Amphi 2 : triangulations de Delaunay (I) Steve Oudot INF562 G eom - - PowerPoint PPT Presentation

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Amphi 2 : triangulations de Delaunay (I) Steve Oudot INF562 G eom - - PowerPoint PPT Presentation

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INF562 G eom etrie Algorithmique et Applications Amphi 2 : triangulations de Delaunay (I) Steve Oudot INF562 G eom etrie Algorithmique et Applications Amphi 2 : triangulations de Delaunay (I) Steve Oudot (certains slides

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SLIDE 1

INF562 – G´ eom´ etrie Algorithmique et Applications

Amphi 2 : triangulations de Delaunay (I)

Steve Oudot

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SLIDE 2

INF562 – G´ eom´ etrie Algorithmique et Applications

Amphi 2 : triangulations de Delaunay (I)

Steve Oudot

(certains slides sont repris du cours de O. Devillers)

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Delaunay Triangulations

  • 1. Definitions and Examples
  • 2. Basic Properties and applications
  • 3. Construction
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Definitions

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looking for nearest neighbor

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looking for nearest neighbor

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Voronoi

Georgy F. Voronoi (1868–1908)

Vi = {q : |qpi| ≤ |qpj| ∀j}

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Voronoi Delaunay

Boris N. Delaunay (1890–1980)

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Voronoi Delaunay Voronoi ↔ geometry Delaunay ↔ connectivity (nerve)

Boris N. Delaunay (1890–1980)

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faces of the Voronoi diagram Voronoi

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faces of the Voronoi diagram Voronoi

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faces of the Voronoi diagram Voronoi

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faces of the Voronoi diagram Voronoi

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Voronoi is everywhere

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Voronoi Empty sphere property

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Voronoi Empty sphere property

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Voronoi

Delaunay

Empty sphere property

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Voronoi

Delaunay

Empty sphere property

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p⋆ = (xp, yp, x2

p + y2 p)

P : z = x2 + y2

point / sphere duality

R2

p = (xp, yp)

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C : x2 + y2 − 2ax − 2by + c = 0 C∗ : z − 2ax − 2by + c = 0 point / sphere duality

R2

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p ∈ C p⋆ ∈ C∗ point / sphere duality

R2

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p ∈ int(C) p⋆ below C∗ point / sphere duality

R2

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p / ∈ int(C) p⋆ above C∗ point / sphere duality

R2

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SLIDE 24
  • rientation predicate

in-sphere predicate

point / sphere duality

R2

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point / sphere duality

R2 Lower CH Delaunay ⇒ Delaunay is a triangulation

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Basic properties and applications

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nearest neighbor graph p q q nearest neighbor of p ⇒ pq Delaunay edge

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nearest neighbor graph

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k nearest neighbors

query point k − 1 nearest neighbors kth nearest neighbor

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k nearest neighbors

query point k − 1 nearest neighbors kth nearest neighbor

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Largest empty circle

(centered in the convex hull)

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Largest empty circle

(centered in the convex hull)

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Minimum Spanning Tree

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p q

Minimum Spanning Tree

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p q x y ∀[pq] ∈ A, p − q = min{x − y | x ∈ Ap, y ∈ Aq}

Minimum Spanning Tree

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p q

Minimum Spanning Tree

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Mesh generation

(bottom image courtesy of J.-P. Pons)

Applications down the road

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Mesh generation / remeshing

(image courtesy of P. Alliez)

Applications down the road

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Mesh generation / remeshing Reconstruction

(cf. s´ eance 6)

Applications down the road

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Mesh generation / remeshing Reconstruction Path planning Applications down the road

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Mesh generation / remeshing Reconstruction Path planning and many more... Applications down the road

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Properties specific to 2D Delaunay

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Euler formula f: number of facets (except ∞) e: number of edges v: number of vertices f − e + v = 1

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Euler formula f: number of facets (except ∞) e: number of edges v: number of vertices 1 − 3 + 3 = 1 f − e + v = 1

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Euler formula f: number of facets (except ∞) e: number of edges v: number of vertices f − e + v = 1 +1 − 2 + 1 = +0

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number of oriented edges in a triangulation: 2e = 3f + k k: size of ∞ facet

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Euler formula f − e + v = 1 Triangulation 2e = 3f + k

f = 2v − 2 − k = O(v) e = 3v − 3 − k = O(v)

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Euler formula f − e + v = 1 Triangulation 2e = 3f + k

f = 2v − 2 − k = O(v) e = 3v − 3 − k = O(v) 2D Delaunay has linear size!

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Delaunay maximizes the smallest angle

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Delaunay maximizes the smallest angle

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Delaunay maximizes the smallest angle

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... but the converse is false

Delaunay maximizes the smallest angle

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SLIDE 53
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Delaunay maximizes the sequence of angles in lexicographical order

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Local optimality vs global optimality

highlighted triangle is only locally Delaunay

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Theorem Locally Delaunay everywhere Globally Delaunay

⇐ ⇒

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Theorem Locally Delaunay everywhere Globally Delaunay

⇐ ⇒

Will be useful for computing Delaunay in 2D

(Guibas/Stolfi variant of incremental algorithm)

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Proof:

Let t0 be locally Delaunay, but not globally Delaunay v Let v ∈ disk(t) (v / ∈ t)

t0

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Proof:

Let t0 be locally Delaunay, but not globally Delaunay v Let v ∈ disk(t) (v / ∈ t)

t0

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Proof:

Let t0 be locally Delaunay, but not globally Delaunay v Let v ∈ disk(t) (v / ∈ t)

t0 t1

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Proof:

Let t0 be locally Delaunay, but not globally Delaunay v Let v ∈ disk(t) (v / ∈ t)

t0 t1

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Proof:

Let t0 be locally Delaunay, but not globally Delaunay v Let v ∈ disk(t) (v / ∈ t)

t0 t1

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Proof:

Let t0 be locally Delaunay, but not globally Delaunay

Since ∃ finitely many triangles, at some point v is a vertex of ti

v Let v ∈ disk(t) (v / ∈ t)

t0 t1 t2

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Local optimality and smallest angle Case of 4 points Lemma: For any 4 points in convex position, Delaunay ⇐ ⇒ smallest angle maximized

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Local optimality and smallest angle Case of 4 points

δ

Let δ be the smallest angle

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Local optimality and smallest angle Case of 4 points

p q r s δ

Let δ be the smallest angle

≤ δ iff r / ∈ disk(pqs)

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Algorithm for making a triangulation Delaunay

while ∃ pairs of adjacent triangles that are not locally Delaunay pick an arbitrary pair and flip common edge

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Algorithm for making a triangulation Delaunay

while ∃ pairs of adjacent triangles that are not locally Delaunay pick an arbitrary pair and flip common edge Theorem: whatever the choice of pairs, the algorithm terminates → proof: each flip increases smallest angle in quad ⇒ cannot be undone

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Algorithm for making a triangulation Delaunay

while ∃ pairs of adjacent triangles that are not locally Delaunay pick an arbitrary pair and flip common edge Theorem: whatever the choice of pairs, the algorithm terminates → output is (globally) Delaunay → proof: each flip increases smallest angle in quad ⇒ cannot be undone

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Algorithm for making a triangulation Delaunay

while ∃ pairs of adjacent triangles that are not locally Delaunay pick an arbitrary pair and flip common edge Theorem: whatever the choice of pairs, the algorithm terminates → output is (globally) Delaunay → proof: each flip increases smallest angle in quad ⇒ cannot be undone does not work in higher dimensions (several types of flips possible)

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Theorem Delaunay ⇒ maximum smallest angle Local optimality and smallest angle

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Theorem Delaunay ⇒ maximum smallest angle Let T triangulation Proof: Local optimality and smallest angle

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Theorem Delaunay ⇒ maximum smallest angle Let T triangulation Apply flipping algorithm on T → output is Delaunay Proof: Local optimality and smallest angle

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Theorem Delaunay ⇒ maximum smallest angle Let T triangulation Apply flipping algorithm on T → output is Delaunay Each flip increases angles within quadrangle Proof: Local optimality and smallest angle → output has larger smallest angle

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Computing Delaunay in the Plane Lower bound

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Lower bound for Delaunay Delaunay can be used to sort numbers

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Lower bound for Delaunay Delaunay can be used to sort numbers

Take an instance of sort Assume one can compute Delaunay in R2 Use Delaunay to solve this instance of sort

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Let x1, x2, . . . , xn ∈ R, to be sorted

x1 Lower bound for Delaunay

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Let x1, x2, . . . , xn ∈ R, to be sorted

x1 (x1, x2

1) (x1, x2

1), . . . , (xn, x2 n)

n points

Lower bound for Delaunay

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Let x1, x2, . . . , xn ∈ R, to be sorted

x1

(x1, x2

1), . . . , (xn, x2 n)

n points

Delaunay → order in x Lower bound for Delaunay

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Let x1, x2, . . . , xn ∈ R, to be sorted

x1

(x1, x2

1), . . . , (xn, x2 n)

n points

Delaunay → order in x

O(n) O(n) f(n) O(n) + f(n) ∈ Ω(n log n)

Lower bound for Delaunay

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⇒ f(n) ∈ Ω(n log n) Lower bound for Delaunay

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Computing Delaunay Incremental algorithm

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Algorithm overview

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Algorithm overview

p

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  • Find triangles in conflict with p

Algorithm overview

p

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  • Find triangles in conflict with p

Algorithm overview

p

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  • Find triangles in conflict with p

Algorithm overview

  • Delete triangles in conflict

p

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  • Find triangles in conflict with p

Algorithm overview

  • Delete triangles in conflict
  • Re-triangulate hole w.r.t. p

p

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Why (and how) it works

Property 1: the conflict zone is connected → compute conflict zone by dfs in dual graph

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Why (and how) it works

Property 1: the conflict zone is connected → compute conflict zone by dfs in dual graph Property 2: conflict zone is starred with respect to p → re-triangulating the conflict zone w.r.t. p gives a triangulation

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Why (and how) it works

Property 1: the conflict zone is connected → compute conflict zone by dfs in dual graph Property 2: conflict zone is starred with respect to p Property 3: every new Delaunay triangle is incident to p → re-triangulating the conflict zone w.r.t. p gives a Delaunay tr. → re-triangulating the conflict zone w.r.t. p gives a triangulation

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The Guibas/Stolfi variant

  • Locate point in triangulation
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The Guibas/Stolfi variant

  • Locate point in triangulation
  • Star triangle
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The Guibas/Stolfi variant

  • Locate point in triangulation
  • Star triangle
  • Apply flipping algo.
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The Guibas/Stolfi variant

  • Locate point in triangulation
  • Star triangle
  • Apply flipping algo.
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The Guibas/Stolfi variant

  • Locate point in triangulation
  • Star triangle
  • Apply flipping algo.
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The Guibas/Stolfi variant

  • Locate point in triangulation
  • Star triangle
  • Apply flipping algo.
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The Guibas/Stolfi variant

  • Locate point in triangulation
  • Star triangle
  • Apply flipping algo.

→ implemented in the Jcg library

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Delaunay triangulation with constraints

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Point insertion with constraints

  • Locate point in triangulation
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Point insertion with constraints

  • Locate point in triangulation
  • Star triangle
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Point insertion with constraints

  • Locate point in triangulation
  • Star triangle
  • Apply conservative flipping
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Point insertion with constraints

  • Locate point in triangulation
  • Star triangle
  • Apply conservative flipping
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Point insertion with constraints

  • Locate point in triangulation
  • Star triangle
  • Apply conservative flipping
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Point insertion with constraints

  • Locate point in triangulation
  • Star triangle
  • Apply conservative flipping
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Constraint insertion

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Constraint insertion

  • Insert vertices (if not there)
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Constraint insertion

  • Insert vertices (if not there)
  • If edge is there, return
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Constraint insertion

  • Insert vertices (if not there)
  • If edge is there, return
  • Else, recurse with midpoint
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Pour le TD d’aujourd’hui

  • representation face/sommet de la triangulation

f f.vertex(0) f.vertex(1) f.vertex(2) f f.neighbor(2) f.neighbor(0) f.neighbor(1)

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Pour le TD d’aujourd’hui

  • representation face/sommet de la triangulation

f f.vertex(0) f.vertex(1) f.vertex(2) f HalfedgeHandle(f,0) HalfedgeHandle(f,2) HalfedgeHandle(f,1)

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Pour le TD d’aujourd’hui

  • representation face/sommet de la triangulation

f f.vertex(0) f.vertex(1) f.vertex(2) f.isMarked(2) = true f f.isMarked(1) = false f.isMarked(0) = false

  • representation des arˆ

etes de contrainte