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

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 M.T.Goodrich, R.Tamassia. Dynamic ray shooting and shortest paths in planar subdivisions via balanced geodesic triangulations visibility M.Pocchiola, G.Vegter. Minimal tangent visibility graphs M.Pocchiola, G.Vegter. Topologically sweeping visibility complexes via pseudo-triangulations kinetic collision detection P.K.Agarwal, J.Basch, L.J.Guibas, J.Hershberger, L.Zhang. Deformable free space tilings for kinetic collision detection D.Kirkpatrick, J.Snoeyink, B.Speckmann. Kinetic collision detection for simple polygons D.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 G.Rote, F.Santos, I.Streinu. Expansive motions and the polytope of pointed pseudo-triangulations R.Haas, F.Santos, B.Servatius, D.Souvaine, I.Streinu, W.Whiteley. Planar minimally rigid graphs have pseudo-triangular embeddings guarding M.Pocchiola, G.Vegter. On polygon covers B.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 projective edges of surface project vertically to edges of graph regular surface 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 surface

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

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.

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

Stability

remove both pending points: no valid pseudo-triangulation

Stability remove right pending point: no valid pseudo-triangulation

Stability remove left pending point: status changes

Stability  stable

Stability  not stable

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

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

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 to reach optimality

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

Polytope Representation convex polytope: all regular pseudo-triangulations constrained by an admissible planar straight-line graph generalization of associahedron (secondary polytope) for regular triangulations

Spatial Embedding of Pseudo-Triangulations Thank you!