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Subquadratic nonobtuse triangulation of convex polygons
A convex polygon with n sides can be triangulated by O(n^1.85) triangles, without any obtuse angles. The construction uses a novel form of geometric divide and conquer
Mathematics and Morphogenesis of the City: A Geometrical Approach
Cities are living organisms. They are out of equilibrium, open systems that
never stop developing and sometimes die. The local geography can be compared to
a shell constraining its development. In brief, a city's current layout is a
step in a running morphogenesis process. Thus cities display a huge diversity
of shapes and none of traditional models from random graphs, complex networks
theory or stochastic geometry takes into account geometrical, functional and
dynamical aspects of a city in the same framework. We present here a global
mathematical model dedicated to cities that permits describing, manipulating
and explaining cities' overall shape and layout of their street systems. This
street-based framework conciliates the topological and geometrical sides of the
problem. From the static analysis of several French towns (topology of first
and second order, anisotropy, streets scaling) we make the hypothesis that the
development of a city follows a logic of division / extension of space. We
propose a dynamical model that mimics this logic and which from simple general
rules and a few parameters succeeds in generating a large diversity of cities
and in reproducing the general features the static analysis has pointed out.Comment: 13 pages, 13 figure
Embedding the dual complex of hyper-rectangular partitions
A rectangular partition is the partition of an (axis-aligned) rectangle into
interior-disjoint rectangles. We ask whether a rectangular partition permits a
"nice" drawing of its dual, that is, a straight-line embedding of it such that
each dual vertex is placed into the rectangle that it represents. We show that
deciding whether such a drawing exists is NP-complete. Moreover, we consider
the drawing where a vertex is placed in the center of the represented rectangle
and consider sufficient conditions for this drawing to be nice. This question
is studied both in the plane and for the higher-dimensional generalization of
rectangular partitions
Contact Representations of Graphs in 3D
We study contact representations of graphs in which vertices are represented
by axis-aligned polyhedra in 3D and edges are realized by non-zero area common
boundaries between corresponding polyhedra. We show that for every 3-connected
planar graph, there exists a simultaneous representation of the graph and its
dual with 3D boxes. We give a linear-time algorithm for constructing such a
representation. This result extends the existing primal-dual contact
representations of planar graphs in 2D using circles and triangles. While
contact graphs in 2D directly correspond to planar graphs, we next study
representations of non-planar graphs in 3D. In particular we consider
representations of optimal 1-planar graphs. A graph is 1-planar if there exists
a drawing in the plane where each edge is crossed at most once, and an optimal
n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a
linear-time algorithm for representing optimal 1-planar graphs without
separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph
admits a representation with boxes. Hence, we consider contact representations
with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a
quadratic-time algorithm for representing optimal 1-planar graph with L-shaped
polyhedra
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