427 research outputs found
Circumscribing Polygons and Polygonizations for Disjoint Line Segments
Given a planar straight-line graph G=(V,E) in R^2, a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P.
We prove that every arrangement of n disjoint line segments in the plane has a subset of size Omega(sqrt{n}) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R^3.
We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization
Partitioning Regular Polygons into Circular Pieces II:Nonconvex Partitions
We explore optimal circular nonconvex partitions of regular k-gons. The
circularity of a polygon is measured by its aspect ratio: the ratio of the
radii of the smallest circumscribing circle to the largest inscribed disk. An
optimal circular partition minimizes the maximum ratio over all pieces in the
partition. We show that the equilateral triangle has an optimal 4-piece
nonconvex partition, the square an optimal 13-piece nonconvex partition, and
the pentagon has an optimal nonconvex partition with more than 20 thousand
pieces. For hexagons and beyond, we provide a general algorithm that approaches
optimality, but does not achieve it.Comment: 13 pages, 11 figure
Conforming restricted Delaunay mesh generation for piecewise smooth complexes
A Frontal-Delaunay refinement algorithm for mesh generation in piecewise
smooth domains is described. Built using a restricted Delaunay framework, this
new algorithm combines a number of novel features, including: (i) an
unweighted, conforming restricted Delaunay representation for domains specified
as a (non-manifold) collection of piecewise smooth surface patches and curve
segments, (ii) a protection strategy for domains containing curve segments that
subtend sharply acute angles, and (iii) a new class of off-centre refinement
rules designed to achieve high-quality point-placement along embedded curve
features. Experimental comparisons show that the new Frontal-Delaunay algorithm
outperforms a classical (statically weighted) restricted Delaunay-refinement
technique for a number of three-dimensional benchmark problems.Comment: To appear at the 25th International Meshing Roundtabl
Relative Convex Hull Determination from Convex Hulls in the Plane
A new algorithm for the determination of the relative convex hull in the
plane of a simple polygon A with respect to another simple polygon B which
contains A, is proposed. The relative convex hull is also known as geodesic
convex hull, and the problem of its determination in the plane is equivalent to
find the shortest curve among all Jordan curves lying in the difference set of
B and A and encircling A. Algorithms solving this problem known from
Computational Geometry are based on the triangulation or similar decomposition
of that difference set. The algorithm presented here does not use such
decomposition, but it supposes that A and B are given as ordered sequences of
vertices. The algorithm is based on convex hull calculations of A and B and of
smaller polygons and polylines, it produces the output list of vertices of the
relative convex hull from the sequence of vertices of the convex hull of A.Comment: 15 pages, 4 figures, Conference paper published. We corrected two
typing errors in Definition 2: has to be defined based on , and
has to be defined based on (not just using ). These errors
appeared in the text of the original conference paper, which also contained
the pseudocode of an algorithm where and appeared as correctly
define
On Chaotic Dynamics in Rational Polygonal Billiards
We discuss the interplay between the piece-line regular and vertex-angle
singular boundary effects, related to integrability and chaotic features in
rational polygonal billiards. The approach to controversial issue of regular
and irregular motion in polygons is taken within the alternative deterministic
and stochastic frameworks. The analysis is developed in terms of the
billiard-wall collision distribution and the particle survival probability,
simulated in closed and weakly open polygons, respectively. In the multi-vertex
polygons, the late-time wall-collision events result in the circular-like
regular periodic trajectories (sliding orbits), which, in the open billiard
case are likely transformed into the surviving collective excitations
(vortices). Having no topological analogy with the regular orbits in the
geometrically corresponding circular billiard, sliding orbits and vortices are
well distinguished in the weakly open polygons via the universal and
non-universal relaxation dynamics.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
- …