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Delaunay Mesh Generation. Tamal K. Dey The Ohio State University. Delaunay Mesh Generation. Automatic mesh generation with good quality. Delaunay refinements: The Delaunay triangulation lends to a proof structure . - PowerPoint PPT Presentation

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Delaunay Mesh GenerationTamal K. Dey The Ohio State University#/43Department of Computer Science and Engineering#/43Department of Computer Science and Engineering1Delaunay Mesh GenerationAutomatic mesh generation with good quality.

Delaunay refinements:The Delaunay triangulation lends to a proof structure.And it naturally optimizes certain geometric properties such as min angle.

#/43Department of Computer Science and EngineeringInput/OutputPoints P sampled from a surface S in 3D (dont know S) Reconstruct S : A simplicial complex K, (i) K has a geometric realization in 3D(ii) |K| homeomorphic to S, (iii) Hausdorff distance between |K| and S is small

A smooth surface S(or a compact set):Generate a point sample P from SGenerate a simplicial complex K with vert K=P and satisfying (i), (ii), (iii).#/43Department of Computer Science and EngineeringSurface Reconstruction`

Point Cloud

Surface Reconstruction

#/43Department of Computer Science and EngineeringMedial Axis

#/43Department of Computer Science and EngineeringLocal Feature Size (Smooth)Local feature size is calculated using the medial axis of a smooth shape.

f(x) is the distance from a point to the medial axis

#/43Department of Computer Science and Engineering Each x has a sample within f(x) distance-Sample[ABE98]x#/43Department of Computer Science and EngineeringVoronoi/Delaunay

#/43Department of Computer Science and EngineeringNormal and Voronoi Cells(3D) [Amenta-Bern SoCG98]

#/43Department of Computer Science and EngineeringPoles

P+P-#/43Department of Computer Science and EngineeringNormal LemmaThe angle between the pole vector vp and the normal np is O().

P+P-npvp#/43Department of Computer Science and EngineeringRestricted DelaunayIf the point set is sampled from a domain D.

We can define the restricted Delaunay triangulation, denoted Del P|D.Each simplex Del P|D is the dual of a Voronoi face V that has a nonempty intersection with the domain D.

#/43Department of Computer Science and Engineering

Topological Ball Property (TBP)P has the TBP for a manifold S if each k-face in Vor P either does not intersect S or intersects in a topological (k-1)-ball.Thm (Edelsbrunner-Shah97 ) If P has the TBP then Del P|S is homeomorphic to S.

#/43Department of Computer Science and EngineeringCocone (Amenta-Choi-D.-Leekha)

vp= p+ - p is the pole vectorSpace spanned by vectors within the Voronoi cell making angle > 3/8 with vp or -vp#/43Department of Computer Science and EngineeringCocone Algorithm

#/43Department of Computer Science and EngineeringCocone GuaranteesTheorem: Any point x S is within O(e)f(x) distance from a point in the output. Conversely, any point of output surface has a point x S within O(e)f(x) distance. Triangle normals make O(e) angle with true normals at vertices.Theorem: The output surface computed by Cocone from an e-sample is homeomorphic to the sampled surface for sufficiently small e ( l.Refine a triangle or a ball if disk condition is violatedRefine a ball if it is too big.

Return i Deli S|Di#/43Department of Computer Science and EngineeringGuarantees for DelPSCManifoldFor each D2, triangles in Del S| are a manifold with vertices only in . Further, their boundary is homeomorphic to bd with vertices only in .GranularityThere exists some > 0 so that the output of DelPSC(D, ) is homeomorphic to D.This homeomorphism respects stratification, For 0 i 2, and Di, Del S| is homemorphic to too.

#/43Department of Computer Science and EngineeringReducing

#/43Department of Computer Science and EngineeringExamples

#/43Department of Computer Science and EngineeringExamples

#/43Department of Computer Science and EngineeringExamples

#/43Department of Computer Science and EngineeringExamples

#/43Department of Computer Science and EngineeringSome ResourcesSoftware available from http://www.cse.ohio-state.edu/~tamaldey/cocone.html http://www.cse.ohio-state.edu/~tamaldey/delpsc.html http://www.cse.ohio-state.edu/~tamaldey/locdel.htmlOpen : Reconstruct piecewise smooth surfaces, non-manifoldsOpen: Guarantee quality of all tetrahedra in volume meshingA book Delaunay Mesh Generation: w/ S.-W. Cheng, J. Shewchuk (2012)

#/43Department of Computer Science and EngineeringThank You!#/43Department of Computer Science and Engineering#/43Department of Computer Science and Engineering