Walking Faster in a Triangulation

Point location in a triangulation is one of the most studied problems in computational geometry. For a single query, stochastic walk is a good practical strategy. In this work, we propose two approaches improving the performance of the stochastic walk. The first improvement is based on a relaxation of the exactness of the predicate, whereas the second is based on termination guessing.La localisation d'un point dans une triangulation est un des problèmes les plus étudiés en géométrie algorithmique. Pour un petit nombre de requêtes, la marche stochastique est une bonne stratégie en pratique. Dans ce travail, nous proposons deux idées qui améliorent les performances de la marche stochastique. La première est basée sur une relaxation de l'exactitude du prédicat d'orientation, tandis que la deuxième est basée sur lune tentative de divination de la longueur de cette marche

One machine, one minute, three billion tetrahedra

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multi-threaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator which takes as input the triangulated surface boundary of the volume to mesh

The centers of gravity of the associahedron and of the permutahedron are the same

In this article, we show that Loday's realization of the associahedron has the the same center of gravity than the permutahedron. This proves an observation made by F. Chapoton. We also prove that this result holds for the associahedron and the cyclohedron as realized by the first author and C. Lange

Angular Scale Expansion Theory And The Misperception Of Egocentric Distance In Locomotor Space

Perception is crucial for the control of action, but perception need not be scaled accurately to produce accurate actions. This paper reviews evidence for an elegant new theory of locomotor space perception that is based on the dense coding of angular declination so that action control may be guided by richer feedback. The theory accounts for why so much direct-estimation data suggests that egocentric distance is underestimated despite the fact that action measures have been interpreted as indicating accurate perception. Actions are calibrated to the perceived scale of space and thus action measures are typically unable to distinguish systematic (e.g., linearly scaled) misperception from accurate perception. Whereas subjective reports of the scaling of linear extent are difficult to evaluate in absolute terms, study of the scaling of perceived angles (which exist in a known scale, delimited by vertical and horizontal) provides new evidence regarding the perceptual scaling of locomotor space