Gaia is a cornerstone mission in the science programme of the European Space Agency (ESA). The spacecraft construction was approved in 2006, following a study in which the original interferometric concept was changed to a direct-imaging approach. Both the spacecraft and the payload were built by European industry. The involvement of the scientific community focusses on data processing for which the international Gaia Data Processing and Analysis Consortium (DPAC) was selected in 2007. Gaia was launched on 19 December 2013 and arrived at its operating point, the second Lagrange point of the Sun-Earth-Moon system, a few weeks later. The commissioning of the spacecraft and payload was completed on 19 July 2014. The nominal five-year mission started with four weeks of special, ecliptic-pole scanning and subsequently transferred into full-sky scanning mode. We recall the scientific goals of Gaia and give a description of the as-built spacecraft that is currently (mid-2016) being operated to achieve these goals. We pay special attention to the payload module, the performance of which is closely related to the scientific performance of the mission. We provide a summary of the commissioning activities and findings, followed by a description of the routine operational mode. We summarise scientific performance estimates on the basis of in-orbit operations. Several intermediate Gaia data releases are planned and the data can be retrieved from the Gaia Archive, which is available through the Gaia home page.

Context. At about 1000 days after the launch of Gaia we present the first Gaia data release, Gaia DR1, consisting of astrometry and photometry for over 1 billion sources brighter than magnitude 20.7. Aims. A summary of Gaia DR1 is presented along with illustrations of the scientific quality of the data, followed by a discussion of the limitations due to the preliminary nature of this release. Methods. The raw data collected by Gaia during the first 14 months of the mission have been processed by the Gaia Data Processing and Analysis Consortium (DPAC) and turned into an astrometric and photometric catalogue. Results. Gaia DR1 consists of three components: a primary astrometric data set which contains the positions, parallaxes, and mean proper motions for about 2 million of the brightest stars in common with the Hipparcos and Tycho-2 catalogues -a realisation of the Tycho-Gaia Astrometric Solution (TGAS) -and a secondary astrometric data set containing the positions for an additional 1.1 billion sources. The second component is the photometric data set, consisting of mean G-band magnitudes for all sources. The G-band light curves and the characteristics of ∼3000 Cepheid and RR Lyrae stars, observed at high cadence around the south ecliptic pole, form the third component. For the primary astrometric data set the typical uncertainty is about 0.3 mas for the positions and parallaxes, and about 1 mas yr −1 for the proper motions. A systematic component of ∼0.3 mas should be added to the parallax uncertainties. For the subset of ∼94 000 Hipparcos stars in the primary data set, the proper motions are much more precise at about 0.06 mas yr −1 . For the secondary astrometric data set, the typical uncertainty of the positions is ∼10 mas. The median uncertainties on the mean G-band magnitudes range from the mmag level to ∼0.03 mag over the magnitude range 5 to 20.7. Conclusions. Gaia DR1 is an important milestone ahead of the next Gaia data release, which will feature five-parameter astrometry for all sources. Extensive validation shows that Gaia DR1 represents a major advance in the mapping of the heavens and the availability of basic stellar data that underpin observational astrophysics. Nevertheless, the very preliminary nature of this first Gaia data release does lead to a number of important limitations to the data quality which should be carefully considered before drawing conclusions from the data.

Quadtree representation of two-dimensional objects is performed with a tree that describes the recursive subdivision of the more complex parts of a picture until the desired resolution is reached. At the end, all the leaves of the tree are square cells that lie completely inside or outside the object. There are two great disadvantages in the use of quadtrees as a representation scheme for objects in geometric modeling system: The amount of memory required for polygonal objects is too great, and it is difficult to recompute the boundary representation of the object after some Boolean operations have been performed. In the present paper a new class of quadtrees, in which nodes may contain zero or one edge, is introduced. By using these quadtrees, storage requirements are reduced and it is possible to obtain the exact backward conversion to boundary representation. Algorithms for the generation of the quadtree, Boolean operations, and recomputation of the boundary representation are presented, and their complexities in time and space are discussed. Three-dimensional algorithms working on octrees are also presented. Their use in the geometric modeling of three-dimensional polyhedral objects is discussed.

Solid modelers must be based on reliable and fast algorithms for Boolean operations. The octree model, as well as several generalizations (polytrees, integrated polytrees, extended octrees), is specially well suited for these algorithms and can be used either as a primary or as a secondary model in solid modeling systems. This paper is concerned with a precise definition of the extended octree model that allows the representation of nonmanifold objects with planar faces and, consequently, is closed under Boolean operations on polyhedrons. Boolean nodes and nearly vertex nodes are introduced, and the model is discussed in comparison with related representations. A fast algorithm for the direct generation of the extended octree from the geometry of the base polygon in extrusion solids is presented, and its complexity is studied. Boolean operation algorithms are introduced.