1,189 research outputs found
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
Lines pinning lines
A line g is a transversal to a family F of convex polytopes in 3-dimensional
space if it intersects every member of F. If, in addition, g is an isolated
point of the space of line transversals to F, we say that F is a pinning of g.
We show that any minimal pinning of a line by convex polytopes such that no
face of a polytope is coplanar with the line has size at most eight. If, in
addition, the polytopes are disjoint, then it has size at most six. We
completely characterize configurations of disjoint polytopes that form minimal
pinnings of a line.Comment: 27 pages, 10 figure
Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces
We state that any constant curvature Riemannian metric with conical
singularities of constant sign curvature on a compact (orientable) surface
can be realized as a convex polyhedron in a Riemannian or Lorentzian)
space-form. Moreover such a polyhedron is unique, up to global isometries,
among convex polyhedra invariant under isometries acting on a totally umbilical
surface. This general statement falls apart into 10 different cases. The cases
when is the sphere are classical.Comment: Survey paper. No proof. 10 page
Flipping Cubical Meshes
We define and examine flip operations for quadrilateral and hexahedral
meshes, similar to the flipping transformations previously used in triangular
and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th
International Meshing Roundtable. This version removes some unwanted
paragraph breaks from the previous version; the text is unchange
Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
Positive geometries encode the physics of scattering amplitudes in flat
space-time and the wavefunction of the universe in cosmology for a large class
of models. Their unique canonical forms, providing such quantum mechanical
observables, are characterised by having only logarithmic singularities along
all the boundaries of the positive geometry. However, physical observables have
logarithmic singularities just for a subset of theories. Thus, it becomes
crucial to understand whether a similar paradigm can underlie their structure
in more general cases. In this paper we start a systematic investigation of a
geometric-combinatorial characterisation of differential forms with
non-logarithmic singularities, focusing on projective polytopes and related
meromorphic forms with multiple poles. We introduce the notions of covariant
forms and covariant pairings. Covariant forms have poles only along the
boundaries of the given polytope; moreover, their leading Laurent coefficients
along any of the boundaries are still covariant forms on the specific boundary.
Whereas meromorphic forms in covariant pairing with a polytope are associated
to a specific (signed) triangulation, in which poles on spurious boundaries do
not cancel completely, but their order is lowered. These meromorphic forms can
be fully characterised if the polytope they are associated to is viewed as the
restriction of a higher dimensional one onto a hyperplane. The canonical form
of the latter can be mapped into a covariant form or a form in covariant
pairing via a covariant restriction. We show how the geometry of the higher
dimensional polytope determines the structure of these differential forms.
Finally, we discuss how these notions are related to Jeffrey-Kirwan residues
and cosmological polytopes.Comment: 47 pages, figures in Tik
Overlapping Unit Cells in 3d Quasicrystal Structure
A 3-dimensional quasiperiodic lattice, with overlapping unit cells and
periodic in one direction, is constructed using grid and projection methods
pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are
the vertices of a convex polytope P, and 4 are interior points also shared with
other neighboring unit cells. Using Kronecker's theorem the frequencies of all
possible types of overlapping are found.Comment: LaTeX2e, 11 pages, 5 figures (8 eps files), uses iopart.class. Final
versio
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