Методы построение сеток в трехмерных областях

The methods of discretisation are considered in three-dimensional areas. Direct and iteration methods are described, and similarly a method is built on a akvidistetion net. A template is created with high exactness of description of akvidistetion netРассмотрены методы дискретизации в трехмерных областях. Описаны прямые и итерационные методы, а так же метод построен на эквидистантной сетке. Создан шаблон с высокой точностью описания эквидистантной сеткиРозглянуті методи дискретизації в тривимірних областях. Описані прямі і ітераційні методи, а так само метод побудований на еквідистантній сітці. Створений шаблон з високою точністю опису еквідистантної сітк

Методы построение сеток в трехмерных областях

The methods of discretisation are considered in three-dimensional areas. Direct and iteration methods are described, and similarly a method is built on a akvidistetion net. A template is created with high exactness of description of akvidistetion netРассмотрены методы дискретизации в трехмерных областях. Описаны прямые и итерационные методы, а так же метод построен на эквидистантной сетке. Создан шаблон с высокой точностью описания эквидистантной сеткиРозглянуті методи дискретизації в тривимірних областях. Описані прямі і ітераційні методи, а так само метод побудований на еквідистантній сітці. Створений шаблон з високою точністю опису еквідистантної сітк

XOR-Based Compact Triangulations

Media, image processing, and geometric-based systems and applications need data structures to model and represent different geometric entities and objects. These data structures have to be time efficient and compact in term of space. Many structures in use are proposed to satisfy those constraints. This paper introduces a novel compact data structure inspired by the XOR-linked lists. The subject of this paper concerns the triangular data structures. Nevertheless, the underlying idea could be used for any other geometrical subdivision. The ability of the bitwise XOR operator to reduce the number of references is used to model triangle and vertex references. The use of the XOR combined references needs to define a context from which the triangle is accessed. The direct access to any triangle is not possible using only the XOR-linked scheme. To allow the direct access, additional information are added to the structure. This additional information permits a constant time access to any element of the triangulation using a local resolution scheme. This information represents an additional cost to the triangulation, but the gain is still maintained. This cost is reduced by including this additional information to a local sub-triangulation and not to each triangle. Sub-triangulations are calculated implicitly according to the catalog-based structure. This approach could be easily extended to other representation models, such as vertex-based structures or edge-based structures. The obtained results are very interesting since the theoretical gain is estimated to 38 % and the practical gain obtained from sample benches is about 34 %

The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website

Validation and automatic repair of planar partitions using a constrained triangulation

Planar partitions are frequently used to model, among others, land cover, cadastral parcels and administrative boundaries. In practice, they are often stored as a set of individual polygons to which attributes are attached (e.g. with the Simple Features paradigm), causing different errors and inconsistencies (e.g. gaps, overlaps and disconnected polygons), which are introduced during their creation, manipulation and exchange. These errors severely hamper the use of planar partitions in other software (e.g. due to false assumptions causing erroneous calculations). Existing approaches to validate planar partitions involve first building a planar graph of the polygons and enforcing constraints, then repair is done by snapping vertices and edges of this graph. We argue that these approaches have many shortcomings in terms of complexity, numerical robustness and difficulty of implementation, and do not guarantee valid results. Furthermore, they are semi-automatic, requiring manual user intervention. We propose in this paper a novel method to validate and automatically repair planar partitions. It uses a constrained triangulation of the polygons as a base-which by definition is a planar partition-and only simple operations are needed (i.e. labelling of triangles) to both validate and repair. Perhaps the biggest advantage of our method is that we can guarantee that a planar partition is valid after repair. In the paper we describe the details of our method, our implementation, and the experiments we have done with real-world datasets. We show that our implementation scales to big datasets and that it offers better capabilities and overall performance than existing solutions

Erzeugung von 3D-Netzmodellen in der Produktentwicklung durch Deformation initialer 3D-Netzmodelle

Mit 3D-Netzmodellen werden Objekte der materiellen Welt oder unserer Vorstellung computergestützt abgebildet. In digitalen Produktentwicklungsprozessen werden mit ihnen sowohl die Objektgestalt als auch anwendungsspezifische Informationen von Objekten und von Prozessen definiert. Mit flächenhaften Netzen (z. B. Dreiecksnetze) wird die Oberfläche von Objekten in diskreter Form repräsentiert, mit volumenhaften Netzen (z. B. Tetraedernetze) zusätzlich das Objektinnere. 3D-Netzmodelle werden bei der Erzeugung und der Manipulation, der Analyse und der Validierung, in fertigungsvorbereitenden Prozessen sowie zur Präsentation digitaler 3D-Objekte angewandt

Compact Representations of Simplicial Meshes in Two and Three Dimensions

We describe data structures for representing simplicial meshes compactly while supporting online queries and updates efficiently. Our representation requires about a factor of five times less memory than the most efficient standard representations of triangular or tetrahedral meshes, while efficiently supporting traversal among simpices, storing data on simpices, and insertion and deletion of simplices. Our implementation of the data structures uses about 5 bytes/triangle in two dimensions (2D) and 7.5 bytes/tetrahedron in three dimensions (3D). We use the representations to implement 2D and 3D incremental algorithms for generating a Delaunay mesh. The 3D algorithm can generate 100 Million tetrahedrons with 1 Gbyte of memory, including the space for the coordinates and all data used by the algorithm. The runtime of the algorithm is as fast a Shewchuk's Pyramid code, the most efficient we know of, and uses a factor of 3.5 less memory overall

Compact representations of simplicial meshes in two and three dimensions

We describe data structures for representing simplicial meshes compactly while supporting online queries and updates efficiently. Our data structure requires about a factor of five less memory than the most efficient standard data structures for triangular or tetrahedral meshes, while efficiently supporting traversal among simplices, storing data on simplices, and insertion and deletion of simplices. Our implementation of the data structures uses about 5 bytes/triangle in two dimensions (2D) and 7.5 bytes/tetrahedron in three dimensions (3D). We use the data structures to implement 2D and 3D incremental algorithms for generating a Delaunay mesh. The 3D algorithm can generate 100 Million tetrahedrons with 1 Gbyte of memory, including the space for the coordinates and all data used by the algorithm. The runtime of the algorithm is as fast as Shewchuk’s Pyramid code, the most efficient we know of, and uses a factor of 3.5 less memory overall.

Computations of Delaunay and Higher Order Triangulations, with Applications to Splines

Digital data that consist of discrete points are frequently captured and processed by scientific and engineering applications. Due to the rapid advance of new data gathering technologies, data set sizes are increasing, and the data distributions are becoming more irregular. These trends call for new computational tools that are both efficient enough to handle large data sets and flexible enough to accommodate irregularity. A mathematical foundation that is well-suited for developing such tools is triangulation, which can be defined for discrete point sets with little assumption about their distribution. The potential benefits from using triangulation are not fully exploited. The challenges fundamentally stem from the complexity of the triangulation structure, which generally takes more space to represent than the input points. This complexity makes developing a triangulation program a delicate task, particularly when it is important that the program runs fast and robustly over large data. This thesis addresses these challenges in two parts. The first part concentrates on techniques designed for efficiently and robustly computing Delaunay triangulations of three kinds of practical data: the terrain data from LIDAR sensors commonly found in GIS, the atom coordinate data used for biological applications, and the time varying volume data generated from from scientific simulations. The second part addresses the problem of defining spline spaces over triangulations in two dimensions. It does so by generalizing Delaunay configurations, defined as follows. For a given point set P in two dimensions, a Delaunay configuration is a pair of subsets (T, I) from P, where T, called the boundary set, is a triplet and I, called the interior set, is the set of points that fall in the circumcircle through T. The size of the interior set is the degree of the configuration. As recently discovered by Neamtu (2004), for a chosen point set, the set of all degree k Delaunay configurations can be associated with a set of degree k plus 1 splines that form the basis of a spline space. In particular, for the trivial case of k equals 0, the spline space coincides with the PL interpolation functions over the Delaunay triangulation. Neamtu’s definition of the spline space relies only on a few structural properties of the Delaunay configurations. This raises the question whether there exist other sets of configurations with identical structural properties. If there are, then these sets of configurations—let us call them generalized configurations from hereon—can be substituted for Delaunay configurations in Neamtu’s definition of spline space thereby yielding a family of splines over the same point set

Compact Representations of Simplicial Meshes in Two and Three Dimensions

We describe data structures for representing simplicial meshes compactly while supporting online queries and updates eciently. Our representation requires about a factor of ve less memory than the most ecient standard representations of triangular or tetrahedral meshes, while eciently supporting traversal among simplices, storing data on simplices, and insertion and deletion of simplices