Searchable Sky Coverage of Astronomical Observations: Footprints and Exposures

Sky coverage is one of the most important pieces of information about astronomical observations. We discuss possible representations, and present algorithms to create and manipulate shapes consisting of generalized spherical polygons with arbitrary complexity and size on the celestial sphere. This shape specification integrates well with our Hierarchical Triangular Mesh indexing toolbox, whose performance and capabilities are enhanced by the advanced features presented here. Our portable implementation of the relevant spherical geometry routines comes with wrapper functions for database queries, which are currently being used within several scientific catalog archives including the Sloan Digital Sky Survey, the Galaxy Evolution Explorer and the Hubble Legacy Archive projects as well as the Footprint Service of the Virtual Observatory.Comment: 11 pages, 7 figures, submitted to PAS

Flip Distance Between Triangulations of a Planar Point Set is APX-Hard

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set $S$ in the Euclidean plane and two triangulations $T_1$ and $T_2$ of $S$, it is an APX-hard problem to minimize the number of edge flips to transform $T_1$ to $T_2$.Comment: A previous version only showed NP-completeness of the corresponding decision problem. The current version is the one of the accepted manuscrip

A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

The Footprint Database and Web Services of the Herschel Space Observatory

Data from the Herschel Space Observatory is freely available to the public but no uniformly processed catalogue of the observations has been published so far. To date, the Herschel Science Archive does not contain the exact sky coverage (footprint) of individual observations and supports search for measurements based on bounding circles only. Drawing on previous experience in implementing footprint databases, we built the Herschel Footprint Database and Web Services for the Herschel Space Observatory to provide efficient search capabilities for typical astronomical queries. The database was designed with the following main goals in mind: (a) provide a unified data model for meta-data of all instruments and observational modes, (b) quickly find observations covering a selected object and its neighbourhood, (c) quickly find every observation in a larger area of the sky, (d) allow for finding solar system objects crossing observation fields. As a first step, we developed a unified data model of observations of all three Herschel instruments for all pointing and instrument modes. Then, using telescope pointing information and observational meta-data, we compiled a database of footprints. As opposed to methods using pixellation of the sphere, we represent sky coverage in an exact geometric form allowing for precise area calculations. For easier handling of Herschel observation footprints with rather complex shapes, two algorithms were implemented to reduce the outline. Furthermore, a new visualisation tool to plot footprints with various spherical projections was developed. Indexing of the footprints using Hierarchical Triangular Mesh makes it possible to quickly find observations based on sky coverage, time and meta-data. The database is accessible via a web site (http://herschel.vo.elte.hu) and also as a set of REST web service functions.Comment: Accepted for publication in Experimental Astronom

Quasi-Fuchsian manifolds with particles

We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $\pi$: any first-order deformation changes either one of those angles or the conformal structure at infinity, with marked points corresponding to the endpoints of the singular lines. Moreover, any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone-manifold structure.Comment: Now 48 pages, no figure. v2: new title, various corrections, results extended to include graph singularities ("interacting particles"). v3: various corrections/improvements, in particular thanks to comments by an anonymous refere