2,878 research outputs found
Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension
We study the combinatorial complexity of D-dimensional polyhedra defined as
the intersection of n halfspaces, with the property that the highest dimension
of any bounded face is much smaller than D. We show that, if d is the maximum
dimension of a bounded face, then the number of vertices of the polyhedron is
O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For
inputs in general position the number of bounded faces is O(n^d). For any fixed
d, we show how to compute the set of all vertices, how to determine the maximum
dimension of a bounded face of the polyhedron, and how to compute the set of
bounded faces in polynomial time, by solving a polynomial number of linear
programs
Computing largest circles separating two sets of segments
A circle separates two planar sets if it encloses one of the sets and its
open interior disk does not meet the other set. A separating circle is a
largest one if it cannot be locally increased while still separating the two
given sets. An Theta(n log n) optimal algorithm is proposed to find all largest
circles separating two given sets of line segments when line segments are
allowed to meet only at their endpoints. In the general case, when line
segments may intersect times, our algorithm can be adapted to
work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n)
represents the extremely slowly growing inverse of the Ackermann function.Comment: 14 pages, 3 figures, abstract presented at 8th Canadian Conference on
Computational Geometry, 199
Geometric Rounding and Feature Separation in Meshes
Geometric rounding of a mesh is the task of approximating its vertex
coordinates by floating point numbers while preserving mesh structure.
Geometric rounding allows algorithms of computational geometry to interface
with numerical algorithms. We present a practical geometric rounding algorithm
for 3D triangle meshes that preserves the topology of the mesh. The basis of
the algorithm is a novel strategy: 1) modify the mesh to achieve a feature
separation that prevents topology changes when the coordinates change by the
rounding unit; and 2) round each vertex coordinate to the closest floating
point number. Feature separation is also useful on its own, for example for
satisfying minimum separation rules in CAD models. We demonstrate a robust,
accurate implementation
Searching for integrable Hamiltonian systems with Platonic symmetries
In this paper we try to find examples of integrable natural Hamiltonian
systems on the sphere with the symmetries of each Platonic polyhedra.
Although some of these systems are known, their expression is extremely
complicated; we try here to find the simplest possible expressions for this
kind of dynamical systems. Even in the simplest cases it is not easy to prove
their integrability by direct computation of the first integrals, therefore, we
make use of numerical methods to provide evidences of integrability; namely, by
analyzing their Poincar\'e sections (surface sections). In this way we find
three systems with platonic symmetries, one for each class of equivalent
Platonic polyhedra: tetrahedral, exahedral-octahedral,
dodecahedral-icosahedral, showing evidences of integrability. The proof of
integrability and the construction of the first integrals are left for further
works. As an outline of the possible developments if the integrability of these
systems will be proved, we show how to build from them new integrable systems
in dimension three and, from these, superintegrable systems in dimension four
corresponding to superintegrable interactions among four points on a line, in
analogy with the systems with dihedral symmetry treated in a previous article.
A common feature of these possibly integrable systems is, besides to the rich
symmetry group on the configuration manifold, the partition of the latter into
dynamically separated regions showing a simple structure of the potential in
their interior. This observation allows to conjecture integrability for a class
of Hamiltonian systems in the Euclidean spaces.Comment: 22 pages; 4 figure
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