Spatial Embedding of Pseudo-Triangulations

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

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. 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

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.

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

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 viS

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 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 remove edges flip 3: planarizing

Optimality Theorem optimum

Spatial Embedding of Pseudo-Triangulations