Delaunay Triangulation: Applications Reconstruction Meshing 1 - - PowerPoint PPT Presentation

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Delaunay Triangulation: Applications Reconstruction Meshing

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Reconstruction

From points

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Reconstruction

From points to shape

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Reconstruction

From points

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Reconstruction

From points to shape

2 - 5 From points

Reconstruction

2 - 6 From points to shape

Reconstruction

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Reconstruction

Context Delaunay is a good start

(wanted result ⇢ Delaunay)

Crust 2D 0.4 sample ) wanted result ⇢ crust 0.25 sample ) crust ⇢ wanted result Algorithm 3D

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Reconstruction

Context Sensor Point set (no structure or unknown)

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Reconstruction

Context Medical Images

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Reconstruction

Context Medical Images

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Reconstruction

Context Childbirth simulation Surgery planning Radiotherapy planing Endoscopy simulation

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Reconstruction

Context Childbirth simulation Surgery planning Radiotherapy planing Endoscopy simulation

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Reconstruction

Context Sensor Point set (no structure or unknown) Scanner

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Reconstruction

Context Sensor Point set (no structure or unknown) Scanner Endoscope

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Reconstruction

Context Cultural heritage

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Reconstruction

Context Cultural heritage

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Context

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Reconstruction

Context Reverse engineering

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Reconstruction

Context Reverse engineering Prototyping (3D print) Quality control

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Reconstruction

Context Sensor Point set (no structure or unknown)

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Reconstruction

Context Sensor Point set (no structure or unknown) Laser illuminate in a plane

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Reconstruction

Context Sensor Point set (no structure or unknown) Laser illuminate in a plane Camera

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Reconstruction

Context Sensor Point set (no structure or unknown) Laser illuminate in a plane Camera Image

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Reconstruction

Context Sensor Point set (no structure or unknown) Laser illuminate in a plane Camera Get 3D position

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Reconstruction

Context Geology

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Reconstruction

Context Sensor Point set (no structure or unknown) Geology

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section Can be solve using Voronoi diagrams

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Context Sensor Point set (no structure or unknown) Abstract 3D problem that we can solve in 2D section

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Reconstruction

Delaunay is a good start Medial axis of a curve (surface in 3D) Locus of center of bitangent spheres

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Reconstruction

Delaunay is a good start Medial axis of a curve (surface in 3D) Locus of center of bitangent spheres

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Reconstruction

Delaunay is a good start Medial axis of a curve (surface in 3D) Locus of center of bitangent spheres

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Reconstruction

Delaunay is a good start Medial axis of a curve (surface in 3D) Locus of center of bitangent spheres

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Reconstruction

Delaunay is a good start Medial axis of a curve (surface in 3D) Locus of center of bitangent spheres

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Reconstruction

Delaunay is a good start Medial axis of a curve (surface in 3D) Locus of center of bitangent spheres

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Reconstruction

Delaunay is a good start Medial axis of a curve (surface in 3D) Locus of center of bitangent spheres

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Reconstruction

Delaunay is a good start Medial axis of a curve (surface in 3D) Locus of center of bitangent spheres

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Reconstruction

Delaunay is a good start ✏-sample of a curve Local feature size:

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Reconstruction

Delaunay is a good start ✏-sample of a curve Local feature size: x lfs(x) =

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Reconstruction

Delaunay is a good start ✏-sample of a curve Local feature size: x lfs(x) = distance(x, medial axis) lfs(x)

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Reconstruction

Delaunay is a good start ✏-sample of a curve Local feature size: Sample is an

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Reconstruction

Delaunay is a good start ✏-sample of a curve Local feature size: x lfs(x) = distance(x, medial axis) lfs(x) Sample is an if 8x, Disk(x, ✏·lfs(x))\Sample6= ;

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Reconstruction

Delaunay is a good start 8 Disk, Disk\Curve has a single connected component

• r Disk\Medial axis6= ;

Lemma:

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Reconstruction

Delaunay is a good start 8 Disk, Disk\Curve has a single connected component

• r Disk\Medial axis6= ;

Lemma:

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Reconstruction

Delaunay is a good start 8 Disk, Disk\Curve has a single connected component

• r Disk\Medial axis6= ;

Lemma:

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Reconstruction

Delaunay is a good start 8 Disk, Disk\Curve has a single connected component

• r Disk\Medial axis6= ;

Lemma: Disk\Curve has 2 cc A and B A B c

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Reconstruction

Delaunay is a good start 8 Disk, Disk\Curve has a single connected component

• r Disk\Medial axis6= ;

Lemma: Disk\Curve has 2 cc A and B A B a = closest of c on Curve(wlog on A) b = closest of c on B b a c

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Reconstruction

Delaunay is a good start 8 Disk, Disk\Curve has a single connected component

• r Disk\Medial axis6= ;

Lemma: Disk\Curve has 2 cc A and B A B a = closest of c on Curve(wlog on A) b = closest of c on B b a Moving from c to a dist to B % c

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Reconstruction

Delaunay is a good start 8 Disk, Disk\Curve has a single connected component

• r Disk\Medial axis6= ;

Lemma: Disk\Curve has 2 cc A and B A B a = closest of c on Curve(wlog on A) b = closest of c on B b a Moving from c to a dist to B % reach center of bitangent disk a0 c

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Reconstruction

Delaunay is a good start If Sample is a ✏-sample, ✏ < 1 neighboring points along Curve are Delaunay neighbors Theorem x0

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Reconstruction

Delaunay is a good start If Sample is a ✏-sample, ✏ < 1 neighboring points along Curve are Delaunay neighbors Theorem disks centered on Curve, through x x0 Two neighboring points along curve

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Reconstruction

Delaunay is a good start If Sample is a ✏-sample, ✏ < 1 neighboring points along Curve are Delaunay neighbors Theorem disks centered on Curve, through x x0 Two neighboring points along curve

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Reconstruction

Delaunay is a good start If Sample is a ✏-sample, ✏ < 1 neighboring points along Curve are Delaunay neighbors Theorem disks centered on Curve, through x x0 Two neighboring points along curve

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Reconstruction

Delaunay is a good start If Sample is a ✏-sample, ✏ < 1 neighboring points along Curve are Delaunay neighbors Theorem x0 xx0 neighbors on curve ) no points on cc xx0 in

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Reconstruction

Delaunay is a good start If Sample is a ✏-sample, ✏ < 1 neighboring points along Curve are Delaunay neighbors Theorem 1-sampling ) ⇢ x0 xx0 neighbors on curve ) no points on cc xx0 in

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Reconstruction

Delaunay is a good start If Sample is a ✏-sample, ✏ < 1 neighboring points along Curve are Delaunay neighbors Theorem 1-sampling ) ⇢ x0 xx0 neighbors on curve ) no points on cc xx0 in Lemma }) no other cc \

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Reconstruction

Delaunay is a good start If Sample is a ✏-sample, ✏ < 1 neighboring points along Curve are Delaunay neighbors Theorem 1-sampling ) ⇢ x0 xx0 neighbors on curve ) no points on cc xx0 in Lemma }) no other cc \ ) empty

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Reconstruction

Delaunay is a good start Given a sampling

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Reconstruction

Delaunay is a good start Given a sampling Compute Delaunay

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Reconstruction

Delaunay is a good start Given a sampling Compute Delaunay Search the good sequence of edges there

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Reconstruction

Delaunay is a good start 1-sample is not enough

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Reconstruction

Delaunay is a good start 1-sample is not enough

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Reconstruction

Delaunay is a good start 1-sample is not enough

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Reconstruction

Delaunay is a good start 1-sample is not enough

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D Algorithm Compute Voronoi diagram

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Reconstruction

Crust 2D Algorithm Keep Voronoi vertices

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Reconstruction

Crust 2D Algorithm Keep Voronoi vertices Compute Delaunay triangulation

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Reconstruction

Crust 2D Algorithm Keep Voronoi vertices Compute Delaunay triangulation Keep edges between original points

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Reconstruction

Crust 2D Algorithm Keep edges between original points

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D Algorithm

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem:

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma)

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓  lfs

13 - 7 tangent disk is empty

Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓  lfs

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓  lfs ✏ wlog lfs=1 and r  ✏

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓  lfs ✏ 1 r wlog lfs=1 and r  ✏

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓ 

↵ 2

r r = 2 sin ↵

2

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓ 

↵ 2

x0 x

• = ⇡ ⇡↵

2

1 r r r = 2 sin ↵

2

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

 ⇡

2 + arcsin r 2

✓ 

↵ 2

x0 x

• = ⇡ ⇡↵

2

r r = 2 sin ↵

2

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓  ⇡

2 + arcsin r 2

 +

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓  ⇡

2 + arcsin r 2

 +  r + 2r sin ⇡

4 + 1 2arcsin r 2

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓  ⇡

2 + arcsin r 2

 +  r + 2r sin ⇡

4 + 1 2arcsin r 2

• lfs= 1 contradiction is reached

if

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem: x x0 x, x0 two neighboring points on Curve Curve Circle thru x and x0 centered on Curve By contradiction assume v 2 v intersects another cc of curve

(by Lemma) ✓

R  2r sin ✓

2

✓  ⇡

2 + arcsin r 2

 +  r + 2r sin ⇡

4 + 1 2arcsin r 2

• lfs= 1 contradiction is reached

if

r + 2r sin ⇡

4 + 1 2arcsin r 2

• Plot

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Reconstruction

Crust 2D

0.4 sample ) wanted result ⇢ crust

0.4 sample ) wanted result ⇢ crust

Theorem:

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0 Assume empty circle

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0 Assume empty circle No Voronoi vertices there = )

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0 Assume empty circle No Voronoi vertices there = ) No sample points there

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0 Assume empty circle

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

 ⇡

2 + 2arcsin r 2 ↵ 4

= ⇡ ⇡↵

2

r ' 2 sin ↵

4

r

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

 ⇡

2 + 2arcsin r 2 ↵ 4

= ⇡ ⇡↵

2

r ' 2 sin ↵

4

r

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0 Assume empty circle ⇡

4 2 arcsin r 2

biggest of two angles

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0 Assume empty circle ⇡

4 2 arcsin r 2

biggest of two angles 2 y kxyk 2 sin

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0 Assume empty circle ⇡

4 2 arcsin r 2

biggest of two angles 2 y kxyk 2 sin By Lemma, circle xx0yintersects medial axis

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Reconstruction

Crust 2D

0.25 sample ) crust ⇢ wanted result

Theorem:

0.25 sample ) crust ⇢ wanted result

x x0 Assume empty circle ⇡

4 2 arcsin r 2

biggest of two angles 2 y kxyk 2 sin Compute ✏ to ensure that

1 ✏ ⇥

encloses

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Reconstruction

3D

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Reconstruction

3D Difficulty: sliver

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Reconstruction

3D Difficulty: sliver small sphere

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Reconstruction

3D Difficulty: sliver small sphere four sample points

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Reconstruction

3D Difficulty: sliver small sphere four sample points almost flat Delaunay tetrahedron

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Reconstruction

3D Difficulty: sliver small sphere four sample points almost flat Delaunay tetrahedron Which triangle belongs to reconstruction ?

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Reconstruction

3D Difficulty: sliver small sphere four sample points almost flat Delaunay tetrahedron Which triangle belongs to reconstruction ? Crust: Voronoi vertices may kill useful triangles

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Reconstruction

3D

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Reconstruction

3D

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Reconstruction

3D

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Reconstruction

3D Pole = farthest of seed

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Reconstruction

3D Pole = farthest of seed 2nd pole= farthest of 1st pole

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Reconstruction

3D Pole = farthest of seed 2nd pole= farthest of 1st pole Approximate normal Approximate medial axis ! crust

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Reconstruction

3D Pole = farthest of seed 2nd pole= farthest of 1st pole Approximate normal Approximate medial axis ! crust Do not kill slivers

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