Section:
Research Program
Non-Euclidean computational geometry
Figure
2. Left: 3D mesh of a gyroid (triply periodic
surface) [43].
Right: Simulation of a periodic Delaunay triangulation of the hyperbolic plane [15].
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Triangulations, in particular Delaunay triangulations, in the
Euclidean space have been extensively studied
throughout the 20th century and they are still a very active research
topic. Their mathematical properties are now well understood, many
algorithms to construct them have been proposed and analyzed
(see the book of Aurenhammer et al.
[14]). Some members of Gamble have been contributing to these algorithmic advances
(see, e.g. [18], [51], [29], [17]); they have also
contributed robust and efficient triangulation packages through the
state-of-the-art Computational Geometry Algorithms Library
Cgal , (http://www.cgal.org/) whose impact extends far
beyond computational geometry.
Application fields include particle physics, fluid dynamics, shape
matching, image processing, geometry processing, computer graphics,
computer vision, shape reconstruction, mesh generation, virtual
worlds, geophysics, and medical
imaging. (See http://www.cgal.org/projects.html for details.)
It is fair to say that little has been done on non-Euclidean spaces,
in spite of the large number of questions raised by application
domains. Needs for simulations or modeling in a variety of
domains (See
http://www.loria.fr/~teillaud/PeriodicSpacesWorkshop/,http://www.lorentzcenter.nl/lc/web/2009/357/info.php3?wsid=357 andhttp://neg15.loria.fr/.)
ranging from the infinitely small (nuclear matter, nano-structures,
biological data) to the infinitely large (astrophysics) have led us to
consider 3D periodic Delaunay triangulations, which can be seen
as Delaunay triangulations in the 3D flat torus, quotient of
under the action of some group of translations
[24]. This work has already yielded a fruitful
collaboration with astrophysicists
[37], [52] and new collaborations with
physicists are emerging. To the best of our knowledge, our Cgal package [23] is the only publicly available software
that computes Delaunay triangulations of
a 3D flat torus, in the special case where the domain is cubic. This case, although restrictive is already useful. (See examples at
http://www.cgal.org/projects.html)
We have also generalized this algorithm
to the case of general -dimensional
compact flat manifolds [25]. As far as non-compact
manifolds are concerned, past approaches, limited to the
two-dimensional case, have stayed theoretical [42].
Interestingly, even for the simple case of triangulations on the sphere, the software
packages that are
currently
available are far from offering satisfactory solutions in terms of
robustness and efficiency [22].
Moreover, while our solution for computing triangulations in
hyperbolic spaces can be considered as ultimate [15], the case
of hyperbolic manifolds has hardly been explored. Hyperbolic manifolds are
quotients of a hyperbolic space by some group of hyperbolic
isometries. Their triangulations can be seen as hyperbolic
periodic triangulations. Periodic hyperbolic triangulations and
meshes appear for instance in geometric modeling
[44], neuromathematics [27], or physics
[47]. Even the simplest possible case (a surface
homeomorphic to the torus with two
handles)
shows strong mathematical
difficulties [16], [49].
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