Master internship (M1): Meshing Symmetric Hyperbolic Surfaces

Supervision and Contact: Vincent Despré 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

Tools

• A theoretical study of triangle groups and hyperbolic isometries representing symmetries of hyperbolic surfaces will be performed.
• The two CGAL packages [R,BIT] will be interfaced. Such an interface is non-trivial, as the package for meshing was designed to work with Euclidean triangulations. Its extension to hyperbolic triangulations will be studies.
• The mesh refinement algorithm constructs vertices whose coordinates are algebraic numbers. To avoid an explosion of the algebraic degree of the polynomials arising in computations, these coordinates will be rounded. However, symbolic information will be used to ensure consistency, through exact geometric computing.

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.