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

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

3. 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.

4. 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)

5. 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).

6. 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].

7. 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.

8. 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].

9. 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.

10. 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].

11. 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||}}.

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. 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.

20. 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.

21. 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

22. 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.

23. 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.

24. 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.

25. 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.

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

27. 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

28. 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

29. 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.

30. 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.

31. 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.

32. 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.

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

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

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46. 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?