Delaunay Triangulation: Applications Reconstruction Meshing 1 - PowerPoint PPT Presentation

Reconstruction Delaunay is a good start 1-sample is not enough 10 - 4

Reconstruction Crust 2D Algorithm 11 - 1

Reconstruction Crust 2D Algorithm Compute Voronoi diagram 11 - 2

Reconstruction Crust 2D Algorithm Keep Voronoi vertices 11 - 3

Reconstruction Crust 2D Algorithm Keep Voronoi vertices Compute Delaunay triangulation 11 - 4

Reconstruction Crust 2D Algorithm Keep Voronoi vertices Compute Delaunay triangulation Keep edges between original points 11 - 5

Reconstruction Crust 2D Algorithm Keep edges between original points 11 - 6

Reconstruction Crust 2D Algorithm 12 - 1

Reconstruction Crust 2D Algorithm 12 - 2

Reconstruction Crust 2D Algorithm 12 - 3

Reconstruction Crust 2D Algorithm 12 - 4

Reconstruction Crust 2D Algorithm 12 - 5

Reconstruction Crust 2D Algorithm 12 - 6

Reconstruction Crust 2D Algorithm 12 - 7

Reconstruction Crust 2D Algorithm 12 - 8

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: 13 - 1

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve Curve x x 0 13 - 2

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 v Curve x x 0 13 - 3

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x x 0 13 - 4

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ ✓ 2 x 0 13 - 5

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ lfs ✓ 2 ✓  x 0 13 - 6

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ lfs ✓ 2 ✓  x 0 tangent disk is empty 13 - 7

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ ✓ 2 ✏ ✓  x 0 lfs wlog lfs=1 and r  ✏ 13 - 8

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ r ✓ 2 ✏ ✓  x 0 lfs wlog lfs=1 and r  ✏ 1 13 - 9

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ ✓ 2 r ✓  x 0 ↵ 2 r = 2 sin ↵ 2 13 - 10

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ x r ✓ � 2 r � = ⇡ � ⇡ � ↵ ✓  x 0 x 0 2 ↵ 2 r = 2 sin ↵ 1 2 13 - 11

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ x ✓ � 2 r � = ⇡ � ⇡ � ↵ ✓  x 0 x 0 2  ⇡ 2 + arcsin r ↵ 2 2 r = 2 sin ↵ 2 13 - 12

Reconstruction Crust 2D 0.4 sample ) wanted result ⇢ crust 0.4 sample ) wanted result ⇢ crust Theorem: x , x 0 two neighboring points on Curve Circle thru x and x 0 centered on Curve By contradiction assume v 2 intersects another cc of curve v Curve (by Lemma) x R  2 r sin ✓ ✓ 2 ✓  ⇡ 2 + arcsin r x 0 2  + 13 - 13