s w s = strong witness w = weak witness Theorem [de Silva 03] w - PowerPoint PPT Presentation

Delaunay triangulation s w ◮ s = strong witness ◮ w = weak witness

Theorem [de Silva 03] w ab b s w bc w abc a c w ca

Motivation Delaunay triangulation Restricted Delaunay triangulation

Motivation Delaunay triangulation Restricted Delaunay triangulation Witness complexes approximation of restricted Delaunay triangulation?

Triangles b a

Triangles b a

Conclusion ◮ Witness complexes approximate restricted Delaunay triangulations for curves and surfaces. √ ◮ ε 1 = 3. √ 1 5 ≤ ε 2 ≤ 2. ◮ √

Conclusion ◮ Witness complexes approximate restricted Delaunay triangulations for curves and surfaces. √ ◮ ε 1 = 3. √ 1 5 ≤ ε 2 ≤ 2. ◮ √ ◮ For k -manifolds with k ≥ 3, situation more complicated: ◮ ε k = 0 for k ≥ 3 → counterexample by Oudot uses slivers

Conclusion ◮ Witness complexes approximate restricted Delaunay triangulations for curves and surfaces. √ ◮ ε 1 = 3. √ 1 5 ≤ ε 2 ≤ 2. ◮ √ ◮ For k -manifolds with k ≥ 3, situation more complicated: ◮ ε k = 0 for k ≥ 3 → counterexample by Oudot uses slivers ◮ Boissonnat et al. assign weights to landmarks to eliminate slivers