1. 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

2. 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

3. Delaunay Refinement Large meshes  Memory thrashing
Localized framework avoids this

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

5. 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

6. 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

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

8. Refinement Criteria
Ball connectivity
Patch vertices
Disk
Size λ

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

10. Reprocessing
Refining ν
Do we affect other local meshes? δg

11. 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

12. 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

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

14. 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

15. Results

16. Software & Results

17. Thank You!
Questions?