Master internship (M1): Meshing Symmetric Hyperbolic Surfaces

Supervision and Contact: Vincent Despré, Benedikt Kolbe, and Monique Teillaud
Location: Gamble group, INRIA Nancy - Grand Est, LORIA

Context

Tesselations of space from repeating motifs have a long and involved history in mathematics, art, engineering, and natural sciences. The mathematical literature has traditionally focused on patterns in Euclidean spaces but the role of hyperbolic geometry in the natural sciences is increasingly recognized.
Meshes of symmetric hyperbolic surfaces are used in a wide range of fields [ABS,B-M,BV,CFF,STV]. Hyperbolic surfaces exhibiting the symmetries of the triangle group T(2,4,6) are ubiquitous in nature, where they appear as triply-periodic minimal surfaces [ES].

The CGAL library has recently extended its scope to Delaunay triangulations in the hyperbolic plane [BDT,BIT]. This package is providing us with new opportunities to improve on early work [ST] that was aiming at efficiently computing meshes of the regular hyperbolic octagon, which involves the triangle group T(2,3,8).


(figure from [ST])

Goal of the internship

The objective is to compute meshes of hyperbolic surfaces presenting symmetries related to general Schwarz triangles T(k,l,m), i.e., hyperbolic triangles with angles π/k, π/l, π/m, with 1/k + 1/l + 1/m < 1 [C].

Tools

Mathematical, algorithmic, and software tools will be used.

Required knowledge and skills

- C++: generic programming through templates, etc,
- Algorithms, preferably geometric algorithms,
- Strong interest for the mathematical aspects of the topic, in particular (hyperbolic) geometry.

References

[BDT] Mikhail Bogdanov, Olivier Devillers, and Monique Teillaud. Hyperbolic Delaunay complexes and Voronoi diagrams made practical. Journal of Computational Geometry, 5:56-85, 2014.
[BIT] Mikhail Bogdanov, Iordan Iordanov, and Monique Teillaud. 2D Hyperbolic Delaunay Triangulations. CGAL, 2019.
[C] H.S.M. Coxeter. Regular Polytopes. Dover Publications, 3rd edition, 1973. See also Wikipedia.
[R] Laurent Rineau. 2D Conforming Triangulations and Meshes. CGAL, 2004.
[ST] Mathieu Schmitt and Monique Teillaud. Meshing the hyperbolic octagon. Preprint, 2012.

Motivation
[ABS] R. Aurich, E. B. Bogomolny, and F. Steiner. Periodic orbits on the regular hyperbolic octagon. Physica D: Nonlinear Phenomena, 48(1):91-101, 1991.
[B-M] A. Bachelot-Motet. Wave computation on the hyperbolic double doughnut. Journal of Computational Mathematics, 28:1-17, 2010.
[BV] N.L. Balazs and A. Voros. Chaos on the pseudosphere. Physics Reports, 143(3):109-240, 1986.
[CFF] P. Chossat, G. Faye, and O. Faugeras. Bifurcation of hyperbolic planforms. Journal of Nonlinear Science, 21:465-498, 2011.
[ES] M.E. Evans and G.E. Schröder-Turk. In a material world: Hyperbolic geometry in biological materials. Asia Pacific Mathematics Newsletter, 5(2):21-30, 2015.
[STV] F. Sausset, G. Tarjus, and P. Viot. Tuning the fragility of a glassforming liquid by curving space. Physical Review Letters, 101:155701(1)-155701(4), 2008.


Last modified: Tue Sep 15 13:58:28 CEST 2020