29,596 research outputs found
Convex Hulls: Complexity and Applications (a Survey)
Computational geometry is, in brief, the study of algorithms for geometric problems. Classical study of geometry and geometric objects, however, is not well-suited to efficient algorithms techniques. Thus, for the given geometric problems, it becomes necessary to identify properties and concepts that lend themselves to efficient computation. The primary focus of this paper will be on one such geometric problems, the Convex Hull problem
Aspects Of Combinatorial Geometry
This thesis presents solutions to various problems in the expanding field of combinatorial geometry. Chapter 1 gives an introduction to the theory of the solution of an integer programming problem, that is maximising a linear form with integer variables subject to a number of constraints. Since the maximum value of the linear form occurs at a vertex of the convex hull of integer points defined by the constraints, it is of interest to estimate the number of these vertices. Chapter 2 describes the application of certain geometrical interpretations of number theory to the solution of integer programming problems in the plane. By using, in part, the well-known Klein interpretation of continued fractions, a method of constructing the vertices of the convex hull of integer points defined by particular constraints is developed. Bounds for the number of these vertices and properties of certain special cases are given. Chapter 3 considers the general d-dimensional integer programming problem. Upper and lower bounds are presented for the number of vertices of the convex hull of integer points defined by particular constraints. Chapter 4 is concerned with the approximation of convex sets by convex polytopes. First, a detailed description of recent work on minimal circumscribing triangles for convex polygons and the extension to minimal circumscribing equilateral triangles is given. This leads to a new approach to constructing a Borsuk Division and finding a regular hexagon circumscribing a convex polygon. Then, a method of approximating general convex sets by convex polytopes is presented, leading to consideration of the problem of a d-simplex approximating a d-ball. Chapter 5 develops algorithms for finding points with particular combinatorial properties, using containment objects such as balls, closed half-spaces and ellipsoids. Chapter 6 gives a new approach to the problem of inscribing a square in a convex polygon, leading to possible ideas for an algorithm
Slime mould computes planar shapes
Computing a polygon defining a set of planar points is a classical problem of
modern computational geometry. In laboratory experiments we demonstrate that a
concave hull, a connected alpha-shape without holes, of a finite planar set is
approximated by slime mould Physarum polycephalum. We represent planar points
with sources of long-distance attractants and short-distance repellents and
inoculate a piece of plasmodium outside the data set. The plasmodium moves
towards the data and envelops it by pronounced protoplasmic tubes
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Given a set of spheres in , with and
odd, having a fixed number of distinct radii , we
show that the worst-case combinatorial complexity of the convex hull
of is
, where
is the number of spheres in with radius .
To prove the lower bound, we construct a set of spheres in
, with odd, where spheres have radius ,
, and , such that their convex hull has combinatorial
complexity
.
Our construction is then generalized to the case where the spheres have
distinct radii.
For the upper bound, we reduce the sphere convex hull problem to the problem
of computing the worst-case combinatorial complexity of the convex hull of a
set of -dimensional convex polytopes lying on parallel hyperplanes
in , where odd, a problem which is of independent
interest. More precisely, we show that the worst-case combinatorial complexity
of the convex hull of a set
of -dimensional convex polytopes lying on parallel hyperplanes of
is
, where
is the number of vertices of .
We end with algorithmic considerations, and we show how our tight bounds for
the parallel polytope convex hull problem, yield tight bounds on the
combinatorial complexity of the Minkowski sum of two convex polytopes in
.Comment: 22 pages, 5 figures, new proof of upper bound for the complexity of
the convex hull of parallel polytopes (the new proof gives upper bounds for
all face numbers of the convex hull of the parallel polytopes
Geometric Reasoning with polymake
The mathematical software system polymake provides a wide range of functions
for convex polytopes, simplicial complexes, and other objects. A large part of
this paper is dedicated to a tutorial which exemplifies the usage. Later
sections include a survey of research results obtained with the help of
polymake so far and a short description of the technical background
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