1. Department of Computer Science and Engineering Practical Algorithm for a Large Class of Domains Tamal K. Dey and Joshua A. Levine The Ohio State University Joint Work: Siu-Wing Cheng, HKUST ADelaunay Meshing

2. 2/16 Department of Computer Science and Engineering PSCs – A Large Input Class Piecewise smooth complexes (PSCs) include: Polyhedra Smooth Surfaces Piecewise-smooth Surfaces Non-manifolds

3. 3/16 Department of Computer Science and Engineering PSCs – A Large Class A domain D is a PSC if: Each k-dimensional element is a manifold and compact subset of a smooth (C 2 ) k- manifold, 0≤k≤3. The k-th stratum, D k, is the set of k-dim elements of D (k-faces). D satisfies usual reqs for being a complex. Element interiors are disjoint and for σ  D, bd σ  D. For any σ,   D, either σ   =  or σ    D. D 1 is set of elements which are sharp or non-manifold features (highlighted in red)

4. 4/16 Department of Computer Science and Engineering Delaunay Refinement History Polyhedral Domains (Input conforming): Angle restricted: Chew89, Ruppert92, Miller-Talmor-Teng- Walkington95, Shewchuk98. Small angles allowed: Shewchuk00, Cohen-Steiner-Verdiere- Yvinec02, Cheng-Dey-Ramos-Ray04, Pav-Walkington04. Smooth Surfaces (Topology conforming): Chew93 (w/out guarantee), Cheng-Dey-Edelsbrunner-Sullivan01 (skin surfaces), Boissonnat-Oudot03 and Cheng-Dey-Ramos- Ray04, Oudot-Rineau-Yvinec06 (Volumes). Non-smooth: Boissonnat-Oudot06 (Lipschitz surfaces). Cheng-Dey-Ramos07 (PSCs).

5. 5/16 Department of Computer Science and Engineering Basics of Delaunay Refinement Pioneered by Chew89, Ruppert92, Shewchuck98 To mesh some domain D, 1.Initialize a discrete set of points V  D, compute Del V. 2.If some property is not satisfied, insert a point c from |D| into V and repeat step 2. 3.Return Del V. Burden is to show that the algorithm terminates (shown by a packing argument). Chosen property and output often use Del V| D. Each simplex   Del V| D is the dual of a Voronoi face V  that has a nonempty intersection with D.

6. 6/16 Department of Computer Science and Engineering Challenges for PSCs Requirements: 1.Preserve nonsmooth and nonmanifold features. Elements of D 1. 2.Faithful topology.

7. 7/16 Department of Computer Science and Engineering Delaunay Refinement of PSCs D 1 preserved by sampling with protecting balls (turned into weighted vertices) To satisfy topology, apply the extended topological ball property [Edels.-Shah97]

8. 8/16 Department of Computer Science and Engineering Making it Practical Algorithm of Cheng-Dey-Ramos07 requires four properties to be satisfied (for eTBP): 1.Voronoi edges intersect |D| only once or not at all. 2.Normals on curves/surfaces vary a bounded amount within each Voronoi cell. 3.Delaunay edges do not connect vertices from different surface patches. 4.The set of restricted Delaunay triangles incident to each point form topological disks.

9. 9/16 Department of Computer Science and Engineering Skeletons 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|D i =  σ  Di Skl i S| σ.

10. 10/16 Department of Computer Science and Engineering A New 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

11. 11/16 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 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.

12. 12/16 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.

13. 13/16 Department of Computer Science and Engineering DelPSC Terminates Lemma: Let p in S be a point on a 2-face σ, Let σ’ be the connected component in V p | σ containing p. There exists a constant λ > 0 so that the following holds: If some edge of V p intersects σ and size(t, σ) < λ for each triangle t in Skl 2 S| D2 incident to p, then i.There is no 2-face  where p   and  intersects a Voronoi edge in V p ; ii.σ’ = V p  B  σ where B = B(p,2w(p) + 2λ) and w(p) is the weight at p; iii.σ’ is a 2-disk; iv.Any edge of V p intersects σ’ at most once; v.And any facet of V p intersects σ’ in an empty set or an open curve.

14. 14/16 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.

15. 15/16 Department of Computer Science and Engineering Reducing λ

16. 16/16 Department of Computer Science and Engineering Conclusions Delaunay meshing for PSCs which guarantees topology and preserves features Made practical by simplifying algorithm of Cheng-Dey-Ramos07. Bounded aspect ratio added for triangles/tetrahedra Time/Space complexity?

17. 17/16 Department of Computer Science and Engineering Examples

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27. 27/16 Department of Computer Science and Engineering Sharp Example