3,581 research outputs found
On polygons enclosing point sets II
Let R and B be disjoint point sets such that is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements
in R belong to the interior of P.
In this paper we prove that if the vertices of the convex hull of belong to B, and
|R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a
previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices \emph{p_1},...,\emph{p_n} and S a set of m points contained in the interior of P, m ≤ n−1. Then there is a convex decomposition {,...,} of P such that all points from S
lie on the boundaries of ,...,, and each contains a whole edge of P on its boundary.Postprint (published version
On Polygons Excluding Point Sets
By a polygonization of a finite point set in the plane we understand a
simple polygon having as the set of its vertices. Let and be sets
of blue and red points, respectively, in the plane such that is in
general position, and the convex hull of contains interior blue points
and interior red points. Hurtado et al. found sufficient conditions for the
existence of a blue polygonization that encloses all red points. We consider
the dual question of the existence of a blue polygonization that excludes all
red points . We show that there is a minimal number , which is
polynomial in , such that one can always find a blue polygonization
excluding all red points, whenever . Some other related problems are
also considered.Comment: 14 pages, 15 figure
On the geometric dilation of closed curves, graphs, and point sets
The detour between two points u and v (on edges or vertices) of an embedded
planar graph whose edges are curves is the ratio between the shortest path in
in the graph between u and v and their Euclidean distance. The maximum detour
over all pairs of points is called the geometric dilation. Ebbers-Baumann,
Gruene and Klein have shown that every finite point set is contained in a
planar graph whose geometric dilation is at most 1.678, and some point sets
require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower
bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem
of packing and covering the plane by circular disks.
The proof relies on halving pairs, pairs of points dividing a given closed
curve C in two parts of equal length, and their minimum and maximum distances h
and H. Additionally, we analyze curves of constant halving distance (h=H),
examine the relation of h to other geometric quantities and prove some new
dilation bounds.Comment: 31 pages, 16 figures. The new version is the extended journal
submission; it includes additional material from a conference submission
(ref. [6] in the paper
Strictly convex drawings of planar graphs
Every three-connected planar graph with n vertices has a drawing on an O(n^2)
x O(n^2) grid in which all faces are strictly convex polygons. These drawings
are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids.
More generally, a strictly convex drawing exists on a grid of size O(W) x
O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds
are obtained when the faces have fewer sides.
In the proof, we derive an explicit lower bound on the number of primitive
vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The
revision includes numerous small additions, corrections, and improvements, in
particular: - a discussion of the constants in the O-notation, after the
statement of thm.1. - a different set-up and clarification of the case
distinction for Lemma
Maximal Area Triangles in a Convex Polygon
The widely known linear time algorithm for computing the maximum area
triangle in a convex polygon was found incorrect recently by Keikha et.
al.(arXiv:1705.11035). We present an alternative algorithm in this paper.
Comparing to the only previously known correct solution, ours is much simpler
and more efficient. More importantly, our new approach is powerful in solving
related problems
Green's functions for multiply connected domains via conformal mapping
A method is described for the computation of the Green's function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz-Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K-1 slits. From the Green's function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iteration. By making the end of the strip jagged, the method can be generalised to weighted Green's functions and weighted approximations
Hamiltonian mappings and circle packing phase spaces
We introduce three area preserving maps with phase space structures which
resemble circle packings. Each mapping is derived from a kicked Hamiltonian
system with one of three different phase space geometries (planar, hyperbolic
or spherical) and exhibits an infinite number of coexisting stable periodic
orbits which appear to `pack' the phase space with circular resonances.Comment: 23 pages including 12 figures, REVTEX
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