Delaunay triangulations on hyperbolic surfaces Iordan Iordanov - PowerPoint PPT Presentation

Delaunay triangulations on hyperbolic surfaces Iordan Iordanov Monique Teillaud Astonishing Workshop 25 September 2017 Nancy, France I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 1 / 30

Outline 1 Introduction 1.1 Motivation 1.2 The Bolza Surface 1.3 Background from [BTV, SoCG’16] 2 Implementation 2.1 Data Structure 2.2 Incremental Insertion 2.3 Results 3 Future work I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 2 / 30

Introduction Outline 1 Introduction 1.1 Motivation 1.2 The Bolza Surface 1.3 Background from [BTV, SoCG’16] 2 Implementation 2.1 Data Structure 2.2 Incremental Insertion 2.3 Results 3 Future work I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 3 / 30

Introduction Motivation Motivation Periodic triangulations in the Euclidean plane I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 4 / 30

Introduction Motivation Motivation Periodic triangulations in the hyperbolic plane I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 4 / 30

Introduction Motivation Motivation Applications (d) 200 segments [Sausset, Tarjus, Viot] [Chossat, Faye, Faugeras] [Balazs, Voros] I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 5 / 30

Introduction Motivation Motivation Beautiful groups Fuchsian groups finitely presented groups triangle groups . . . I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 6 / 30

Introduction Motivation State of the art Closed Euclidean manifolds Algorithms 2D [Mazón, Recio], 3D [Dolbilin, Huson], d D [Caroli, Teillaud, DCG’16] Software (square/cubic flat torus) 2D [Kruithof], 3D [Caroli, Teillaud] Closed hyperbolic manifolds Algorithms 2D, genus 2 [Bogdanov, Teillaud, Vegter, SoCG’16] Software (Bolza surface) [Iordanov, Teillaud, SoCG’17] I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 7 / 30

Introduction The Bolza Surface Poincaré model of the hyperbolic plane H 2 H ∞ I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 8 / 30

Introduction The Bolza Surface Hyperbolic translations a ( q ) > ℓ ( a ) q a ( p ) ℓ ( a ) X a p special case axis = diameter I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 8 / 30

Introduction The Bolza Surface Hyperbolic translations non-commutative! X b q X a ab ( q ) ba ( q ) I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 8 / 30

Introduction The Bolza Surface Bolza surface What is it? Closed, compact, orientable surface of genus 2. Constant negative curvature − → locally hyperbolic metric. The most symmetric of all genus-2 surfaces. I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 9 / 30

Introduction The Bolza Surface Bolza surface c ¯ b Fuchsian group G with finite presentation ¯ d � � G = a , b , c , d | abcdabcd O a a ¯ G contains only translations (and 1 ) Bolza surface d b M = H 2 / G ¯ c with projection map π M : H 2 → M I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 10 / 30

Introduction The Bolza Surface Bolza surface c ¯ b Fuchsian group G with finite presentation ¯ d � � G = a , b , c , d | abcdabcd O a a ¯ G contains only translations (and 1 ) Bolza surface d b M = H 2 / G ¯ c with projection map π M : H 2 → M � � A = � g 0 , g 1 , ..., g 7 � a , b , c , d , a , b , c , d = � � √ β k = e ik π/ 4 √ α β k g k ( z ) = α z + β k g k = , β k z + α , α = 1 + 2 , 2 α β k α I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 10 / 30

Introduction The Bolza Surface Bolza surface c ¯ b ¯ d O a ¯ a d b c ¯ ↓ c ¯ b ¯ d O a ¯ a d b ¯ c I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 11 / 30

Introduction The Bolza Surface Hyperbolic octagon Voronoi diagram of G O I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 12 / 30

Introduction The Bolza Surface Hyperbolic octagon Fundamental domain D O = Dirichlet region of O I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 12 / 30

Introduction The Bolza Surface Hyperbolic octagon “Original” domain D : contains exactly one point of each orbit I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 12 / 30

Introduction Background from [BTV, SoCG’16] Criterion Systole sys ( M ) = minimum length of a non-contractible loop on M S set of points in D O ⊂ H 2 diameter of largest disks in H 2 δ S = not containing any point of G S δ S < 1 2 sys ( M ) = ⇒ π M ( DT H ( G S ) ) = DT M ( S ) is a simplicial complex = ⇒ The usual incremental algorithm can be used [Bowyer] I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 13 / 30

Introduction Background from [BTV, SoCG’16] Criterion Systole sys ( M ) = minimum length of a non-contractible loop on M S set of points in H 2 diameter of largest disks in H 2 δ S = not containing any point of G S δ S < 1 2 sys ( M ) = ⇒ π M ( DT H ( G S ) ) = DT M ( S ) is a simplicial complex = ⇒ The usual incremental algorithm can be used [Bowyer] I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 13 / 30

Introduction Background from [BTV, SoCG’16] Criterion Systole sys ( M ) = minimum length of a non-contractible loop on M S set of points in H 2 diameter of largest disks in H 2 δ S = not containing any point of G S δ S < 1 2 sys ( M ) = ⇒ π M ( DT H ( G S ) ) = DT M ( S ) is a simplicial complex = ⇒ The usual incremental algorithm can be used [Bowyer] I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 13 / 30

Introduction Background from [BTV, SoCG’16] How can we satisfy δ S < 1 2 sys ( M )? Two ways: 1 Covering spaces effect: increase the systole take copies of the fundamental domain with input points new: 32 < number of sheets ≤ 128 new: 34 < number of sheets [Ebbens, 2017] I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 14 / 30

Introduction Background from [BTV, SoCG’16] How can we satisfy δ S < 1 2 sys ( M )? Two ways: 1 Covering spaces effect: increase the systole take copies of the fundamental domain with input points new: 32 < number of sheets ≤ 128 new: 34 < number of sheets [Ebbens, 2017] 2 Dummy points effect: artificially satisfy the condition in the 1-cover set of points given for the Bolza surface more appealing computationally I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 14 / 30

Introduction Background from [BTV, SoCG’16] How can we satisfy δ S < 1 2 sys ( M )? Two ways: 1 Covering spaces effect: increase the systole take copies of the fundamental domain with input points new: 32 < number of sheets ≤ 128 new: 34 < number of sheets [Ebbens, 2017] 2 Dummy points effect: artificially satisfy the condition in the 1-cover set of points given for the Bolza surface more appealing computationally We adopt the second approach. I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 14 / 30

Introduction Background from [BTV, SoCG’16] Systole on the octagon 1 2 sys( M ) � �� � � �� � sys( M ) I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 15 / 30

Introduction Background from [BTV, SoCG’16] Set of dummy points I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 15 / 30

Introduction Background from [BTV, SoCG’16] Set of dummy points vs. criterion I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 15 / 30

Introduction Background from [BTV, SoCG’16] Delaunay triangulation of the dummy points I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 15 / 30

Introduction Background from [BTV, SoCG’16] Delaunay triangulation of the Bolza surface Algorithm: 1 initialize with dummy points 2 insert points in S 3 remove dummy points I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 15 / 30

Implementation Outline 1 Introduction 1.1 Motivation 1.2 The Bolza Surface 1.3 Background from [BTV, SoCG’16] 2 Implementation 2.1 Data Structure 2.2 Incremental Insertion 2.3 Results 3 Future work I. Iordanov & M. Teillaud Delaunay triangulations on hyperbolic surfaces 16 / 30