90 research outputs found
Universality theorems for inscribed polytopes and Delaunay triangulations
We prove that every primary basic semialgebraic set is homotopy equivalent to
the set of inscribed realizations (up to M\"obius transformation) of a
polytope. If the semialgebraic set is moreover open, then, in addition, we
prove that (up to homotopy) it is a retract of the realization space of some
inscribed neighborly (and simplicial) polytope. We also show that all algebraic
extensions of are needed to coordinatize inscribed polytopes.
These statements show that inscribed polytopes exhibit the Mn\"ev universality
phenomenon.
Via stereographic projections, these theorems have a direct translation to
universality theorems for Delaunay subdivisions. In particular, our results
imply that the realizability problem for Delaunay triangulations is
polynomially equivalent to the existential theory of the reals.Comment: 15 pages, 2 figure
Witness (Delaunay) Graphs
Proximity graphs are used in several areas in which a neighborliness
relationship for input data sets is a useful tool in their analysis, and have
also received substantial attention from the graph drawing community, as they
are a natural way of implicitly representing graphs. However, as a tool for
graph representation, proximity graphs have some limitations that may be
overcome with suitable generalizations. We introduce a generalization, witness
graphs, that encompasses both the goal of more power and flexibility for graph
drawing issues and a wider spectrum for neighborhood analysis. We study in
detail two concrete examples, both related to Delaunay graphs, and consider as
well some problems on stabbing geometric objects and point set discrimination,
that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200
There are 174 Subdivisions of the Hexahedron into Tetrahedra
This article answers an important theoretical question: How many different
subdivisions of the hexahedron into tetrahedra are there? It is well known that
the cube has five subdivisions into 6 tetrahedra and one subdivision into 5
tetrahedra. However, all hexahedra are not cubes and moving the vertex
positions increases the number of subdivisions. Recent hexahedral dominant
meshing methods try to take these configurations into account for combining
tetrahedra into hexahedra, but fail to enumerate them all: they use only a set
of 10 subdivisions among the 174 we found in this article.
The enumeration of these 174 subdivisions of the hexahedron into tetrahedra
is our combinatorial result. Each of the 174 subdivisions has between 5 and 15
tetrahedra and is actually a class of 2 to 48 equivalent instances which are
identical up to vertex relabeling. We further show that exactly 171 of these
subdivisions have a geometrical realization, i.e. there exist coordinates of
the eight hexahedron vertices in a three-dimensional space such that the
geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for
these configurations and show in particular subdivisions of hexahedra with 15
tetrahedra that have a strictly positive Jacobian
A Finite Algorithm for the Realizabilty of a Delaunay Triangulation
The \emph{Delaunay graph} of a point set is the
plane graph with the vertex-set and the edge-set that contains
if there exists a disc whose intersection with is exactly .
Accordingly, a triangulated graph is \emph{Delaunay realizable} if there
exists a triangulation of the Delaunay graph of some , called a \emph{Delaunay triangulation} of , that is
isomorphic to . The objective of \textsc{Delaunay Realization} is to compute
a point set that realizes a given graph (if such
a exists). Known algorithms do not solve \textsc{Delaunay Realization} as
they are non-constructive. Obtaining a constructive algorithm for
\textsc{Delaunay Realization} was mentioned as an open problem by Hiroshima et
al.~\cite{hiroshima2000}. We design an -time constructive
algorithm for \textsc{Delaunay Realization}. In fact, our algorithm outputs
sets of points with {\em integer} coordinates
Six topics on inscribable polytopes
Inscribability of polytopes is a classic subject but also a lively research
area nowadays. We illustrate this with a selection of well-known results and
recent developments on six particular topics related to inscribable polytopes.
Along the way we collect a list of (new and old) open questions.Comment: 11 page
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