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Spatial Embedding of Pseudo-Triangulations

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## Spatial Embedding of Pseudo-Triangulations

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**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 Institut für Informatik Freie Universität Berlin Berlin, Germany supported by Apart, FWF, DFG**non-corners**3 corners Pseudo-Triangle**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 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 guarding M.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**regular**surface is in convex position Projectivity projective edges of surface project vertically to edges of graph**Pseudo-Triangulations**more general: polygon with interior points set of points in the plane partition points rigid points corner in all incident pseudo-triangles pending points non-corner in one incident pseudo-triangle**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**surface surface**linear system:**Surface Theorem sketch of proof: rigid points: fixed height pending points: co-planar with 3 corners**Surface Theorem**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.**not projective edges**Projectivity**Projectivity**• A pseudo-triangulation is stable • if no subset of pending points can be • eliminated with their incident edges s.t. • 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**triangulations: tetrahedral flips, Lawson flips edge-exchanging point 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 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 edge-removing/inserting**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 points h … 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.**flip**Optimality Theorem initial surface**Optimality Theorem**flip flip 1: convexifying**Optimality Theorem**flip flip 2: planarizing**Optimality Theorem**flip flip 3: planarizing**Optimality Theorem**flip 4: convexifying optimum**reflex**convex Optimality Theorem tetrahedral flips are not sufficient to reach optimality 0 1 1 1 0 0**Optimality Theorem**initial triangulation**Optimality Theorem**lifted surface**Optimality Theorem**flip lifted surface**Optimality Theorem**flip flip 1: planarizing**Optimality Theorem**flip flip 2: planarizing**Optimality Theorem**remove edges flip 3: planarizing**Optimality Theorem**optimum