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s w s = strong witness w = weak witness Theorem [de Silva 03] w - - PowerPoint PPT Presentation
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
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Delaunay triangulation
s w
◮ s = strong witness ◮ w = weak witness
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Theorem [de Silva 03]
a b c s w
ab
w
bc abc
w
ca
w
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Motivation
Delaunay triangulation Restricted Delaunay triangulation
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Motivation
Delaunay triangulation Restricted Delaunay triangulation Witness complexes approximation of restricted Delaunay triangulation?
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Triangles
b a
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Triangles
b a
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Conclusion
◮ Witness complexes approximate restricted Delaunay
triangulations for curves and surfaces.
◮ ε1 =
√ 3.
◮
1 √ 5 ≤ ε2 ≤
√ 2.
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Conclusion
◮ Witness complexes approximate restricted Delaunay
triangulations for curves and surfaces.
◮ ε1 =
√ 3.
◮
1 √ 5 ≤ ε2 ≤
√ 2.
◮ For k-manifolds with k ≥ 3, situation more complicated:
◮ εk = 0 for k ≥ 3 → counterexample by Oudot uses slivers
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Conclusion
◮ Witness complexes approximate restricted Delaunay
triangulations for curves and surfaces.
◮ ε1 =
√ 3.
◮
1 √ 5 ≤ ε2 ≤
√ 2.
◮ For k-manifolds with k ≥ 3, situation more complicated:
◮ εk = 0 for k ≥ 3 → counterexample by Oudot uses slivers ◮ Boissonnat et al. assign weights to landmarks to eliminate
slivers
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