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Delaunay Meshing for Piecewise Smooth Complexes Tamal K. Dey The Ohio State U. Joint work: Siu-Wing Cheng, Joshua Levine, Edgar A. Ramos

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2/22 Department of Computer Science and Engineering Piecewise Smooth Complexes Sharp EdgesNon-manifold

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3/22 Department of Computer Science and Engineering Piecewise Smooth Complexes D is a piecewise smooth complex (PSC) if Each k-dimensional element is a manifold and compact subset of a smooth (C 2 ) k-manifold, 0≤k≤2. The k-th stratum, D k : set of k-dim elements of D. D 0 – vertices, D 1 – 1-faces, D 2 – 2-faces. D ≤k = D 0 … D k. D satisfies usual reqs for being a complex. Interiors of elements are disjoint and for σ D, bd σ D. For any σ, D, either σ = or σ D.

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4/22 Department of Computer Science and Engineering Delaunay refinement : History Chew89, Ruppert92, Shewchuk98 (Linear domains with no small angle) Cohen-Steiner-Verdiere-Yvinec02, Cheng-Dey-Ramos- Ray04 (polyhedral domains with small angle) Chew93 (surface without guarantees) Cheng-Dey-Edelsbrunner-Sullivan01 (skin surfaces) Boissonnat-Oudot03 and Cheng-Dey-Ramos-Ray04 (smooth surface) Boissonnat-Oudot06 (Lipschitz surfaces) Oudot-Rineau-Yvinec06 (Volumes)

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5/22 Department of Computer Science and Engineering Basics of Delaunay Refinement Chew 89, Ruppert 92, Shewchuk 98 Maintain a Delaunay triangulation of the current set of vertices. If some property is not satisfied by the current triangulation, insert a new point which is locally farthest. Burden is on showing that the algorithm terminates (shown by packing argument).

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6/22 Department of Computer Science and Engineering Challenges for PSC Topology Polyhedral case (input conformity,topology trivial). Curved elements (topology is an issue). Topological Ball Property (TBP) was used for smooth manifolds [BO03,CDRR04]. We need extended TBP for nonmanifolds. Nonsmoothness Lipschitz surfaces [BO06], Remeshing [DLR05]. Small angles Delaunay refinement is hard [CP03, CDRR05, PW04].

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7/22 Department of Computer Science and Engineering Topological Ball Property For a weighted point set S, let Vor S and Del S denote the weighted Voronoi and Delaunay diagrams. S has the TBP for σ D i if σ intersects any k-face in Vor S either in emptyset or in a closed topological (i+k-3)-ball.

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8/22 Department of Computer Science and Engineering CW-Complexes A CW-complex R is a collection of closed (topological) balls whose interiors are pairwise disjoint and whose boundaries are the union of other closed balls in R. Our algorithm builds a CW-complex, Vor S| |D|, to satisfy an extended TBP[ES97].

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9/22 Department of Computer Science and Engineering Extended TBP S |D| has the extended TBP (eTBP) for D if there is a CW- complex R with |R| = |D| s.t. (C1) The restricted Voronoi face F |D| is the underlying space of a CW-complex R’ R. (C2) The closed balls in R’ are incident to a unique closed ball b F R. (C3) If b F is a j-ball then b F bd F is a (j-1)-sphere. (C4) Each k-ball in R’, except b F, intersects bd F in a (k-1)-ball.

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10/22 Department of Computer Science and Engineering Extended TBP For a 1- or 2-face σ, let Del S| σ denote the Delaunay subcomplex restricted to σ. Del S| |D i | = σ D i Del S| σ. Del S| |D| = σ D Del S| σ. Theorem. If S has the eTBP for D then the underlying space of Del S| |D| is homeomorphic to |D| [ES97].

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11/22 Department of Computer Science and Engineering Feature Size For analysis, we require a feature size which is 1- Lipschitz and non-zero. For any x |D|, let f(x) = min{m(x), g(x)}. For any σ D, f() is 1-Lipschitz over int σ. For δ (0,1] and x |D|, if x D 0, lfs δ (x) = δf(x). if x int |D i |, for i ≥ 1, lfs δ (x) = max{δf(x), max y bd|D i | {lfs δ (y)-||x-y||}}.

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12/22 Department of Computer Science and Engineering Protecting D 1 1.Any 2 adjacent balls on a 1-face must overlap significantly without containing each others centers 2.No 3 balls have a common intersection 3.For a point p σ D 1, if we enlarge any protecting ball B p by a factor c ≤ 8, forming B’: 1.B’ intersects σ in a single curve, and intersects all D 2 adjacent to σ in a topological disk. 2.For any q in B’ σ, the tangent variation between p and q is bounded. 3.For any q in B’ ( D 2 adjacent to σ), the normal variation between p and q is bounded.

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13/22 Department of Computer Science and Engineering Admissible Point Sets Protecting balls are turned into weighted points We call a point set S admissible if S contains all weighted points placed on D 1. Other points in S are unweighted and they lie outside of the protecting balls (the weighted points). We maintain an admissible point set at each step of the algorithm.

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14/22 Department of Computer Science and Engineering D 1 conformation Lemma. Let S is an admissible point set. For a 1-face σ, if p and q are adjacent weighted vertices spanning segment σ pq on σ then V pq is the only Voronoi facet which intersects σ pq and it does so exactly once.

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15/22 Department of Computer Science and Engineering Meshing PSCs Meshing algorithm uses four tests to detect eTBP violations. Upon violation, we insert points outside of protected balls of weighted vertices.

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16/22 Department of Computer Science and Engineering Test 1: Multi-Intersection(q,σ) For a point q S on a 2- face σ, find a triangle t Del S| σ incident to q s.t. V t intersects σ multiple times. If no t exists, return null, otherwise return the furthest (weighted) intersection point from q.

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17/22 Department of Computer Science and Engineering Test 2: Normal-Deviation(q,σ,Θ) For a point q S on a 2-face σ, check n σ (p), n σ (q) < Θ for all points p V q | σ. 2ω ≤ Θ ≤ /6. If so return null. Otherwise return a point p where n σ (p), n σ (q) = Θ.

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18/22 Department of Computer Science and Engineering Test 3: Infringement(q,σ) For q S σ, return null if q is not infringed, otherwise let pq be the infringing edge. If the boundary edges of V pq intersect int σ, return any intersection point. Else, V pq σ is a collection of closed curves, return a critical point of V pq σ in a direction parallel to V pq. We say q is infringed w.r.t. σ if σ is a 2-face containing q s.t. pq Del S| σ for some p σ. σ is a 2-face and there is a 1-face in bd σ containing q and a non- adjacent vertex p s.t. pq Del S| σ.

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19/22 Department of Computer Science and Engineering Test 4: No-Disk(q,σ) If the star of q in Del S| σ is a topological disk, return null. Otherwise, find the triangle t Del S| σ incident to q which has the furthest (weighted) intersection point in V t | σ from q and return the intersection point.

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20/22 Department of Computer Science and Engineering Meshing Algorithm 1.Protect elements in D ≤1 with weighted points. Insert a point in each element of D 2 outside of protected regions. Let S be this point set. 2.For any σ D 2 and point q S σ: If Infringed(q,σ), Multi-Intersection(q,σ), Normal-Deviation(q,σ,Θ), or No-Disk(q,σ) (checked in that order) return a point x, insert x into S. 3.Repeat 2. until no points are inserted. 4.Return Del S| D.

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21/22 Department of Computer Science and Engineering Admissibility is Invariant Lemma. The algorithm never attempts to insert a point in any protecting ball Since no 3 weighted points intersect, all surface points (intersections of dual Voronoi edges and D) lie outside of every protecting ball

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22/22 Department of Computer Science and Engineering Initialization The algorithm must initialize with a few points from each patch in D 2 Otherwise, components can be missed.

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23/22 Department of Computer Science and Engineering Termination Each point x inserted is Ω(lfs δ (x)) away from all other points. Standard packing argument follows.

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24/22 Department of Computer Science and Engineering Topology Preservation To satisfy C1-C4 of eTBP, we show each Voronoi k-face F = V p 1 … V p (4-k) has: (P1) If F σ ≠ , for σ D j, the intersection is a (k+j-3)-ball (P2) There is a unique lowest dimensional σ F s.t. p 1, …, p (4-k) σ F. (P3) F intersects σ F and only incident elements of σ F. Theorem. If S satisfies P1-P3 then S satisfies C1-C4 of eTBP.

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25/22 Department of Computer Science and Engineering Feature Preservation h:|D| |Del S| D | can be constructed which respects each D i [ES97]. Thus h i :|D i | |Del S| D i | also a homeomorphism with vertex restrictions, ensuring that the nonsmooth features are preserved.

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Delaunay Refinement made practical for PSCs S.-W. Cheng, Tamal K. Dey, Joshua Levine

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27/22 Department of Computer Science and Engineering Definitions For a patch σ D i, When sampled with S Del S| σ is the Delaunay subcomplex restricted to σ Skl i S| σ is the i-dimensional subcomplex of Del S| σ, Skl i S| σ = closure { t | t Del S| σ is an i-simplex} Skl i S| Di = σ Di Skl i S| σ

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28/22 Department of Computer Science and Engineering Disk Condition For a point p on a 2-face σ, Umb D (p) is the set of triangles in Skl 2 S| D2 incident to p. Umb σ (p) is the set of triangles in Skl 2 S| σ incident to p. Disk_Condition(p) requires: i.Umb D (p) = σ, p σ Umb σ (p) ii.For each σ containing p, Umb σ (p) is a 2-disk where p is in the interior iff p int σ

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29/22 Department of Computer Science and Engineering Meshing Algorithm DelPSC(D, r) 1.Protect elements of D ≤1. 2.Mesh2Complex – Repeatedly insert surface points for triangles in Skl 2 S| σ for some σ if either 1.Disk_Condition(p) violated for p σ, or 2.A triangle has orthoradius > r. 3.Mesh3Complex – Repeatedly insert orthocenters of tetrahedra in Skl 3 S| σ for some σ if 1.A tetrahedra has orthoradius > r and its orthocenter does not encroach any surface triangle in Skl 2 S| D2. 4.Return i Skl i S| Di.

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30/22 Department of Computer Science and Engineering Termination Properties 1.Curve Preservation For each σ D 1, Skl 1 S| σ σ. Two vertices are joined by an edge in Skl 1 S| σ iff they were adjacent in σ. 2.Manifold For 0 ≤ i ≤ 2, and σ D i, Skl i S| σ is a manifold with vertices only in σ. Further, bd Skl i S| σ = Skl i-1 S| bd σ. For i=3, the above holds when Skl i S| σ is nonempty after Mesh2Complex. 3.Strata Preservation There exists some r > 0 so that the output of DelPSC(D, r) is homeomorphic to D. This homeomorphism respects stratification.

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31/22 Department of Computer Science and Engineering Voronoi Cells Intersect “Discly” Given a vertex p on a 2-face σ, if Triangles incident to p in Skl 2 S| σ are small enough. Then, V p | σ is a topological disk, Any edge of V p | σ intersects σ at most once, and Any facet of V p | σ which intersects σ does so in an open curve.

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32/22 Department of Computer Science and Engineering TBP holds globally if All triangles incident in Skl 2 S| σ are smaller than a bound for all 2-faces, Then TBP holds globally This leads to the proof of ETBP and more…topic of a new unpublished paper.

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33/22 Department of Computer Science and Engineering Adjusting MaxRad Example

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34/22 Department of Computer Science and Engineering Adjusting MaxRad Example

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35/22 Department of Computer Science and Engineering Examples

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36/22 Department of Computer Science and Engineering Examples

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38/22 Department of Computer Science and Engineering Examples

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42/22 Department of Computer Science and Engineering Examples

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44/22 Department of Computer Science and Engineering Examples

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45/22 Department of Computer Science and Engineering Sharp Example

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46/22 Department of Computer Science and Engineering Conclusions Delaunay meshing for PSC with guarantees. Feature preservation is an extra `feature’. Making computations easier, faster? Analyzing size complexity?

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