63 research outputs found
Triangulating the Real Projective Plane
We consider the problem of computing a triangulation of the real projective
plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a
triangulation of P2 always exists if at least six points in S are in general
position, i.e., no three of them are collinear. We also design an algorithm for
triangulating P2 if this necessary condition holds. As far as we know, this is
the first computational result on the real projective plane
Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions
We describe the implementation of algorithms to construct and maintain
three-dimensional dynamic Delaunay triangulations with kinetic vertices using a
three-simplex data structure. The code is capable of constructing the geometric
dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points
is triangulated. Time evolution of the triangulation is not only governed by
kinetic vertices but also by a changing number of vertices. We use
three-dimensional simplex flip algorithms, a stochastic visibility walk
algorithm for point location and in addition, we propose a new simple method of
deleting vertices from an existing three-dimensional Delaunay triangulation
while maintaining the Delaunay property. The dual Dirichlet tessellation can be
used to solve differential equations on an irregular grid, to define partitions
in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly
nomenclature), referee suggestions included, typos corrected, bibliography
update
The Delaunay Hierarchy
International audienceWe propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the level below. Point location is done by walking in a triangulation to determine the nearest neighbor of the query at that level, then the walk restarts from that neighbor at the level below. Using a small subset (3%) to sample a level allows a small memory occupation; the walk and the use of the nearest neighbor to change levels quickly locate the query
Towards a Scalable Dynamic Spatial Database System
With the rise of GPS-enabled smartphones and other similar mobile devices,
massive amounts of location data are available. However, no scalable solutions
for soft real-time spatial queries on large sets of moving objects have yet
emerged. In this paper we explore and measure the limits of actual algorithms
and implementations regarding different application scenarios. And finally we
propose a novel distributed architecture to solve the scalability issues.Comment: (2012
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Routing with Guaranteed Delivery on Virtual Coordinates
We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for non-triangular graphs, and their virtual coordinates are easy to construct.Engineering and Applied Science
Formal Verification of a Geometry Algorithm: A Quest for Abstract Views and Symmetry in Coq Proofs
This extended abstract is about an effort to build a formal description of a
triangulation algorithm starting with a naive description of the algorithm
where triangles, edges, and triangulations are simply given as sets and the
most complex notions are those of boundary and separating edges. When
performing proofs about this algorithm, questions of symmetry appear and this
exposition attempts to give an account of how these symmetries can be handled.
All this work relies on formal developments made with Coq and the mathematical
components library
Implementing Delaunay Triangulations of the Bolza Surface
The CGAL library offers software packages to compute Delaunay triangulations of the (flat) torus of genus one in two and three dimensions. To the best of our knowledge, there is no available software for the simplest possible extension, i.e., the Bolza surface, a hyperbolic manifold homeomorphic to a torus of genus two.
In this paper, we present an implementation based on the theoretical results and the incremental algorithm proposed last year at SoCG by Bogdanov, Teillaud, and Vegter. We describe the representation of the triangulation, we detail the different steps of the algorithm, we study predicates, and report experimental results
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