

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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 be flipped to regularity with surface flips generalization of situation for Delaunay triangulation (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 not continuous 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 regular triangulations

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Spatial Embedding of Pseudo-Triangulations Thank you!

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