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therefore a measure of the local density of the point distribution. This property of the
Delaunay tessellation is exploited in step 2 of the DTFE, in which the local density is estimated at the locations of the sampling points. For this purpose the density is defined at the location of each sampling point as the inverse of the area of its surrounding Delaunay triangles (times a normalization constant, see figure, lower right-hand frame).
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of the point distribution is constructed. This is a volume-covering division of space into triangles (tetrahedra in three dimensions), whose vertices are formed by the point distribution (see figure, upper right-hand frame). The
Delaunay tessellation is defined such that inside the interior of the
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and improving computer simulation programs of cosmic structure formation. It has been developed by Willem Schaap and Rien van de
Weijgaert. The main advantage of the DTFE is that it automatically adapts to (strong) variations in density and geometry. It is therefore very well suited for studies of
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The DTFE has been designed for reconstructing density or intensity fields from a discrete set of irregularly distributed points sampling this field. However, it can also be used to reconstruct other continuous fields which have been sampled at the locations of these points, for example the cosmic
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The
Delaunay tessellation forms the heart of the DTFE. In the figure it is clearly visible that the tessellation automatically adapts to both the local density and geometry of the point distribution: where the density is high, the triangles are small and vice versa. The size of the triangles is
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The starting point is a given discrete point distribution. In the upper left-hand frame of the figure, a point distribution is plotted in which at the center of the frame an object is located whose density diminishes radially outwards. In the first step of the DTFE, the
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velocity field. The use of the DTFE for this purpose has the same advantages as it has for reconstructing density fields. The fields are reconstructed locally without the application of an artificial or user-dependent
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In step 3 these density estimates are interpolated to any other point, by assuming that inside each
Delaunay triangle the density field varies linearly (see figure, lower left-hand frame).
162:(SPH) density estimation procedure. Replacing it by the DTFE density estimate will yield a major improvement for simulations incorporating feedback processes, which play a major role in
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Most algorithms for simulating cosmic structure formation are particle hydrodynamics codes. At the core of these codes is the
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One of the main applications of the DTFE is the rendering of our cosmic neighborhood. Below the DTFE reconstruction of the
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effects. The estimated quantities are volume-covering and allow for a direct comparison with theoretical predictions.
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circumcircle of each
Delaunay triangle no other points from the defining point distribution are present.
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DTFE velocity field reconstructions of superclusters and voids in the large scale galaxy distribution.
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NASA Astronomy
Picture of the Day: The Sloan Great Wall: Largest Known Structure? (7 November 2007)
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is shown, revealing an impressive view on the cosmic structures in the nearby universe. Several
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The reconstruction of a density field from a discrete set of points sampling this field.
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The DTFE has been specifically designed for describing the complex properties of the
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Mathematical tool for reconstructing a density field from a discrete point set
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DTFE reconstruction of the inner parts of the 2dF Galaxy
Redshift Survey
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Probing cosmic velocity flows in the local universe
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247:DTFE: the Delaunay Tessellation Field Estimator
266:, Rien van de Weygaert and Willem Schaap, 2004
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154:Numerical simulations of structure formation
29:Delaunay tessellation field estimator (DTFE)
33:Delone tessellation field estimator (DTFE)
203:Evolution and dynamics of the cosmic web
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67:Overview of the DTFE procedure.
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