Localized Delaunay Refinement For Piecewise-Smooth Complexes Andrew G. Slatton Joint work with Tamal K. Dey The Ohio State University Department of Computer Science and Engineering

The Problem Input: Piecewise-smooth complex (PSC) D Output: Triangular mesh approximating D Constraint: Use localized Delaunay framework Generate many local sub-meshes Ignore mesh structure at global level (mostly) PIC OF PSC  MESHED PSC

Delaunay Refinement Large meshes  Memory thrashing Localized framework avoids this

Sharp Features Preserve via weighted points protecting balls [Cheng, Dey, Ramos 2008] weighted points protecting balls

Localization & Protection Localization [Dey, Levine, Slatton 2010] Divide sample in octree Refine one node at a time Protection [Cheng, Dey, Shewchuk 2012] Preserve sharp features via protecting balls Refine protecting balls that are too large

Difficulties An assimilation of previous works… So where’s the challenge? Is ball refinement local? Yes! – this must be proven What is its radius of operation? Max distance at which we insert/delete samples Need this to initialize a local triangulation

Node Processing Split Refine When |Pν|> κ Pν δg Nν gathering distance δg Nν

Refinement Criteria Ball connectivity Patch vertices Disk Size λ

Point Insertion Inter-point distance LB  Termination Must avoid arbitrarily close insertions

Reprocessing Refining ν Do we affect other local meshes? δg

Ball Refinement Refining b: Radius of operation ≤ δg Remove contiguous set of balls containing b Cover exposed segment with smaller balls Remove zero-weighted points inside new balls Radius of operation ≤ δg Refining b∈ν, all affected points lie in PνUNν

Potential Complications Why is radius of operation important? δg too large increased overhead Too small: May preclude lower bound on inter-ball distance May allow zero-weighted points to lie inside a ball Would falsify some lemmas leading to termination

Local Ball Refinement Theorem Proven by showing radius of operation ≤ δg

Guarantees Termination Subcomplex of restricted Delaunay Del(P )|D Each point in output lies close to D For sufficiently small λ Output homeomorphic to input

Results

Software & Results http://www.cse.ohio-state.edu/~tamaldey/locpsc.html

Thank You! Questions?