Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Panoramic optical and near-infrared SETI instrument: optical and structural design concepts

We propose a novel instrument design to greatly expand the current optical and near-infrared SETI search parameter space by monitoring the entire observable sky during all observable time. This instrument is aimed to search for technosignatures by means of detecting nano- to micro-second light pulses that could have been emitted, for instance, for the purpose of interstellar communications or energy transfer. We present an instrument conceptual design based upon an assembly of 198 refracting 0.5-m telescopes tessellating two geodesic domes. This design produces a regular layout of hexagonal collecting apertures that optimizes the instrument footprint, aperture diameter, instrument sensitivity and total field-of-view coverage. We also present the optical performance of some Fresnel lenses envisaged to develop a dedicated panoramic SETI (PANOSETI) observatory that will dramatically increase sky-area searched (pi steradians per dome), wavelength range covered, number of stellar systems observed, interstellar space examined and duration of time monitored with respect to previous optical and near-infrared technosignature finders.Comment: 14 pages, 5 figures, 3 table

Ramification conjecture and Hirzebruch's property of line arrangements

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on the complex projective plane with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a CAT[0] ramification and prove this in several cases. In the latter case we prove that the ramification is CAT[0] if the metric is non-negatively curved. We deduce that complex line arrangements in the complex projective plane studied by Hirzebruch have aspherical complement.Comment: 19 pages 1 figur

Attaching handlebodies to 3-manifolds

The main theorem of this paper is a generalisation of well known results about Dehn surgery to the case of attaching handlebodies to a simple 3-manifold. The existence of a finite set of `exceptional' curves on the boundary of the 3-manifold is established. Provided none of these curves is attached to the boundary of a disc in a handlebody, the resulting manifold is shown to be word hyperbolic and `hyperbolike'. We then give constructions of gluing maps satisfying this condition. These take the form of an arbitrary gluing map composed with powers of a suitable homeomorphism of the boundary of the handlebodies.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper26.abs.htm

Reappraising the distribution of the number of edge crossings of graphs on a sphere

Many real transportation and mobility networks have their vertices placed on the surface of the Earth. In such embeddings, the edges laid on that surface may cross. In his pioneering research, Moon analyzed the distribution of the number of crossings on complete graphs and complete bipartite graphs whose vertices are located uniformly at random on the surface of a sphere assuming that vertex placements are independent from each other. Here we revise his derivation of that variance in the light of recent theoretical developments on the variance of crossings and computer simulations. We show that Moon's formulae are inaccurate in predicting the true variance and provide exact formulae.Comment: Corrected mistakes in equation 31. Added new figure (7). Added acknowledgements to J. W. Moon. Other minor changes. Updated figures. Minor changes in the last updat