Delaunay triangulations of symmetric hyperbolic surfaces Matthijs - PowerPoint PPT Presentation

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 symmetric hyperbolic surfaces 1 / 44

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

Why this topic? Motivation (d) 200 segments [Sausset, Tarjus, Viot ’08] [Chossat, Faye, Faugeras ’11] [Balazs, Voros ’86] M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 3 / 44

Why this topic? State of the art Closed Euclidean manifolds Algorithms 2D [Maz´ on, Recio ’97], 3D [Dolbilin, Huson ’97], d D [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

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

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

Why this topic? Delaunay triangulations in the Euclidean plane triangulation = simplicial complex! M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 7 / 44

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

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

What is a hyperbolic surface? e model of the hyperbolic plane H 2 Poincar´ H ∞ M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 10 / 44

What is a hyperbolic surface? Hyperbolic translations > ℓ ( a ) a ( q ) q Special case: axis = diameter a ( p ) ℓ ( a ) p M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 11 / 44

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

What is a hyperbolic surface? Hyperbolic translations q Non-commutative! a ab ( q ) b ba ( q ) M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 13 / 44

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

What is a hyperbolic surface? Tiling of the hyperbolic plane with octagons c d b a a b d c M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

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

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

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

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

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

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

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

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

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

What is a hyperbolic surface? Tiling of the hyperbolic plane with octagons D N M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 15 / 44

What is a hyperbolic surface? The flat torus and the Bolza surface b Euclidean: translation group � � � abab = ✶ Γ 1 = a , b � O a a Flat torus: M 1 = E 2 / Γ 1 with projection map π 1 : E 2 → M 1 b c Hyperbolic: Fuchsian group ¯ b ¯ d � � Γ 2 = a , b , c , d | abcdabcd = ✶ ¯ O a a Bolza surface: M 2 = H 2 / Γ 2 d b with projection map π 2 : H 2 → M 2 ¯ c M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 16 / 44

What is a hyperbolic surface? The flat torus and the Bolza surface b Euclidean: translation group v 2 v 1 � � � abab = ✶ Γ 1 = a , b � O a a Flat torus: M 1 = E 2 / Γ 1 v 0 v 3 with projection map π 1 : E 2 → M 1 b c Hyperbolic: Fuchsian group ¯ b ¯ d � � Γ 2 = a , b , c , d | abcdabcd = ✶ ¯ O a a Bolza surface: M 2 = H 2 / Γ 2 d b with projection map π 2 : H 2 → M 2 ¯ c M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 16 / 44

What is a hyperbolic surface? Symmetric hyperbolic surfaces of genus g ≥ 2 g = 2 g = 3 g = 4 g = 5 angle sum = 2 π for all 4 g -gons! Let Γ g : Fuchsian group with finite presentation similar to Bolza → 2 g generators, single relation Symmetric hyperbolic surface: M g = H 2 / Γ g , g ≥ 2 with natural projection mapping π g : H 2 → M g M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 17 / 44

What is a hyperbolic surface? Dirichlet regions Voronoi diagram of Γ g O for g = 2 M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 18 / 44

What is a hyperbolic surface? Dirichlet regions angle sum = 2 π c d b a a d b c Fundamental domain D g = Dirichlet region of O for Γ g here for g = 2 M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 19 / 44

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

How to triangulate a hyperbolic surface? Validity condition [BTV16] S set of points in D g M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 21 / 44

How to triangulate a hyperbolic surface? Validity condition [BTV16] orbits Γ g S in H 2 M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 22 / 44

How to triangulate a hyperbolic surface? Validity condition [BTV16] Delaunay triangulation in H 2 DT H (Γ g S ) M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 23 / 44

How to triangulate a hyperbolic surface? Validity condition [BTV16] projection of DT H (Γ g S ) on the surface M g → not necessarily a simplicial complex! double edges double edges and/or loops M. Ebbens, I. Iordanov , M. Teillaud, G. Vegter Delaunay triangulations of symmetric hyperbolic surfaces 24 / 44