The supercover of an m-flat is a discrete analytical object

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Digital objects in rhombic dodecahedron grid

Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system

Efficient voxelization using projected optimal scanline

In the paper, we propose an efficient algorithm for the surface voxelization of 3D geometrically complex models. Unlike recent techniques relying on triangle-voxel intersection tests, our algorithm exploits the conventional parallel-scanline strategy. Observing that there does not exist an optimal scanline interval in general 3D cases if one wants to use parallel voxelized scanlines to cover the interior of a triangle, we subdivide a triangle into multiple axis-aligned slices and carry out the scanning within each polygonal slice. The theoretical optimal scanline interval can be obtained to maximize the efficiency of the algorithm without missing any voxels on the triangle. Once the collection of scanlines are determined and voxelized, we obtain the surface voxelization. We fine tune the algorithm so that it only involves a few operations of integer additions and comparisons for each voxel generated. Finally, we comprehensively compare our method with the state-of-the-art method in terms of theoretical complexity, runtime performance and the quality of the voxelization on both CPU and GPU of a regular desktop PC, as well as on a mobile device. The results show that our method outperforms the existing method, especially when the resolution of the voxelization is high

A fast voxelization algorithm for trilinearly interpolated isosurfaces

International audienceIn this work we propose a new method for a fast incremental voxelization of isosurfaces obtained by the trilinear interpolation of 3D data. Our objective consists in the fast generation of subvoxelized iso-surfaces extracted by a point-based technique similar to the Dividing Cubes algorithm. Our technique involves neither an exhaustive scan search process nor a graph-based search approach when generating iso-surface points. Instead an optimized incremental approach is adopted here for a rapid isosurface extraction. With a sufficient sampling subdivision criteria around critical points, the extracted isosurface is both correct and topologically consistent with respect to the piece-wise trilinear interpolant. Furthermore, the discretiza-tion scheme used in our method ensures obtaining thin-one voxel width-isosurfaces as compared to the given by the Dividing Cubes algorithm. The resultant sub-voxelized isosurfaces are efficiently tested against all possible configurations of the trilinear interpolant and real-world datasets

PlantGL : a Python-based geometric library for 3D plant modelling at different scales

In this paper, we present PlantGL, an open-source graphic toolkit for the creation, simulation and analysis of 3D virtual plants. This C++ geometric library is embedded in the Python language which makes it a powerful user-interactive platform for plant modelling in various biological application domains. PlantGL makes it possible to build and manipulate geometric models of plants or plant parts, ranging from tissues and organs to plant populations. Based on a scene graph augmented with primitives dedicated to plant representation, several methods are provided to create plant architectures from either field measurements or procedural algorithms. Because they reveal particularly useful in plant design and analysis, special attention has been paid to the definition and use of branching system envelopes. Several examples from different modelling applications illustrate how PlantGL can be used to construct, analyse or manipulate geometric models at different scales

A Stronger Strong Schottky Lemma for Euclidean Buildings

We provide a criterion for two hyperbolic isometries of a Euclidean building to generate a free group of rank two. In particular, we extend the application of a Strong Schottky Lemma to buildings given by Alperin, Farb and Noskov. We then use this extension to obtain an infinite family of matrices that generate a free group of rank two. In doing so, we also introduce an algorithm that terminates in finite time if the lemma is applicable for pairs of certain kinds of matrices acting on the Euclidean building for the special linear group over certain discretely valued fields