Delaunay triangulations of symmetric hyperbolic surfaces Matthijs - - PowerPoint PPT Presentation

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Delaunay triangulations of symmetric hyperbolic surfaces Matthijs - - PowerPoint PPT Presentation

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Delaunay triangulations of symmetric hyperbolic surfaces Matthijs Ebbens Iordan Iordanov Monique Teillaud Gert Vegter Curves and Surfaces 2018 Arcachon, France M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of

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SLIDE 1

Delaunay triangulations

  • f symmetric hyperbolic surfaces

Matthijs Ebbens Iordan Iordanov Monique Teillaud Gert Vegter

Curves and Surfaces 2018

Arcachon, France

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 1 / 44

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SLIDE 2

Why this topic?

Outline

1 Why this topic? 2 What is a hyperbolic surface? 3 How to triangulate a hyperbolic surface? 4 How is the triangulation represented? 5 What is needed for a triangulation in higher genus? 6 What results do we have so far? 7 What comes next?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 2 / 44

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SLIDE 3

Why this topic?

Motivation

[Sausset, Tarjus, Viot ’08] [Chossat, Faye, Faugeras ’11]

(d)

200 segments

[Balazs, Voros ’86]

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 3 / 44

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SLIDE 4

Why this topic?

State of the art

Closed Euclidean manifolds Algorithms

2D [Maz´

  • n, Recio ’97], 3D [Dolbilin, Huson ’97], dD [Caroli, Teillaud ’16]

Software (square/cubic flat torus)

2D [Kruithof ’13], 3D [Caroli, Teillaud ’09]

Closed hyperbolic manifolds Algorithms

2D, genus 2 [Bogdanov, Teillaud, Vegter, SoCG’16]

Software (Bolza surface)

[I., Teillaud, SoCG’17] → submitted to

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 4 / 44

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SLIDE 5

Why this topic?

Delaunay triangulations in the Euclidean plane

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 5 / 44

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SLIDE 6

Why this topic?

Delaunay triangulations in the Euclidean plane

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 6 / 44

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SLIDE 7

Why this topic?

Delaunay triangulations in the Euclidean plane

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 7 / 44

triangulation = simplicial complex!

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SLIDE 8

Why this topic?

Delaunay triangulations in the Euclidean plane

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 8 / 44

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SLIDE 9

What is a hyperbolic surface?

Outline

1 Why this topic? 2 What is a hyperbolic surface? 3 How to triangulate a hyperbolic surface? 4 How is the triangulation represented? 5 What is needed for a triangulation in higher genus? 6 What results do we have so far? 7 What comes next?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 9 / 44

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SLIDE 10

What is a hyperbolic surface?

Poincar´ e model of the hyperbolic plane H2

H∞

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 10 / 44

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SLIDE 11

What is a hyperbolic surface?

Hyperbolic translations

Special case: axis = diameter

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 11 / 44

p a(p) q a(q) ℓ(a) > ℓ(a)

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SLIDE 12

What is a hyperbolic surface?

Hyperbolic translations

Side-pairing transformation

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 12 / 44

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SLIDE 13

What is a hyperbolic surface?

Hyperbolic translations

Non-commutative!

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 13 / 44

ba(q) ab(q) a b q

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SLIDE 14

What is a hyperbolic surface?

Tilings of the Euclidean and hyperbolic planes

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 14 / 44

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SLIDE 15

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a a b b c c d d

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 16

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 17

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 18

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a ab

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 19

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a ab a b c

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 20

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a ab a b c abcd = dcba

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 21

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a ab a b c abcd = dcba dcb

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 22

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a ab a b c abcd = dcba dcb dc

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 23

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a ab a b c abcd = dcba dcb dc d

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 24

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

a ab a b c abcd = dcba dcb dc d

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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SLIDE 25

What is a hyperbolic surface?

Tiling of the hyperbolic plane with octagons

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

DN

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What is a hyperbolic surface?

The flat torus and the Bolza surface

Euclidean: translation group Γ1 =

  • a, b
  • abab = ✶
  • Flat torus: M1 = E2/Γ1

with projection map π1 : E2 → M1 Hyperbolic: Fuchsian group Γ2 =

  • a, b, c, d | abcdabcd = ✶
  • Bolza surface: M2 = H2/Γ2

with projection map π2 : H2 → M2

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 16 / 44

O a a b b a ¯ b c ¯ d ¯ a b ¯ c d O

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SLIDE 27

What is a hyperbolic surface?

The flat torus and the Bolza surface

Euclidean: translation group Γ1 =

  • a, b
  • abab = ✶
  • Flat torus: M1 = E2/Γ1

with projection map π1 : E2 → M1 Hyperbolic: Fuchsian group Γ2 =

  • a, b, c, d | abcdabcd = ✶
  • Bolza surface: M2 = H2/Γ2

with projection map π2 : H2 → M2

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 16 / 44

O a a b b v3 v0 v1 v2 a ¯ b c ¯ d ¯ a b ¯ c d O

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What is a hyperbolic surface?

Symmetric hyperbolic surfaces of genus g ≥ 2

Let Γg: Fuchsian group with finite presentation similar to Bolza → 2g generators, single relation Symmetric hyperbolic surface: Mg = H2/Γg, g ≥ 2 with natural projection mapping πg : H2 → Mg

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 17 / 44

g = 2 g = 3 g = 4 g = 5 angle sum = 2π for all 4g-gons!

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SLIDE 29

What is a hyperbolic surface?

Dirichlet regions

Voronoi diagram of ΓgO for g = 2

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 18 / 44

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What is a hyperbolic surface?

Dirichlet regions

a a b b c c d d

Fundamental domain Dg = Dirichlet region of O for Γg

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 19 / 44

angle sum = 2π here for g = 2

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How to triangulate a hyperbolic surface?

Outline

1 Why this topic? 2 What is a hyperbolic surface? 3 How to triangulate a hyperbolic surface? 4 How is the triangulation represented? 5 What is needed for a triangulation in higher genus? 6 What results do we have so far? 7 What comes next?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 20 / 44

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How to triangulate a hyperbolic surface?

Validity condition [BTV16]

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 21 / 44

S set of points in Dg

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SLIDE 33

How to triangulate a hyperbolic surface?

Validity condition [BTV16]

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 22 / 44

  • rbits ΓgS in H2
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SLIDE 34

How to triangulate a hyperbolic surface?

Validity condition [BTV16]

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 23 / 44

Delaunay triangulation in H2 DTH(ΓgS)

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SLIDE 35

How to triangulate a hyperbolic surface?

Validity condition [BTV16]

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 24 / 44

projection of DTH(ΓgS) on the surface Mg → not necessarily a simplicial complex! double edges double edges and/or loops

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SLIDE 36

How to triangulate a hyperbolic surface?

Validity condition [BTV16]

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 25 / 44

projection of DTH(ΓgS) on the surface Mg → not necessarily a simplicial complex! Systole of a surface = minimum length of a non-contractible loop on the surface

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How to triangulate a hyperbolic surface?

Validity condition [BTV16]

δS < 1 2sys(Mg), where DTMg(S) := πg(DTH(ΓgS))

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 26 / 44

projection of DTH(ΓgS) on the surface Mg → is a simplicial complex, if δS = diameter of largest disks in H2 not containing any point of ΓgS

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How to triangulate a hyperbolic surface?

Computing Delaunay triangulations of Mg

Use set of dummy points Qg that satisfies the validity condition: S := Qg

  • P =

⇒ δS < 1 2sys(Mg) always

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 27 / 44

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SLIDE 39

How to triangulate a hyperbolic surface?

Computing Delaunay triangulations of Mg

Use set of dummy points Qg that satisfies the validity condition: S := Qg

  • P =

⇒ δS < 1 2sys(Mg) always Algorithm for computing Delaunay triangulations of Mg

[BTV16]

initialize DTMg with a set Qg that satisfies the validity condition; insert input points P in the triangulation; remove points of Qg from the triangulation, if possible. → condition preserved with insertion of new points → diameter of largest empty disks cannot grow → final triangulation might contain dummy points → if input points too few and/or badly distributed

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 27 / 44

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SLIDE 40

How is the triangulation represented?

Outline

1 Why this topic? 2 What is a hyperbolic surface? 3 How to triangulate a hyperbolic surface? 4 How is the triangulation represented? 5 What is needed for a triangulation in higher genus? 6 What results do we have so far? 7 What comes next?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 28 / 44

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How is the triangulation represented?

Problem statement

To compute DTMg(S), we need to choose what to store.

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 29 / 44

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SLIDE 42

How is the triangulation represented?

Problem statement

To compute DTMg(S), we need to choose what to store. Requirement: all input points lie in Dg → unique representative in Dg ⊂ H2 for each point on Mg

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 29 / 44

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SLIDE 43

How is the triangulation represented?

Problem statement

To compute DTMg(S), we need to choose what to store. Requirement: all input points lie in Dg → unique representative in Dg ⊂ H2 for each point on Mg Question: How to choose a unique representative for each face?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 29 / 44

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SLIDE 44

How is the triangulation represented?

Inclusion property

Let S ⊂ Dg be a point set such that δS < 1

2 sys(Mg).

Let σ be a face of DTH(ΓgS) with at least one vertex in Dg ⇒ σ is contained in DN Proof: for g = 2 → [IT17] for g ≥ 2 → [Ebbens 2018] Matthijs’ talk

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 30 / 44

DN

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SLIDE 45

How is the triangulation represented?

Canonical representatives of faces

Canonical representative: face with at least one vertex in Dg → other vertices will be in DN To make it unique: → choose the face closest to the “first” Dirichlet neighbor

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 31 / 44

abcd

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SLIDE 46

How is the triangulation represented?

Canonical representatives of faces

Canonical representative: face with at least one vertex in Dg → other vertices will be in DN To make it unique: → choose the face closest to the “first” Dirichlet neighbor

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 32 / 44

abcd

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SLIDE 47

How is the triangulation represented?

Canonical representatives of faces

Canonical representative: face with at least one vertex in Dg → other vertices will be in DN To make it unique: → choose the face closest to the “first” Dirichlet neighbor

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 33 / 44

abcd

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SLIDE 48

How is the triangulation represented?

triangulation data structure ν1v1 p1 p2 p0 ν0v0 , ν2v2 f

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 34 / 44

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SLIDE 49

How is the triangulation represented?

Canonical representatives can cross the boundary

p q r a(p) a(q) a(r)

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 35 / 44

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SLIDE 50

How is the triangulation represented?

extended triangulation data structure

a(p) q r

ν0 = 1 ν2 = a ν1 = a q r p

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 36 / 44

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SLIDE 51

What is needed for a triangulation in higher genus?

Outline

1 Why this topic? 2 What is a hyperbolic surface? 3 How to triangulate a hyperbolic surface? 4 How is the triangulation represented? 5 What is needed for a triangulation in higher genus? 6 What results do we have so far? 7 What comes next?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 37 / 44

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SLIDE 52

What is needed for a triangulation in higher genus?

An initial set of dummy points

For M2, a set of dummy points was given [BTV16]. In general?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 38 / 44

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SLIDE 53

What is needed for a triangulation in higher genus?

An initial set of dummy points

For M2, a set of dummy points was given [BTV16]. In general? The idea is to generate dummy points:

1 Start with the set Wg of Weierstrass points for Mg

→ origin, one vertex, and midpoints of half the sides of the 4g-gon

2 Compute the images of these points in DN 3 Compute their hyperbolic Delaunay triangulation in H2 4 Apply Delaunay refinements to satisfy condition

← strategies!

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 38 / 44

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SLIDE 54

What is needed for a triangulation in higher genus?

An initial set of dummy points

For M2, a set of dummy points was given [BTV16]. In general? The idea is to generate dummy points:

1 Start with the set Wg of Weierstrass points for Mg

→ origin, one vertex, and midpoints of half the sides of the 4g-gon

2 Compute the images of these points in DN 3 Compute their hyperbolic Delaunay triangulation in H2 4 Apply Delaunay refinements to satisfy condition

← strategies!

[Ebbens, 2018] sys(Mg) = 2 arcosh

  • 1 + 2 cos
  • π

2g

  • Matthijs’ talk
  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 38 / 44

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SLIDE 55

What results do we have so far?

Outline

1 Why this topic? 2 What is a hyperbolic surface? 3 How to triangulate a hyperbolic surface? 4 How is the triangulation represented? 5 What is needed for a triangulation in higher genus? 6 What results do we have so far? 7 What comes next?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 39 / 44

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SLIDE 56

What results do we have so far?

Implementation

Available code: triangulations in H2 (non-periodic) triangulations of M2 (periodic) generate dummy points with different strategies Todo: Put the pieces together Difficulties: Numerical operations with CORE::Expr

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 40 / 44

  • https://imiordanov.github.io/code
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SLIDE 57

What results do we have so far?

Computer algebraic issues

p q r p q r s

Combinatorial validity of DTMg

  • Exact evaluation of predicates

Assume rational input points, approximate W-points and circumcenters.

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 41 / 44

  • CORE::Expr
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SLIDE 58

What results do we have so far?

Computer algebraic issues

p q r p q r s

Combinatorial validity of DTMg

  • Exact evaluation of predicates

Assume rational input points, approximate W-points and circumcenters. Hyperbolic translations include algebraic numbers: Tk =

 

cot( π

4g )

exp( ikπ

2g )

  • cot2( π

4g ) − 1

exp(−ikπ

2g )

  • cot2( π

4g ) − 1

cot( π

4g )

 

→ images of rational points have algebraic coordinates!

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 41 / 44

  • CORE::Expr
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SLIDE 59

What comes next?

Outline

1 Why this topic? 2 What is a hyperbolic surface? 3 How to triangulate a hyperbolic surface? 4 How is the triangulation represented? 5 What is needed for a triangulation in higher genus? 6 What results do we have so far? 7 What comes next?

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 42 / 44

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SLIDE 60

What comes next?

Future directions

Generalization to arbitrary hyperbolic structures

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 43 / 44

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SLIDE 61

Thank you!

  • M. Ebbens, I. Iordanov, M. Teillaud, G. Vegter

Delaunay triangulations of symmetric hyperbolic surfaces 44 / 44